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.lake/packages/mathlib/Mathlib/Topology/NhdsSet.lean	import Mathlib.Topology.Neighborhoods /-! # Neighborhoods of a set In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all neighborhoods of a set `s`. ## Main Properties There are a couple different notions equivalent to `s ∈ 𝓝ˢ t`: * `s ⊆ interior t` using `subset_interior_iff_mem_nhdsSet` * `∀ x : X, x ∈ t → s ∈ 𝓝 x` using `mem_nhdsSet_iff_forall` * `∃ U : Set X, IsOpen U ∧ t ⊆ U ∧ U ⊆ s` using `mem_nhdsSet_iff_exists` Furthermore, we have the following results: * `monotone_nhdsSet`: `𝓝ˢ` is monotone * In T₁-spaces, `𝓝ˢ`is strictly monotone and hence injective: `strict_mono_nhdsSet`/`injective_nhdsSet`. These results are in `Mathlib/Topology/Separation/Basic.lean`. -/ open Set Filter Topology variable {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X} {s t s₁ s₂ t₁ t₂ : Set X} {x : X} theorem nhdsSet_diagonal (X) [TopologicalSpace (X × X)] : 𝓝ˢ (diagonal X) = ⨆ (x : X), 𝓝 (x, x) := by rw [nhdsSet, ← range_diag, ← range_comp] rfl theorem mem_nhdsSet_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ x : X, x ∈ t → s ∈ 𝓝 x := by simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image] lemma nhdsSet_le : 𝓝ˢ s ≤ f ↔ ∀ x ∈ s, 𝓝 x ≤ f := by simp [nhdsSet] theorem bUnion_mem_nhdsSet {t : X → Set X} (h : ∀ x ∈ s, t x ∈ 𝓝 x) : (⋃ x ∈ s, t x) ∈ 𝓝ˢ s := mem_nhdsSet_iff_forall.2 fun x hx => mem_of_superset (h x hx) <\| subset_iUnion₂ (s := fun x _ => t x) x hx theorem subset_interior_iff_mem_nhdsSet : s ⊆ interior t ↔ t ∈ 𝓝ˢ s := by simp_rw [mem_nhdsSet_iff_forall, subset_interior_iff_nhds] theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl, subset_compl_iff_disjoint_left] theorem disjoint_nhdsSet_principal : Disjoint (𝓝ˢ s) (𝓟 t) ↔ Disjoint s (closure t) := by rw [disjoint_comm, disjoint_principal_nhdsSet, disjoint_comm] theorem mem_nhdsSet_iff_exists : s ∈ 𝓝ˢ t ↔ ∃ U : Set X, IsOpen U ∧ t ⊆ U ∧ U ⊆ s := by rw [← subset_interior_iff_mem_nhdsSet, subset_interior_iff] /-- A proposition is true on a set neighborhood of `s` iff it is true on a larger open set -/ theorem eventually_nhdsSet_iff_exists {p : X → Prop} : (∀ᶠ x in 𝓝ˢ s, p x) ↔ ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x, x ∈ t → p x := mem_nhdsSet_iff_exists /-- A proposition is true on a set neighborhood of `s` iff it is eventually true near each point in the set. -/ theorem eventually_nhdsSet_iff_forall {p : X → Prop} : (∀ᶠ x in 𝓝ˢ s, p x) ↔ ∀ x, x ∈ s → ∀ᶠ y in 𝓝 x, p y := mem_nhdsSet_iff_forall theorem hasBasis_nhdsSet (s : Set X) : (𝓝ˢ s).HasBasis (fun U => IsOpen U ∧ s ⊆ U) fun U => U := ⟨fun t => by simp [mem_nhdsSet_iff_exists, and_assoc]⟩ @[simp] lemma lift'_nhdsSet_interior (s : Set X) : (𝓝ˢ s).lift' interior = 𝓝ˢ s := (hasBasis_nhdsSet s).lift'_interior_eq_self fun _ ↦ And.left lemma Filter.HasBasis.nhdsSet_interior {ι : Sort} {p : ι → Prop} {s : ι → Set X} {t : Set X} (h : (𝓝ˢ t).HasBasis p s) : (𝓝ˢ t).HasBasis p (interior <\| s ·) := lift'_nhdsSet_interior t ▸ h.lift'_interior theorem IsOpen.mem_nhdsSet (hU : IsOpen s) : s ∈ 𝓝ˢ t ↔ t ⊆ s := by rw [← subset_interior_iff_mem_nhdsSet, hU.interior_eq] /-- An open set belongs to its own set neighborhoods filter. -/ theorem IsOpen.mem_nhdsSet_self (ho : IsOpen s) : s ∈ 𝓝ˢ s := ho.mem_nhdsSet.mpr Subset.rfl theorem principal_le_nhdsSet : 𝓟 s ≤ 𝓝ˢ s := fun _s hs => (subset_interior_iff_mem_nhdsSet.mpr hs).trans interior_subset theorem subset_of_mem_nhdsSet (h : t ∈ 𝓝ˢ s) : s ⊆ t := principal_le_nhdsSet h theorem Filter.Eventually.self_of_nhdsSet {p : X → Prop} (h : ∀ᶠ x in 𝓝ˢ s, p x) : ∀ x ∈ s, p x := principal_le_nhdsSet h nonrec theorem Filter.EventuallyEq.self_of_nhdsSet {Y} {f g : X → Y} (h : f =ᶠ[𝓝ˢ s] g) : EqOn f g s := h.self_of_nhdsSet @[simp] theorem nhdsSet_eq_principal_iff : 𝓝ˢ s = 𝓟 s ↔ IsOpen s := by rw [← principal_le_nhdsSet.ge_iff_eq', le_principal_iff, mem_nhdsSet_iff_forall, isOpen_iff_mem_nhds] alias ⟨_, IsOpen.nhdsSet_eq⟩ := nhdsSet_eq_principal_iff @[simp] theorem nhdsSet_interior : 𝓝ˢ (interior s) = 𝓟 (interior s) := isOpen_interior.nhdsSet_eq @[simp] theorem nhdsSet_singleton : 𝓝ˢ {x} = 𝓝 x := by simp [nhdsSet] theorem mem_nhdsSet_interior : s ∈ 𝓝ˢ (interior s) := subset_interior_iff_mem_nhdsSet.mp Subset.rfl @[simp] theorem nhdsSet_empty : 𝓝ˢ (∅ : Set X) = ⊥ := by rw [isOpen_empty.nhdsSet_eq, principal_empty] theorem mem_nhdsSet_empty : s ∈ 𝓝ˢ (∅ : Set X) := by simp @[simp] theorem nhdsSet_univ : 𝓝ˢ (univ : Set X) = ⊤ := by rw [isOpen_univ.nhdsSet_eq, principal_univ] @[gcongr, mono] theorem nhdsSet_mono (h : s ⊆ t) : 𝓝ˢ s ≤ 𝓝ˢ t := sSup_le_sSup <\| image_mono h theorem monotone_nhdsSet : Monotone (𝓝ˢ : Set X → Filter X) := fun _ _ => nhdsSet_mono theorem nhds_le_nhdsSet (h : x ∈ s) : 𝓝 x ≤ 𝓝ˢ s := le_sSup <\| mem_image_of_mem _ h @[simp] theorem nhdsSet_union (s t : Set X) : 𝓝ˢ (s ∪ t) = 𝓝ˢ s ⊔ 𝓝ˢ t := by simp only [nhdsSet, image_union, sSup_union] theorem union_mem_nhdsSet (h₁ : s₁ ∈ 𝓝ˢ t₁) (h₂ : s₂ ∈ 𝓝ˢ t₂) : s₁ ∪ s₂ ∈ 𝓝ˢ (t₁ ∪ t₂) := by rw [nhdsSet_union] exact union_mem_sup h₁ h₂ @[simp] theorem nhdsSet_insert (x : X) (s : Set X) : 𝓝ˢ (insert x s) = 𝓝 x ⊔ 𝓝ˢ s := by rw [insert_eq, nhdsSet_union, nhdsSet_singleton] /- This inequality cannot be improved to an equality. For instance, if `X` has two elements and the coarse topology and `s` and `t` are distinct singletons then `𝓝ˢ (s ∩ t) = ⊥` while `𝓝ˢ s ⊓ 𝓝ˢ t = ⊤` and those are different. -/ theorem nhdsSet_inter_le (s t : Set X) : 𝓝ˢ (s ∩ t) ≤ 𝓝ˢ s ⊓ 𝓝ˢ t := (monotone_nhdsSet (X := X)).map_inf_le s t theorem nhdsSet_iInter_le {ι : Sort} (s : ι → Set X) : 𝓝ˢ (⋂ i, s i) ≤ ⨅ i, 𝓝ˢ (s i) := (monotone_nhdsSet (X := X)).map_iInf_le theorem nhdsSet_sInter_le (s : Set (Set X)) : 𝓝ˢ (⋂₀ s) ≤ ⨅ x ∈ s, 𝓝ˢ x := (monotone_nhdsSet (X := X)).map_sInf_le variable (s) in theorem IsClosed.nhdsSet_le_sup (h : IsClosed t) : 𝓝ˢ s ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓟 (tᶜ) := calc 𝓝ˢ s = 𝓝ˢ (s ∩ t ∪ s ∩ tᶜ) := by rw [Set.inter_union_compl s t] _ = 𝓝ˢ (s ∩ t) ⊔ 𝓝ˢ (s ∩ tᶜ) := by rw [nhdsSet_union] _ ≤ 𝓝ˢ (s ∩ t) ⊔ 𝓝ˢ (tᶜ) := by nth_grw 2 [inter_subset_right] _ = 𝓝ˢ (s ∩ t) ⊔ 𝓟 (tᶜ) := by rw [h.isOpen_compl.nhdsSet_eq] variable (s) in theorem IsClosed.nhdsSet_le_sup' (h : IsClosed t) : 𝓝ˢ s ≤ 𝓝ˢ (t ∩ s) ⊔ 𝓟 (tᶜ) := by rw [Set.inter_comm]; exact h.nhdsSet_le_sup s theorem Filter.Eventually.eventually_nhdsSet {p : X → Prop} (h : ∀ᶠ y in 𝓝ˢ s, p y) : ∀ᶠ y in 𝓝ˢ s, ∀ᶠ x in 𝓝 y, p x := eventually_nhdsSet_iff_forall.mpr fun x x_in ↦ (eventually_nhdsSet_iff_forall.mp h x x_in).eventually_nhds theorem Filter.Eventually.union_nhdsSet {p : X → Prop} : (∀ᶠ x in 𝓝ˢ (s ∪ t), p x) ↔ (∀ᶠ x in 𝓝ˢ s, p x) ∧ ∀ᶠ x in 𝓝ˢ t, p x := by rw [nhdsSet_union, eventually_sup] theorem Filter.Eventually.union {p : X → Prop} (hs : ∀ᶠ x in 𝓝ˢ s, p x) (ht : ∀ᶠ x in 𝓝ˢ t, p x) : ∀ᶠ x in 𝓝ˢ (s ∪ t), p x := Filter.Eventually.union_nhdsSet.mpr ⟨hs, ht⟩ theorem nhdsSet_iUnion {ι : Sort} (s : ι → Set X) : 𝓝ˢ (⋃ i, s i) = ⨆ i, 𝓝ˢ (s i) := by simp only [nhdsSet, image_iUnion, sSup_iUnion (β := Filter X)] theorem eventually_nhdsSet_iUnion₂ {ι : Sort} {p : ι → Prop} {s : ι → Set X} {P : X → Prop} : (∀ᶠ x in 𝓝ˢ (⋃ (i) (_ : p i), s i), P x) ↔ ∀ i, p i → ∀ᶠ x in 𝓝ˢ (s i), P x := by simp only [nhdsSet_iUnion, eventually_iSup] theorem eventually_nhdsSet_iUnion {ι : Sort} {s : ι → Set X} {P : X → Prop} : (∀ᶠ x in 𝓝ˢ (⋃ i, s i), P x) ↔ ∀ i, ∀ᶠ x in 𝓝ˢ (s i), P x := by simp only [nhdsSet_iUnion, eventually_iSup]
.lake/packages/mathlib/Mathlib/Topology/Irreducible.lean	import Mathlib.Order.Minimal import Mathlib.Order.Zorn import Mathlib.Topology.ContinuousOn import Mathlib.Tactic.StacksAttribute /-! # Irreducibility in topological spaces ## Main definitions * `IrreducibleSpace`: a typeclass applying to topological spaces, stating that the space is nonempty and does not admit a nontrivial pair of disjoint opens. * `IsIrreducible`: for a nonempty set in a topological space, the property that the set is an irreducible space in the subspace topology. ## On the definition of irreducible and connected sets/spaces In informal mathematics, irreducible spaces are assumed to be nonempty. We formalise the predicate without that assumption as `IsPreirreducible`. In other words, the only difference is whether the empty space counts as irreducible. There are good reasons to consider the empty space to be “too simple to be simple” See also https://ncatlab.org/nlab/show/too+simple+to+be+simple, and in particular https://ncatlab.org/nlab/show/too+simple+to+be+simple#relationship_to_biased_definitions. -/ open Set Topology variable {X : Type} {Y : Type} [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X} section Preirreducible /-- A preirreducible set `s` is one where there is no non-trivial pair of disjoint opens on `s`. -/ def IsPreirreducible (s : Set X) : Prop := ∀ u v : Set X, IsOpen u → IsOpen v → (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty /-- An irreducible set `s` is one that is nonempty and where there is no non-trivial pair of disjoint opens on `s`. -/ @[stacks 004V "(1) as predicate on subsets of a space"] def IsIrreducible (s : Set X) : Prop := s.Nonempty ∧ IsPreirreducible s theorem IsIrreducible.nonempty (h : IsIrreducible s) : s.Nonempty := h.1 theorem IsIrreducible.isPreirreducible (h : IsIrreducible s) : IsPreirreducible s := h.2 theorem isPreirreducible_empty : IsPreirreducible (∅ : Set X) := fun _ _ _ _ _ ⟨_, h1, _⟩ => h1.elim theorem Set.Subsingleton.isPreirreducible (hs : s.Subsingleton) : IsPreirreducible s := fun _u _v _ _ ⟨_x, hxs, hxu⟩ ⟨y, hys, hyv⟩ => ⟨y, hys, hs hxs hys ▸ hxu, hyv⟩ theorem isPreirreducible_singleton {x} : IsPreirreducible ({x} : Set X) := subsingleton_singleton.isPreirreducible theorem isIrreducible_singleton {x} : IsIrreducible ({x} : Set X) := ⟨singleton_nonempty x, isPreirreducible_singleton⟩ theorem isPreirreducible_iff_closure : IsPreirreducible (closure s) ↔ IsPreirreducible s := forall₄_congr fun u v hu hv => by iterate 3 rw [closure_inter_open_nonempty_iff] exacts [hu.inter hv, hv, hu] @[stacks 004W "(1)"] theorem isIrreducible_iff_closure : IsIrreducible (closure s) ↔ IsIrreducible s := and_congr closure_nonempty_iff isPreirreducible_iff_closure protected alias ⟨_, IsPreirreducible.closure⟩ := isPreirreducible_iff_closure protected alias ⟨_, IsIrreducible.closure⟩ := isIrreducible_iff_closure theorem exists_preirreducible (s : Set X) (H : IsPreirreducible s) : ∃ t : Set X, IsPreirreducible t ∧ s ⊆ t ∧ ∀ u, IsPreirreducible u → t ⊆ u → u = t := let ⟨m, hsm, hm⟩ := zorn_subset_nonempty { t : Set X \| IsPreirreducible t } (fun c hc hcc _ => ⟨⋃₀ c, fun u v hu hv ⟨y, hy, hyu⟩ ⟨x, hx, hxv⟩ => let ⟨p, hpc, hyp⟩ := mem_sUnion.1 hy let ⟨q, hqc, hxq⟩ := mem_sUnion.1 hx Or.casesOn (hcc.total hpc hqc) (fun hpq : p ⊆ q => let ⟨x, hxp, hxuv⟩ := hc hqc u v hu hv ⟨y, hpq hyp, hyu⟩ ⟨x, hxq, hxv⟩ ⟨x, mem_sUnion_of_mem hxp hqc, hxuv⟩) fun hqp : q ⊆ p => let ⟨x, hxp, hxuv⟩ := hc hpc u v hu hv ⟨y, hyp, hyu⟩ ⟨x, hqp hxq, hxv⟩ ⟨x, mem_sUnion_of_mem hxp hpc, hxuv⟩, fun _ hxc => subset_sUnion_of_mem hxc⟩) s H ⟨m, hm.prop, hsm, fun _u hu hmu => (hm.eq_of_subset hu hmu).symm⟩ /-- The set of irreducible components of a topological space. -/ @[stacks 004V "(2)"] def irreducibleComponents (X : Type) [TopologicalSpace X] : Set (Set X) := {s \| Maximal IsIrreducible s} @[stacks 004W "(2)"] theorem isClosed_of_mem_irreducibleComponents (s) (H : s ∈ irreducibleComponents X) : IsClosed s := by rw [← closure_eq_iff_isClosed, eq_comm] exact subset_closure.antisymm (H.2 H.1.closure subset_closure) theorem irreducibleComponents_eq_maximals_closed (X : Type) [TopologicalSpace X] : irreducibleComponents X = { s \| Maximal (fun x ↦ IsClosed x ∧ IsIrreducible x) s} := by ext s constructor · intro H exact ⟨⟨isClosed_of_mem_irreducibleComponents _ H, H.1⟩, fun x h e => H.2 h.2 e⟩ · intro H refine ⟨H.1.2, fun x h e => ?_⟩ have : closure x ≤ s := H.2 ⟨isClosed_closure, h.closure⟩ (e.trans subset_closure) exact le_trans subset_closure this @[stacks 004W "(3)"] lemma exists_mem_irreducibleComponents_subset_of_isIrreducible (s : Set X) (hs : IsIrreducible s) : ∃ u ∈ irreducibleComponents X, s ⊆ u := by obtain ⟨u,hu⟩ := exists_preirreducible s hs.isPreirreducible use u, ⟨⟨hs.left.mono hu.right.left,hu.left⟩,fun _ h hl => (hu.right.right _ h.right hl).le⟩ exact hu.right.left /-- A maximal irreducible set that contains a given point. -/ @[stacks 004W "(4)"] def irreducibleComponent (x : X) : Set X := Classical.choose (exists_preirreducible {x} isPreirreducible_singleton) theorem irreducibleComponent_property (x : X) : IsPreirreducible (irreducibleComponent x) ∧ {x} ⊆ irreducibleComponent x ∧ ∀ u, IsPreirreducible u → irreducibleComponent x ⊆ u → u = irreducibleComponent x := Classical.choose_spec (exists_preirreducible {x} isPreirreducible_singleton) @[stacks 004W "(4)"] theorem mem_irreducibleComponent {x : X} : x ∈ irreducibleComponent x := singleton_subset_iff.1 (irreducibleComponent_property x).2.1 theorem isIrreducible_irreducibleComponent {x : X} : IsIrreducible (irreducibleComponent x) := ⟨⟨x, mem_irreducibleComponent⟩, (irreducibleComponent_property x).1⟩ theorem eq_irreducibleComponent {x : X} : IsPreirreducible s → irreducibleComponent x ⊆ s → s = irreducibleComponent x := (irreducibleComponent_property x).2.2 _ theorem irreducibleComponent_mem_irreducibleComponents (x : X) : irreducibleComponent x ∈ irreducibleComponents X := ⟨isIrreducible_irreducibleComponent, fun _ h₁ h₂ => (eq_irreducibleComponent h₁.2 h₂).le⟩ theorem isClosed_irreducibleComponent {x : X} : IsClosed (irreducibleComponent x) := isClosed_of_mem_irreducibleComponents _ (irreducibleComponent_mem_irreducibleComponents x) /-- A preirreducible space is one where there is no non-trivial pair of disjoint opens. -/ class PreirreducibleSpace (X : Type) [TopologicalSpace X] : Prop where /-- In a preirreducible space, `Set.univ` is a preirreducible set. -/ isPreirreducible_univ : IsPreirreducible (univ : Set X) /-- An irreducible space is one that is nonempty and where there is no non-trivial pair of disjoint opens. -/ @[stacks 004V "(1) as predicate on a space"] class IrreducibleSpace (X : Type) [TopologicalSpace X] : Prop extends PreirreducibleSpace X where toNonempty : Nonempty X -- see Note [lower instance priority] attribute [instance 50] IrreducibleSpace.toNonempty theorem IrreducibleSpace.isIrreducible_univ (X : Type) [TopologicalSpace X] [IrreducibleSpace X] : IsIrreducible (univ : Set X) := ⟨univ_nonempty, PreirreducibleSpace.isPreirreducible_univ⟩ theorem irreducibleSpace_def (X : Type) [TopologicalSpace X] : IrreducibleSpace X ↔ IsIrreducible (⊤ : Set X) := ⟨@IrreducibleSpace.isIrreducible_univ X _, fun h => haveI : PreirreducibleSpace X := ⟨h.2⟩ ⟨⟨h.1.some⟩⟩⟩ theorem nonempty_preirreducible_inter [PreirreducibleSpace X] : IsOpen s → IsOpen t → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty := by simpa only [univ_inter, univ_subset_iff] using @PreirreducibleSpace.isPreirreducible_univ X _ _ s t /-- In a (pre)irreducible space, a nonempty open set is dense. -/ protected theorem IsOpen.dense [PreirreducibleSpace X] (ho : IsOpen s) (hne : s.Nonempty) : Dense s := dense_iff_inter_open.2 fun _t hto htne => nonempty_preirreducible_inter hto ho htne hne theorem IsPreirreducible.image (H : IsPreirreducible s) (f : X → Y) (hf : ContinuousOn f s) : IsPreirreducible (f '' s) := by rintro u v hu hv ⟨_, ⟨⟨x, hx, rfl⟩, hxu⟩⟩ ⟨_, ⟨⟨y, hy, rfl⟩, hyv⟩⟩ rw [← mem_preimage] at hxu hyv rcases continuousOn_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩ rcases continuousOn_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩ have := H u' v' hu' hv' rw [inter_comm s u', ← u'_eq] at this rw [inter_comm s v', ← v'_eq] at this rcases this ⟨x, hxu, hx⟩ ⟨y, hyv, hy⟩ with ⟨x, hxs, hxu', hxv'⟩ refine ⟨f x, mem_image_of_mem f hxs, ?_, ?_⟩ all_goals rw [← mem_preimage] apply mem_of_mem_inter_left show x ∈ _ ∩ s simp [] @[stacks 0379] theorem IsIrreducible.image (H : IsIrreducible s) (f : X → Y) (hf : ContinuousOn f s) : IsIrreducible (f '' s) := ⟨H.nonempty.image _, H.isPreirreducible.image f hf⟩ theorem Subtype.preirreducibleSpace (h : IsPreirreducible s) : PreirreducibleSpace s where isPreirreducible_univ := by rintro _ _ ⟨u, hu, rfl⟩ ⟨v, hv, rfl⟩ ⟨⟨x, hxs⟩, -, hxu⟩ ⟨⟨y, hys⟩, -, hyv⟩ rcases h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ with ⟨x, hxs, ⟨hxu, hxv⟩⟩ exact ⟨⟨x, hxs⟩, ⟨Set.mem_univ _, ⟨hxu, hxv⟩⟩⟩ theorem Subtype.irreducibleSpace (h : IsIrreducible s) : IrreducibleSpace s where isPreirreducible_univ := (Subtype.preirreducibleSpace h.isPreirreducible).isPreirreducible_univ toNonempty := h.nonempty.to_subtype instance (priority := low) [Subsingleton X] : PreirreducibleSpace X := ⟨(Set.subsingleton_univ_iff.mpr ‹_›).isPreirreducible⟩ /-- An infinite type with cofinite topology is an irreducible topological space. -/ instance (priority := 100) {X} [Infinite X] : IrreducibleSpace (CofiniteTopology X) where isPreirreducible_univ u v := by haveI : Infinite (CofiniteTopology X) := ‹_› simp only [CofiniteTopology.isOpen_iff, univ_inter] intro hu hv hu' hv' simpa only [compl_union, compl_compl] using ((hu hu').union (hv hv')).infinite_compl.nonempty toNonempty := (inferInstance : Nonempty X) theorem irreducibleComponents_eq_singleton [IrreducibleSpace X] : irreducibleComponents X = {univ} := Set.ext fun _ ↦ IsGreatest.maximal_iff (s := IsIrreducible (X := X)) ⟨IrreducibleSpace.isIrreducible_univ X, fun _ _ ↦ Set.subset_univ _⟩ /-- A set `s` is irreducible if and only if for every finite collection of open sets all of whose members intersect `s`, `s` also intersects the intersection of the entire collection (i.e., there is an element of `s` contained in every member of the collection). -/ theorem isIrreducible_iff_sInter : IsIrreducible s ↔ ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty := by classical refine ⟨fun h U hu hU => ?_, fun h => ⟨?_, ?_⟩⟩ · induction U using Finset.induction_on with \| empty => simpa using h.nonempty \| insert u U _ IH => rw [Finset.coe_insert, sInter_insert] rw [Finset.forall_mem_insert] at hu hU exact h.2 _ _ hu.1 (U.finite_toSet.isOpen_sInter hu.2) hU.1 (IH hu.2 hU.2) · simpa using h ∅ · intro u v hu hv hu' hv' simpa [] using h {u, v} /-- A set is preirreducible if and only if for every cover by two closed sets, it is contained in one of the two covering sets. -/ theorem isPreirreducible_iff_isClosed_union_isClosed : IsPreirreducible s ↔ ∀ z₁ z₂ : Set X, IsClosed z₁ → IsClosed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂ := by refine compl_surjective.forall.trans <\| forall_congr' fun z₁ => compl_surjective.forall.trans <\| forall_congr' fun z₂ => ?_ simp only [isOpen_compl_iff, ← compl_union, inter_compl_nonempty_iff] refine forall₂_congr fun _ _ => ?_ rw [← and_imp, ← not_or, not_imp_not] /-- A set is irreducible if and only if for every cover by a finite collection of closed sets, it is contained in one of the members of the collection. -/ theorem isIrreducible_iff_sUnion_isClosed : IsIrreducible s ↔ ∀ t : Finset (Set X), (∀ z ∈ t, IsClosed z) → (s ⊆ ⋃₀ ↑t) → ∃ z ∈ t, s ⊆ z := by simp only [isIrreducible_iff_sInter] refine ((@compl_involutive (Set X) _).toPerm _).finsetCongr.forall_congr fun {t} => ?_ simp_rw [Equiv.finsetCongr_apply, Finset.forall_mem_map, Finset.mem_map, Finset.coe_map, sUnion_image, Equiv.coe_toEmbedding, Function.Involutive.coe_toPerm, isClosed_compl_iff, exists_exists_and_eq_and] refine forall_congr' fun _ => Iff.trans ?_ not_imp_not simp only [not_exists, not_and, ← compl_iInter₂, ← sInter_eq_biInter, subset_compl_iff_disjoint_right, not_disjoint_iff_nonempty_inter] /-- A nonempty open subset of a preirreducible subspace is dense in the subspace. -/ theorem subset_closure_inter_of_isPreirreducible_of_isOpen {S U : Set X} (hS : IsPreirreducible S) (hU : IsOpen U) (h : (S ∩ U).Nonempty) : S ⊆ closure (S ∩ U) := by by_contra h' obtain ⟨x, h₁, h₂, h₃⟩ := hS _ (closure (S ∩ U))ᶜ hU isClosed_closure.isOpen_compl h (inter_compl_nonempty_iff.mpr h') exact h₃ (subset_closure ⟨h₁, h₂⟩) /-- If `∅ ≠ U ⊆ S ⊆ t` such that `U` is open and `t` is preirreducible, then `S` is irreducible. -/ theorem IsPreirreducible.subset_irreducible {S U : Set X} (ht : IsPreirreducible t) (hU : U.Nonempty) (hU' : IsOpen U) (h₁ : U ⊆ S) (h₂ : S ⊆ t) : IsIrreducible S := by obtain ⟨z, hz⟩ := hU replace ht : IsIrreducible t := ⟨⟨z, h₂ (h₁ hz)⟩, ht⟩ refine ⟨⟨z, h₁ hz⟩, ?_⟩ rintro u v hu hv ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩ classical obtain ⟨x, -, hx'⟩ : Set.Nonempty (t ∩ ⋂₀ ↑({U, u, v} : Finset (Set X))) := by refine isIrreducible_iff_sInter.mp ht {U, u, v} ?_ ?_ · simp [*] · intro U H simp only [Finset.mem_insert, Finset.mem_singleton] at H rcases H with (rfl \| rfl \| rfl) exacts [⟨z, h₂ (h₁ hz), hz⟩, ⟨x, h₂ hx, hx'⟩, ⟨y, h₂ hy, hy'⟩] replace hx' : x ∈ U ∧ x ∈ u ∧ x ∈ v := by simpa using hx' exact ⟨x, h₁ hx'.1, hx'.2⟩ theorem IsPreirreducible.open_subset {U : Set X} (ht : IsPreirreducible t) (hU : IsOpen U) (hU' : U ⊆ t) : IsPreirreducible U := U.eq_empty_or_nonempty.elim (fun h => h.symm ▸ isPreirreducible_empty) fun h => (ht.subset_irreducible h hU (fun _ => id) hU').2 theorem IsPreirreducible.interior (ht : IsPreirreducible t) : IsPreirreducible (interior t) := ht.open_subset isOpen_interior interior_subset theorem IsPreirreducible.preimage (ht : IsPreirreducible t) {f : Y → X} (hf : IsOpenEmbedding f) : IsPreirreducible (f ⁻¹' t) := by rintro U V hU hV ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩ obtain ⟨_, h₁, ⟨y, h₂, rfl⟩, ⟨y', h₃, h₄⟩⟩ := ht _ _ (hf.isOpenMap _ hU) (hf.isOpenMap _ hV) ⟨f x, hx, Set.mem_image_of_mem f hx'⟩ ⟨f y, hy, Set.mem_image_of_mem f hy'⟩ cases hf.injective h₄ exact ⟨y, h₁, h₂, h₃⟩ section open Set.Notation @[stacks 004Z] lemma IsPreirreducible.preimage_of_dense_isPreirreducible_fiber {V : Set Y} (hV : IsPreirreducible V) (f : X → Y) (hf' : IsOpenMap f) (hf'' : V ⊆ closure (V ∩ { x \| IsPreirreducible (f ⁻¹' {x}) })) : IsPreirreducible (f ⁻¹' V) := by rintro U₁ U₂ hU₁ hU₂ ⟨x, hxV, hxU₁⟩ ⟨y, hyV, hyU₂⟩ obtain ⟨z, hzV, hz₁, hz₂⟩ := hV _ _ (hf' _ hU₁) (hf' _ hU₂) ⟨f x, hxV, x, hxU₁, rfl⟩ ⟨f y, hyV, y, hyU₂, rfl⟩ obtain ⟨z, ⟨⟨z₁, hz₁, e₁⟩, ⟨z₂, hz₂, e₂⟩⟩, hzV, hz⟩ := mem_closure_iff.mp (hf'' hzV) _ ((hf' _ hU₁).inter (hf' _ hU₂)) ⟨hz₁, hz₂⟩ obtain ⟨z₃, hz₃, hz₃'⟩ := hz _ _ hU₁ hU₂ ⟨z₁, e₁, hz₁⟩ ⟨z₂, e₂, hz₂⟩ refine ⟨z₃, show f z₃ ∈ _ from (show f z₃ = z from hz₃) ▸ hzV, hz₃'⟩ lemma IsPreirreducible.preimage_of_isPreirreducible_fiber {V : Set Y} (hV : IsPreirreducible V) (f : X → Y) (hf' : IsOpenMap f) (hf'' : ∀ x, IsPreirreducible (f ⁻¹' {x})) : IsPreirreducible (f ⁻¹' V) := by refine hV.preimage_of_dense_isPreirreducible_fiber f hf' ?_ simp [hf'', subset_closure] variable (f : X → Y) (hf₁ : Continuous f) (hf₂ : IsOpenMap f) variable (hf₃ : ∀ x, IsPreirreducible (f ⁻¹' {x})) (hf₄ : Function.Surjective f) include hf₁ hf₂ hf₃ hf₄ lemma preimage_mem_irreducibleComponents_of_isPreirreducible_fiber {V : Set Y} (hV : V ∈ irreducibleComponents Y) : f ⁻¹' V ∈ irreducibleComponents X := by obtain ⟨Z, hZ, hWZ, H⟩ := exists_preirreducible _ (hV.1.2.preimage_of_isPreirreducible_fiber f hf₂ hf₃) have hZ' : IsIrreducible Z := by obtain ⟨x, hx⟩ := hV.1.1 obtain ⟨x, rfl⟩ := hf₄ x exact ⟨⟨_, hWZ hx⟩, hZ⟩ have hWZ' : f ⁻¹' V = Z := by refine hWZ.antisymm (Set.image_subset_iff.mp ?_) exact hV.2 (IsIrreducible.image hZ' f hf₁.continuousOn) ((Set.image_preimage_eq V hf₄).symm.trans_le (Set.image_mono hWZ)) rw [hWZ'] exact ⟨hZ', fun s hs hs' ↦ (H s hs.2 hs').le⟩ lemma image_mem_irreducibleComponents_of_isPreirreducible_fiber {V : Set X} (hV : V ∈ irreducibleComponents X) : f '' V ∈ irreducibleComponents Y := ⟨hV.1.image _ hf₁.continuousOn, fun Z hZ hWZ ↦ by have := hV.2 ⟨(by obtain ⟨x, hx⟩ := hV.1.1; exact ⟨x, hWZ ⟨x, hx, rfl⟩⟩), hZ.2.preimage_of_isPreirreducible_fiber f hf₂ hf₃⟩ (Set.image_subset_iff.mp hWZ) rw [← Set.image_preimage_eq Z hf₄] exact Set.image_mono this⟩ /-- If `f : X → Y` is continuous, open, and has irreducible fibers, then it induces an bijection between irreducible components -/ @[stacks 037A] def irreducibleComponentsEquivOfIsPreirreducibleFiber : irreducibleComponents Y ≃o irreducibleComponents X where invFun W := ⟨f '' W.1, image_mem_irreducibleComponents_of_isPreirreducible_fiber f hf₁ hf₂ hf₃ hf₄ W.2⟩ toFun W := ⟨f ⁻¹' W.1, preimage_mem_irreducibleComponents_of_isPreirreducible_fiber f hf₁ hf₂ hf₃ hf₄ W.2⟩ right_inv W := Subtype.ext <\| by refine (Set.subset_preimage_image _ _).antisymm' (W.2.2 ?_ (Set.subset_preimage_image _ _)) refine ⟨?_, (W.2.1.image _ hf₁.continuousOn).2.preimage_of_isPreirreducible_fiber _ hf₂ hf₃⟩ obtain ⟨x, hx⟩ := W.2.1.1 exact ⟨_, x, hx, rfl⟩ left_inv _ := Subtype.ext <\| Set.image_preimage_eq _ hf₄ map_rel_iff' {W Z} := by refine ⟨fun H ↦ ?_, Set.preimage_mono⟩ simpa only [Equiv.coe_fn_mk, Set.image_preimage_eq _ hf₄] using Set.image_mono (f := f) H end end Preirreducible
.lake/packages/mathlib/Mathlib/Topology/ExtremallyDisconnected.lean	import Mathlib.Topology.Homeomorph.Lemmas import Mathlib.Topology.Compactification.StoneCech /-! # Extremally disconnected spaces An extremally disconnected topological space is a space in which the closure of every open set is open. Such spaces are also called Stonean spaces. They are the projective objects in the category of compact Hausdorff spaces. ## Main declarations * `ExtremallyDisconnected`: Predicate for a space to be extremally disconnected. * `CompactT2.Projective`: Predicate for a topological space to be a projective object in the category of compact Hausdorff spaces. * `CompactT2.Projective.extremallyDisconnected`: Compact Hausdorff spaces that are projective are extremally disconnected. * `CompactT2.ExtremallyDisconnected.projective`: Extremally disconnected spaces are projective objects in the category of compact Hausdorff spaces. ## References [Gleason, Projective topological spaces][gleason1958] -/ noncomputable section open Function Set universe u variable (X : Type u) [TopologicalSpace X] /-- An extremally disconnected topological space is a space in which the closure of every open set is open. -/ class ExtremallyDisconnected : Prop where /-- The closure of every open set is open. -/ open_closure : ∀ U : Set X, IsOpen U → IsOpen (closure U) theorem extremallyDisconnected_of_homeo {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] [ExtremallyDisconnected X] (e : X ≃ₜ Y) : ExtremallyDisconnected Y where open_closure U hU := by rw [e.symm.isInducing.closure_eq_preimage_closure_image, Homeomorph.isOpen_preimage] exact ExtremallyDisconnected.open_closure _ (e.symm.isOpen_image.mpr hU) section TotallySeparated /-- Extremally disconnected spaces are totally separated. -/ instance [ExtremallyDisconnected X] [T2Space X] : TotallySeparatedSpace X := { isTotallySeparated_univ := by intro x _ y _ hxy obtain ⟨U, V, hUV⟩ := T2Space.t2 hxy refine ⟨closure U, (closure U)ᶜ, ExtremallyDisconnected.open_closure U hUV.1, by simp only [isOpen_compl_iff, isClosed_closure], subset_closure hUV.2.2.1, ?_, by simp only [Set.union_compl_self, Set.subset_univ], disjoint_compl_right⟩ rw [Set.mem_compl_iff, mem_closure_iff] push_neg refine ⟨V, ⟨hUV.2.1, hUV.2.2.2.1, ?_⟩⟩ rw [← Set.disjoint_iff_inter_eq_empty, disjoint_comm] exact hUV.2.2.2.2 } end TotallySeparated section /-- The assertion `CompactT2.Projective` states that given continuous maps `f : X → Z` and `g : Y → Z` with `g` surjective between `t_2`, compact topological spaces, there exists a continuous lift `h : X → Y`, such that `f = g ∘ h`. -/ def CompactT2.Projective : Prop := ∀ {Y Z : Type u} [TopologicalSpace Y] [TopologicalSpace Z], ∀ [CompactSpace Y] [T2Space Y] [CompactSpace Z] [T2Space Z], ∀ {f : X → Z} {g : Y → Z} (_ : Continuous f) (_ : Continuous g) (_ : Surjective g), ∃ h : X → Y, Continuous h ∧ g ∘ h = f variable {X} theorem StoneCech.projective [DiscreteTopology X] : CompactT2.Projective (StoneCech X) := by intro Y Z _tsY _tsZ _csY _t2Y _csZ _csZ f g hf hg g_sur let s : Z → Y := fun z => Classical.choose <\| g_sur z have hs : g ∘ s = id := funext fun z => Classical.choose_spec (g_sur z) let t := s ∘ f ∘ stoneCechUnit have ht : Continuous t := continuous_of_discreteTopology let h : StoneCech X → Y := stoneCechExtend ht have hh : Continuous h := continuous_stoneCechExtend ht refine ⟨h, hh, denseRange_stoneCechUnit.equalizer (hg.comp hh) hf ?_⟩ rw [comp_assoc, stoneCechExtend_extends ht, ← comp_assoc, hs, id_comp] protected theorem CompactT2.Projective.extremallyDisconnected [CompactSpace X] [T2Space X] (h : CompactT2.Projective X) : ExtremallyDisconnected X := by refine { open_closure := fun U hU => ?_ } let Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true} let Z₂ : Set (X × Bool) := closure U ×ˢ {false} let Z : Set (X × Bool) := Z₁ ∪ Z₂ have hZ₁₂ : Disjoint Z₁ Z₂ := disjoint_left.2 fun x hx₁ hx₂ => by cases hx₁.2.symm.trans hx₂.2 have hZ₁ : IsClosed Z₁ := hU.isClosed_compl.prod (T1Space.t1 _) have hZ₂ : IsClosed Z₂ := isClosed_closure.prod (T1Space.t1 false) have hZ : IsClosed Z := hZ₁.union hZ₂ let f : Z → X := Prod.fst ∘ Subtype.val have f_cont : Continuous f := continuous_fst.comp continuous_subtype_val have f_sur : Surjective f := by intro x by_cases hx : x ∈ U · exact ⟨⟨(x, false), Or.inr ⟨subset_closure hx, mem_singleton _⟩⟩, rfl⟩ · exact ⟨⟨(x, true), Or.inl ⟨hx, mem_singleton _⟩⟩, rfl⟩ haveI : CompactSpace Z := isCompact_iff_compactSpace.mp hZ.isCompact obtain ⟨g, hg, g_sec⟩ := h continuous_id f_cont f_sur let φ := Subtype.val ∘ g have hφ : Continuous φ := continuous_subtype_val.comp hg have hφ₁ : ∀ x, (φ x).1 = x := congr_fun g_sec suffices closure U = φ ⁻¹' Z₂ by rw [this, preimage_comp, ← isClosed_compl_iff, ← preimage_compl, ← preimage_subtype_coe_eq_compl Subset.rfl] · exact hZ₁.preimage hφ · rw [hZ₁₂.inter_eq, inter_empty] refine (closure_minimal ?_ <\| hZ₂.preimage hφ).antisymm fun x hx => ?_ · intro x hx have : φ x ∈ Z₁ ∪ Z₂ := (g x).2 rcases this with hφ \| hφ · exact ((hφ₁ x ▸ hφ.1) hx).elim · exact hφ · rw [← hφ₁ x] exact hx.1 end section variable {A D E : Type u} [TopologicalSpace A] [TopologicalSpace D] [TopologicalSpace E] /-- Lemma 2.4 in [Gleason, Projective topological spaces][gleason1958]: a continuous surjection $\pi$ from a compact space $D$ to a Fréchet space $A$ restricts to a compact subset $E$ of $D$, such that $\pi$ maps $E$ onto $A$ and satisfies the "Zorn subset condition", where $\pi(E_0) \ne A$ for any proper closed subset $E_0 \subsetneq E$. -/ lemma exists_compact_surjective_zorn_subset [T1Space A] [CompactSpace D] {X : D → A} (X_cont : Continuous X) (X_surj : X.Surjective) : ∃ E : Set D, CompactSpace E ∧ X '' E = univ ∧ ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → E.restrict X '' E₀ ≠ univ := by -- suffices to apply Zorn's lemma on the subsets of $D$ that are closed and mapped onto $A$ let S : Set <\| Set D := {E : Set D \| IsClosed E ∧ X '' E = univ} suffices ∀ (C : Set <\| Set D) (_ : C ⊆ S) (_ : IsChain (· ⊆ ·) C), ∃ s ∈ S, ∀ c ∈ C, s ⊆ c by rcases zorn_superset S this with ⟨E, E_min⟩ obtain ⟨E_closed, E_surj⟩ := E_min.prop refine ⟨E, isCompact_iff_compactSpace.mp E_closed.isCompact, E_surj, ?_⟩ intro E₀ E₀_min E₀_closed contrapose! E₀_min exact eq_univ_of_image_val_eq <\| E_min.eq_of_subset ⟨E₀_closed.trans E_closed, image_image_val_eq_restrict_image ▸ E₀_min⟩ image_val_subset -- suffices to prove intersection of chain is minimal intro C C_sub C_chain -- prove intersection of chain is closed refine ⟨iInter (fun c : C => c), ⟨isClosed_iInter fun ⟨_, h⟩ => (C_sub h).left, ?_⟩, fun c hc _ h => mem_iInter.mp h ⟨c, hc⟩⟩ -- prove intersection of chain is mapped onto $A$ by_cases hC : Nonempty C · refine eq_univ_of_forall fun a => inter_nonempty_iff_exists_left.mp ?_ -- apply Cantor's intersection theorem refine iInter_inter (ι := C) (X ⁻¹' {a}) _ ▸ IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _ ?_ (fun c => ?_) (fun c => IsClosed.isCompact ?_) (fun c => ?_) · replace C_chain : IsChain (· ⊇ ·) C := C_chain.symm have : ∀ s t : Set D, s ⊇ t → _ ⊇ _ := fun _ _ => inter_subset_inter_left <\| X ⁻¹' {a} exact (directedOn_iff_directed.mp C_chain.directedOn).mono_comp (· ⊇ ·) this · rw [← image_inter_nonempty_iff, (C_sub c.mem).right, univ_inter] exact singleton_nonempty a all_goals exact (C_sub c.mem).left.inter <\| (T1Space.t1 a).preimage X_cont · rw [@iInter_of_empty _ _ <\| not_nonempty_iff.mp hC, image_univ_of_surjective X_surj] /-- Lemma 2.1 in [Gleason, Projective topological spaces][gleason1958]: if $\rho$ is a continuous surjection from a topological space $E$ to a topological space $A$ satisfying the "Zorn subset condition", then $\rho(G)$ is contained in the closure of $A \setminus \rho(E \setminus G)$ for any open set $G$ of $E$. -/ lemma image_subset_closure_compl_image_compl_of_isOpen {ρ : E → A} (ρ_cont : Continuous ρ) (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) {G : Set E} (hG : IsOpen G) : ρ '' G ⊆ closure ((ρ '' Gᶜ)ᶜ) := by -- suffices to prove for nonempty $G$ by_cases G_empty : G = ∅ · simpa only [G_empty, image_empty] using empty_subset _ · -- let $a \in \rho(G)$ intro a ha rw [mem_closure_iff] -- let $N$ be a neighbourhood of $a$ intro N N_open hN -- get $x \in A$ from nonempty open $G \cap \rho^{-1}(N)$ rcases (G.mem_image ρ a).mp ha with ⟨e, he, rfl⟩ have nonempty : (G ∩ ρ⁻¹' N).Nonempty := ⟨e, mem_inter he <\| mem_preimage.mpr hN⟩ have is_open : IsOpen <\| G ∩ ρ⁻¹' N := hG.inter <\| N_open.preimage ρ_cont have ne_univ : ρ '' (G ∩ ρ⁻¹' N)ᶜ ≠ univ := zorn_subset _ (compl_ne_univ.mpr nonempty) is_open.isClosed_compl rcases nonempty_compl.mpr ne_univ with ⟨x, hx⟩ -- prove $x \in N \cap (A \setminus \rho(E \setminus G))$ have hx' : x ∈ (ρ '' Gᶜ)ᶜ := fun h => hx <\| image_mono (by simp) h rcases ρ_surj x with ⟨y, rfl⟩ have hy : y ∈ G ∩ ρ⁻¹' N := by simpa using mt (mem_image_of_mem ρ) <\| mem_compl hx exact ⟨ρ y, mem_inter (mem_preimage.mp <\| mem_of_mem_inter_right hy) hx'⟩ /-- Lemma 2.2 in [Gleason, Projective topological spaces][gleason1958]: in an extremally disconnected space, if $U_1$ and $U_2$ are disjoint open sets, then $\overline{U_1}$ and $\overline{U_2}$ are also disjoint. -/ lemma ExtremallyDisconnected.disjoint_closure_of_disjoint_isOpen [ExtremallyDisconnected A] {U₁ U₂ : Set A} (h : Disjoint U₁ U₂) (hU₁ : IsOpen U₁) (hU₂ : IsOpen U₂) : Disjoint (closure U₁) (closure U₂) := (h.closure_right hU₁).closure_left <\| open_closure U₂ hU₂ private lemma ExtremallyDisconnected.homeoCompactToT2_injective [ExtremallyDisconnected A] [T2Space A] [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ) (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) : ρ.Injective := by -- let $x_1, x_2 \in E$ be distinct points such that $\rho(x_1) = \rho(x_2)$ intro x₁ x₂ hρx by_contra hx -- let $G_1$ and $G_2$ be disjoint open neighbourhoods of $x_1$ and $x_2$ respectively rcases t2_separation hx with ⟨G₁, G₂, G₁_open, G₂_open, hx₁, hx₂, disj⟩ -- prove $A \setminus \rho(E - G_1)$ and $A \setminus \rho(E - G_2)$ are disjoint have G₁_comp : IsCompact G₁ᶜ := IsClosed.isCompact G₁_open.isClosed_compl have G₂_comp : IsCompact G₂ᶜ := IsClosed.isCompact G₂_open.isClosed_compl have G₁_open' : IsOpen (ρ '' G₁ᶜ)ᶜ := (G₁_comp.image ρ_cont).isClosed.isOpen_compl have G₂_open' : IsOpen (ρ '' G₂ᶜ)ᶜ := (G₂_comp.image ρ_cont).isClosed.isOpen_compl have disj' : Disjoint (ρ '' G₁ᶜ)ᶜ (ρ '' G₂ᶜ)ᶜ := by rw [disjoint_iff_inter_eq_empty, ← compl_union, ← image_union, ← compl_inter, disjoint_iff_inter_eq_empty.mp disj, compl_empty, compl_empty_iff, image_univ_of_surjective ρ_surj] -- apply Lemma 2.2 to prove their closures are disjoint have disj'' : Disjoint (closure (ρ '' G₁ᶜ)ᶜ) (closure (ρ '' G₂ᶜ)ᶜ) := disjoint_closure_of_disjoint_isOpen disj' G₁_open' G₂_open' -- apply Lemma 2.1 to prove $\rho(x_1) = \rho(x_2)$ lies in their intersection have hx₁' := image_subset_closure_compl_image_compl_of_isOpen ρ_cont ρ_surj zorn_subset G₁_open <\| mem_image_of_mem ρ hx₁ have hx₂' := image_subset_closure_compl_image_compl_of_isOpen ρ_cont ρ_surj zorn_subset G₂_open <\| mem_image_of_mem ρ hx₂ exact disj''.ne_of_mem hx₁' hx₂' hρx /-- Lemma 2.3 in [Gleason, Projective topological spaces][gleason1958]: a continuous surjection from a compact Hausdorff space to an extremally disconnected Hausdorff space satisfying the "Zorn subset condition" is a homeomorphism. -/ noncomputable def ExtremallyDisconnected.homeoCompactToT2 [ExtremallyDisconnected A] [T2Space A] [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ) (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) : E ≃ₜ A := ρ_cont.homeoOfEquivCompactToT2 (f := Equiv.ofBijective ρ ⟨homeoCompactToT2_injective ρ_cont ρ_surj zorn_subset, ρ_surj⟩) /-- Theorem 2.5 in [Gleason, Projective topological spaces][gleason1958]: in the category of compact spaces and continuous maps, the projective spaces are precisely the extremally disconnected spaces. -/ protected theorem CompactT2.ExtremallyDisconnected.projective [ExtremallyDisconnected A] [CompactSpace A] [T2Space A] : CompactT2.Projective A := by -- let $B$ and $C$ be compact; let $f : B \twoheadrightarrow C$ and $\phi : A \to C$ be continuous intro B C _ _ _ _ _ _ φ f φ_cont f_cont f_surj -- let $D := \{(a, b) : \phi(a) = f(b)\}$ with projections $\pi_1 : D \to A$ and $\pi_2 : D \to B$ let D : Set <\| A × B := {x \| φ x.fst = f x.snd} have D_comp : CompactSpace D := isCompact_iff_compactSpace.mp (isClosed_eq (φ_cont.comp continuous_fst) (f_cont.comp continuous_snd)).isCompact -- apply Lemma 2.4 to get closed $E$ satisfying "Zorn subset condition" let X₁ : D → A := Prod.fst ∘ Subtype.val have X₁_cont : Continuous X₁ := continuous_fst.comp continuous_subtype_val have X₁_surj : X₁.Surjective := fun a => ⟨⟨⟨a, _⟩, (f_surj <\| φ a).choose_spec.symm⟩, rfl⟩ rcases exists_compact_surjective_zorn_subset X₁_cont X₁_surj with ⟨E, _, E_onto, E_min⟩ -- apply Lemma 2.3 to get homeomorphism $\pi_1\|_E : E \to A$ let ρ : E → A := E.restrict X₁ have ρ_cont : Continuous ρ := X₁_cont.continuousOn.restrict have ρ_surj : ρ.Surjective := fun a => by rcases (E_onto ▸ mem_univ a : a ∈ X₁ '' E) with ⟨d, ⟨hd, rfl⟩⟩; exact ⟨⟨d, hd⟩, rfl⟩ let ρ' := ExtremallyDisconnected.homeoCompactToT2 ρ_cont ρ_surj E_min -- prove $\rho := \pi_2\|_E \circ \pi_1\|_E^{-1}$ satisfies $\phi = f \circ \rho$ let X₂ : D → B := Prod.snd ∘ Subtype.val have X₂_cont : Continuous X₂ := continuous_snd.comp continuous_subtype_val refine ⟨E.restrict X₂ ∘ ρ'.symm, ⟨X₂_cont.continuousOn.restrict.comp ρ'.symm.continuous, ?_⟩⟩ suffices f ∘ E.restrict X₂ = φ ∘ ρ' by rw [← comp_assoc, this, comp_assoc, Homeomorph.self_comp_symm, comp_id] ext x exact x.val.mem.symm protected theorem CompactT2.projective_iff_extremallyDisconnected [CompactSpace A] [T2Space A] : Projective A ↔ ExtremallyDisconnected A := ⟨Projective.extremallyDisconnected, fun _ => ExtremallyDisconnected.projective⟩ end -- Note: It might be possible to use Gleason for this instead /-- The sigma-type of extremally disconnected spaces is extremally disconnected. -/ instance instExtremallyDisconnected {ι : Type} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [h₀ : ∀ i, ExtremallyDisconnected (X i)] : ExtremallyDisconnected (Σ i, X i) := by constructor intro s hs rw [isOpen_sigma_iff] at hs ⊢ intro i rcases h₀ i with ⟨h₀⟩ suffices h : Sigma.mk i ⁻¹' closure s = closure (Sigma.mk i ⁻¹' s) by rw [h] exact h₀ _ (hs i) apply IsOpenMap.preimage_closure_eq_closure_preimage · intro U _ rw [isOpen_sigma_iff] intro j by_cases ij : i = j · rwa [← ij, sigma_mk_preimage_image_eq_self] · rw [sigma_mk_preimage_image' ij] exact isOpen_empty · fun_prop end
.lake/packages/mathlib/Mathlib/Topology/OpenPartialHomeomorph.lean	import Mathlib.Logic.Equiv.PartialEquiv import Mathlib.Topology.Homeomorph.Lemmas import Mathlib.Topology.Sets.Opens /-! # Partial homeomorphisms This file defines homeomorphisms between open subsets of topological spaces. An element `e` of `OpenPartialHomeomorph X Y` is an extension of `PartialEquiv X Y`, i.e., it is a pair of functions `e.toFun` and `e.invFun`, inverse of each other on the sets `e.source` and `e.target`. Additionally, we require that these sets are open, and that the functions are continuous on them. Equivalently, they are homeomorphisms there. As in equivs, we register a coercion to functions, and we use `e x` and `e.symm x` throughout instead of `e.toFun x` and `e.invFun x`. ## Main definitions * `Homeomorph.toOpenPartialHomeomorph`: associating an open partial homeomorphism to a homeomorphism, with `source = target = Set.univ`; * `OpenPartialHomeomorph.symm`: the inverse of an open partial homeomorphism * `OpenPartialHomeomorph.trans`: the composition of two open partial homeomorphisms * `OpenPartialHomeomorph.refl`: the identity open partial homeomorphism * `OpenPartialHomeomorph.const`: an open partial homeomorphism which is a constant map, whose source and target are necessarily singleton sets * `OpenPartialHomeomorph.ofSet`: the identity on a set `s` * `OpenPartialHomeomorph.restr s`: restrict an open partial homeomorphism `e` to `e.source ∩ interior s` * `OpenPartialHomeomorph.EqOnSource`: equivalence relation describing the "right" notion of equality for open partial homeomorphisms * `OpenPartialHomeomorph.prod`: the product of two open partial homeomorphisms, as an open partial homeomorphism on the product space * `OpenPartialHomeomorph.pi`: the product of a finite family of open partial homeomorphisms * `OpenPartialHomeomorph.disjointUnion`: combine two open partial homeomorphisms with disjoint sources and disjoint targets * `OpenPartialHomeomorph.lift_openEmbedding`: extend an open partial homeomorphism `X → Y` under an open embedding `X → X'`, to an open partial homeomorphism `X' → Z`. (This is used to define the disjoint union of charted spaces.) ## Implementation notes Most statements are copied from their `PartialEquiv` versions, although some care is required especially when restricting to subsets, as these should be open subsets. For design notes, see `PartialEquiv.lean`. ### Local coding conventions If a lemma deals with the intersection of a set with either source or target of a `PartialEquiv`, then it should use `e.source ∩ s` or `e.target ∩ t`, not `s ∩ e.source` or `t ∩ e.target`. -/ open Function Set Filter Topology variable {X X' : Type} {Y Y' : Type} {Z Z' : Type} [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Y] [TopologicalSpace Y'] [TopologicalSpace Z] [TopologicalSpace Z'] /-- Partial homeomorphisms, defined on open subsets of the space -/ structure OpenPartialHomeomorph (X : Type) (Y : Type) [TopologicalSpace X] [TopologicalSpace Y] extends PartialEquiv X Y where open_source : IsOpen source open_target : IsOpen target continuousOn_toFun : ContinuousOn toFun source continuousOn_invFun : ContinuousOn invFun target @[deprecated (since := "2025-08-29")] alias PartialHomeomorph := OpenPartialHomeomorph namespace OpenPartialHomeomorph variable (e : OpenPartialHomeomorph X Y) /-! Basic properties; inverse (symm instance) -/ section Basic /-- Coercion of an open partial homeomorphisms to a function. We don't use `e.toFun` because it is actually `e.toPartialEquiv.toFun`, so `simp` will apply lemmas about `toPartialEquiv`. While we may want to switch to this behavior later, doing it mid-port will break a lot of proofs. -/ @[coe] def toFun' : X → Y := e.toFun /-- Coercion of an `OpenPartialHomeomorph` to function. Note that an `OpenPartialHomeomorph` is not `DFunLike`. -/ instance : CoeFun (OpenPartialHomeomorph X Y) fun _ => X → Y := ⟨fun e => e.toFun'⟩ /-- The inverse of an open partial homeomorphism -/ @[symm] protected def symm : OpenPartialHomeomorph Y X where toPartialEquiv := e.toPartialEquiv.symm open_source := e.open_target open_target := e.open_source continuousOn_toFun := e.continuousOn_invFun continuousOn_invFun := e.continuousOn_toFun /-- See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections. -/ def Simps.apply (e : OpenPartialHomeomorph X Y) : X → Y := e /-- See Note [custom simps projection] -/ def Simps.symm_apply (e : OpenPartialHomeomorph X Y) : Y → X := e.symm initialize_simps_projections OpenPartialHomeomorph (toFun → apply, invFun → symm_apply) protected theorem continuousOn : ContinuousOn e e.source := e.continuousOn_toFun theorem continuousOn_symm : ContinuousOn e.symm e.target := e.continuousOn_invFun @[simp, mfld_simps] theorem mk_coe (e : PartialEquiv X Y) (a b c d) : (OpenPartialHomeomorph.mk e a b c d : X → Y) = e := rfl @[simp, mfld_simps] theorem mk_coe_symm (e : PartialEquiv X Y) (a b c d) : ((OpenPartialHomeomorph.mk e a b c d).symm : Y → X) = e.symm := rfl theorem toPartialEquiv_injective : Injective (toPartialEquiv : OpenPartialHomeomorph X Y → PartialEquiv X Y) \| ⟨_, _, _, _, _⟩, ⟨_, _, _, _, _⟩, rfl => rfl /- Register a few simp lemmas to make sure that `simp` puts the application of a local homeomorphism in its normal form, i.e., in terms of its coercion to a function. -/ @[simp, mfld_simps] theorem toFun_eq_coe (e : OpenPartialHomeomorph X Y) : e.toFun = e := rfl @[simp, mfld_simps] theorem invFun_eq_coe (e : OpenPartialHomeomorph X Y) : e.invFun = e.symm := rfl @[simp, mfld_simps] theorem coe_coe : (e.toPartialEquiv : X → Y) = e := rfl @[simp, mfld_simps] theorem coe_coe_symm : (e.toPartialEquiv.symm : Y → X) = e.symm := rfl @[simp, mfld_simps] theorem map_source {x : X} (h : x ∈ e.source) : e x ∈ e.target := e.map_source' h /-- Variant of `map_source`, stated for images of subsets of `source`. -/ lemma map_source'' : e '' e.source ⊆ e.target := fun _ ⟨_, hx, hex⟩ ↦ mem_of_eq_of_mem (id hex.symm) (e.map_source' hx) @[simp, mfld_simps] theorem map_target {x : Y} (h : x ∈ e.target) : e.symm x ∈ e.source := e.map_target' h @[simp, mfld_simps] theorem left_inv {x : X} (h : x ∈ e.source) : e.symm (e x) = x := e.left_inv' h @[simp, mfld_simps] theorem right_inv {x : Y} (h : x ∈ e.target) : e (e.symm x) = x := e.right_inv' h theorem eq_symm_apply {x : X} {y : Y} (hx : x ∈ e.source) (hy : y ∈ e.target) : x = e.symm y ↔ e x = y := e.toPartialEquiv.eq_symm_apply hx hy protected theorem mapsTo : MapsTo e e.source e.target := fun _ => e.map_source protected theorem symm_mapsTo : MapsTo e.symm e.target e.source := e.symm.mapsTo protected theorem leftInvOn : LeftInvOn e.symm e e.source := fun _ => e.left_inv protected theorem rightInvOn : RightInvOn e.symm e e.target := fun _ => e.right_inv protected theorem invOn : InvOn e.symm e e.source e.target := ⟨e.leftInvOn, e.rightInvOn⟩ protected theorem injOn : InjOn e e.source := e.leftInvOn.injOn protected theorem bijOn : BijOn e e.source e.target := e.invOn.bijOn e.mapsTo e.symm_mapsTo protected theorem surjOn : SurjOn e e.source e.target := e.bijOn.surjOn end Basic /-- Interpret a `Homeomorph` as an `OpenPartialHomeomorph` by restricting it to an open set `s` in the domain and to `t` in the codomain. -/ @[simps! -fullyApplied apply symm_apply toPartialEquiv, simps! -isSimp source target] def _root_.Homeomorph.toOpenPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : e '' s = t) : OpenPartialHomeomorph X Y where toPartialEquiv := e.toPartialEquivOfImageEq s t h open_source := hs open_target := by simpa [← h] continuousOn_toFun := e.continuous.continuousOn continuousOn_invFun := e.symm.continuous.continuousOn @[deprecated (since := "2025-08-29")] alias _root_.Homeomorph.toPartialHomeomorphOfImageEq := _root_.Homeomorph.toOpenPartialHomeomorphOfImageEq /-- A homeomorphism induces an open partial homeomorphism on the whole space -/ @[simps! (attr := mfld_simps) -fullyApplied] def _root_.Homeomorph.toOpenPartialHomeomorph (e : X ≃ₜ Y) : OpenPartialHomeomorph X Y := e.toOpenPartialHomeomorphOfImageEq univ isOpen_univ univ <\| by rw [image_univ, e.surjective.range_eq] @[deprecated (since := "2025-08-29")] alias _root_.Homeomorph.toPartialHomeomorph := _root_.Homeomorph.toOpenPartialHomeomorph /-- Replace `toPartialEquiv` field to provide better definitional equalities. -/ def replaceEquiv (e : OpenPartialHomeomorph X Y) (e' : PartialEquiv X Y) (h : e.toPartialEquiv = e') : OpenPartialHomeomorph X Y where toPartialEquiv := e' open_source := h ▸ e.open_source open_target := h ▸ e.open_target continuousOn_toFun := h ▸ e.continuousOn_toFun continuousOn_invFun := h ▸ e.continuousOn_invFun theorem replaceEquiv_eq_self (e' : PartialEquiv X Y) (h : e.toPartialEquiv = e') : e.replaceEquiv e' h = e := by cases e subst e' rfl theorem source_preimage_target : e.source ⊆ e ⁻¹' e.target := e.mapsTo theorem eventually_left_inverse {x} (hx : x ∈ e.source) : ∀ᶠ y in 𝓝 x, e.symm (e y) = y := (e.open_source.eventually_mem hx).mono e.left_inv' theorem eventually_left_inverse' {x} (hx : x ∈ e.target) : ∀ᶠ y in 𝓝 (e.symm x), e.symm (e y) = y := e.eventually_left_inverse (e.map_target hx) theorem eventually_right_inverse {x} (hx : x ∈ e.target) : ∀ᶠ y in 𝓝 x, e (e.symm y) = y := (e.open_target.eventually_mem hx).mono e.right_inv' theorem eventually_right_inverse' {x} (hx : x ∈ e.source) : ∀ᶠ y in 𝓝 (e x), e (e.symm y) = y := e.eventually_right_inverse (e.map_source hx) theorem eventually_ne_nhdsWithin {x} (hx : x ∈ e.source) : ∀ᶠ x' in 𝓝[≠] x, e x' ≠ e x := eventually_nhdsWithin_iff.2 <\| (e.eventually_left_inverse hx).mono fun x' hx' => mt fun h => by rw [mem_singleton_iff, ← e.left_inv hx, ← h, hx'] theorem nhdsWithin_source_inter {x} (hx : x ∈ e.source) (s : Set X) : 𝓝[e.source ∩ s] x = 𝓝[s] x := nhdsWithin_inter_of_mem (mem_nhdsWithin_of_mem_nhds <\| IsOpen.mem_nhds e.open_source hx) theorem nhdsWithin_target_inter {x} (hx : x ∈ e.target) (s : Set Y) : 𝓝[e.target ∩ s] x = 𝓝[s] x := e.symm.nhdsWithin_source_inter hx s theorem image_eq_target_inter_inv_preimage {s : Set X} (h : s ⊆ e.source) : e '' s = e.target ∩ e.symm ⁻¹' s := e.toPartialEquiv.image_eq_target_inter_inv_preimage h theorem image_source_inter_eq' (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' s := e.toPartialEquiv.image_source_inter_eq' s theorem image_source_inter_eq (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' (e.source ∩ s) := e.toPartialEquiv.image_source_inter_eq s theorem symm_image_eq_source_inter_preimage {s : Set Y} (h : s ⊆ e.target) : e.symm '' s = e.source ∩ e ⁻¹' s := e.symm.image_eq_target_inter_inv_preimage h theorem symm_image_target_inter_eq (s : Set Y) : e.symm '' (e.target ∩ s) = e.source ∩ e ⁻¹' (e.target ∩ s) := e.symm.image_source_inter_eq _ theorem source_inter_preimage_inv_preimage (s : Set X) : e.source ∩ e ⁻¹' (e.symm ⁻¹' s) = e.source ∩ s := e.toPartialEquiv.source_inter_preimage_inv_preimage s theorem target_inter_inv_preimage_preimage (s : Set Y) : e.target ∩ e.symm ⁻¹' (e ⁻¹' s) = e.target ∩ s := e.symm.source_inter_preimage_inv_preimage _ theorem source_inter_preimage_target_inter (s : Set Y) : e.source ∩ e ⁻¹' (e.target ∩ s) = e.source ∩ e ⁻¹' s := e.toPartialEquiv.source_inter_preimage_target_inter s theorem image_source_eq_target : e '' e.source = e.target := e.toPartialEquiv.image_source_eq_target theorem symm_image_target_eq_source : e.symm '' e.target = e.source := e.symm.image_source_eq_target /-- Two open partial homeomorphisms are equal when they have equal `toFun`, `invFun` and `source`. It is not sufficient to have equal `toFun` and `source`, as this only determines `invFun` on the target. This would only be true for a weaker notion of equality, arguably the right one, called `EqOnSource`. -/ @[ext] protected theorem ext (e' : OpenPartialHomeomorph X Y) (h : ∀ x, e x = e' x) (hinv : ∀ x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' := toPartialEquiv_injective (PartialEquiv.ext h hinv hs) @[simp, mfld_simps] theorem symm_toPartialEquiv : e.symm.toPartialEquiv = e.toPartialEquiv.symm := rfl -- The following lemmas are already simp via `PartialEquiv` theorem symm_source : e.symm.source = e.target := rfl theorem symm_target : e.symm.target = e.source := rfl @[simp, mfld_simps] theorem symm_symm : e.symm.symm = e := rfl theorem symm_bijective : Function.Bijective (OpenPartialHomeomorph.symm : OpenPartialHomeomorph X Y → OpenPartialHomeomorph Y X) := Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩ /-- An open partial homeomorphism is continuous at any point of its source -/ protected theorem continuousAt {x : X} (h : x ∈ e.source) : ContinuousAt e x := (e.continuousOn x h).continuousAt (e.open_source.mem_nhds h) /-- An open partial homeomorphism inverse is continuous at any point of its target -/ theorem continuousAt_symm {x : Y} (h : x ∈ e.target) : ContinuousAt e.symm x := e.symm.continuousAt h theorem tendsto_symm {x} (hx : x ∈ e.source) : Tendsto e.symm (𝓝 (e x)) (𝓝 x) := by simpa only [ContinuousAt, e.left_inv hx] using e.continuousAt_symm (e.map_source hx) theorem map_nhds_eq {x} (hx : x ∈ e.source) : map e (𝓝 x) = 𝓝 (e x) := le_antisymm (e.continuousAt hx) <\| le_map_of_right_inverse (e.eventually_right_inverse' hx) (e.tendsto_symm hx) theorem symm_map_nhds_eq {x} (hx : x ∈ e.source) : map e.symm (𝓝 (e x)) = 𝓝 x := (e.symm.map_nhds_eq <\| e.map_source hx).trans <\| by rw [e.left_inv hx] theorem image_mem_nhds {x} (hx : x ∈ e.source) {s : Set X} (hs : s ∈ 𝓝 x) : e '' s ∈ 𝓝 (e x) := e.map_nhds_eq hx ▸ Filter.image_mem_map hs theorem map_nhdsWithin_eq {x} (hx : x ∈ e.source) (s : Set X) : map e (𝓝[s] x) = 𝓝[e '' (e.source ∩ s)] e x := calc map e (𝓝[s] x) = map e (𝓝[e.source ∩ s] x) := congr_arg (map e) (e.nhdsWithin_source_inter hx _).symm _ = 𝓝[e '' (e.source ∩ s)] e x := (e.leftInvOn.mono inter_subset_left).map_nhdsWithin_eq (e.left_inv hx) (e.continuousAt_symm (e.map_source hx)).continuousWithinAt (e.continuousAt hx).continuousWithinAt theorem map_nhdsWithin_preimage_eq {x} (hx : x ∈ e.source) (s : Set Y) : map e (𝓝[e ⁻¹' s] x) = 𝓝[s] e x := by rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage, e.nhdsWithin_target_inter (e.map_source hx)] theorem eventually_nhds {x : X} (p : Y → Prop) (hx : x ∈ e.source) : (∀ᶠ y in 𝓝 (e x), p y) ↔ ∀ᶠ x in 𝓝 x, p (e x) := Iff.trans (by rw [e.map_nhds_eq hx]) eventually_map theorem eventually_nhds' {x : X} (p : X → Prop) (hx : x ∈ e.source) : (∀ᶠ y in 𝓝 (e x), p (e.symm y)) ↔ ∀ᶠ x in 𝓝 x, p x := by rw [e.eventually_nhds _ hx] refine eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => ?_) rw [hy] theorem eventually_nhdsWithin {x : X} (p : Y → Prop) {s : Set X} (hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) := by refine Iff.trans ?_ eventually_map rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)] theorem eventually_nhdsWithin' {x : X} (p : X → Prop) {s : Set X} (hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p (e.symm y)) ↔ ∀ᶠ x in 𝓝[s] x, p x := by rw [e.eventually_nhdsWithin _ hx] refine eventually_congr <\| (eventually_nhdsWithin_of_eventually_nhds <\| e.eventually_left_inverse hx).mono fun y hy => ?_ rw [hy] /-- This lemma is useful in the manifold library in the case that `e` is a chart. It states that locally around `e x` the set `e.symm ⁻¹' s` is the same as the set intersected with the target of `e` and some other neighborhood of `f x` (which will be the source of a chart on `Z`). -/ theorem preimage_eventuallyEq_target_inter_preimage_inter {e : OpenPartialHomeomorph X Y} {s : Set X} {t : Set Z} {x : X} {f : X → Z} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source) (ht : t ∈ 𝓝 (f x)) : e.symm ⁻¹' s =ᶠ[𝓝 (e x)] (e.target ∩ e.symm ⁻¹' (s ∩ f ⁻¹' t) : Set Y) := by rw [eventuallyEq_set, e.eventually_nhds _ hxe] filter_upwards [e.open_source.mem_nhds hxe, mem_nhdsWithin_iff_eventually.mp (hf.preimage_mem_nhdsWithin ht)] intro y hy hyu simp_rw [mem_inter_iff, mem_preimage, mem_inter_iff, e.mapsTo hy, true_and, iff_self_and, e.left_inv hy, iff_true_intro hyu] theorem isOpen_inter_preimage {s : Set Y} (hs : IsOpen s) : IsOpen (e.source ∩ e ⁻¹' s) := e.continuousOn.isOpen_inter_preimage e.open_source hs theorem isOpen_inter_preimage_symm {s : Set X} (hs : IsOpen s) : IsOpen (e.target ∩ e.symm ⁻¹' s) := e.symm.continuousOn.isOpen_inter_preimage e.open_target hs /-- An open partial homeomorphism is an open map on its source: the image of an open subset of the source is open. -/ lemma isOpen_image_of_subset_source {s : Set X} (hs : IsOpen s) (hse : s ⊆ e.source) : IsOpen (e '' s) := by rw [(image_eq_target_inter_inv_preimage (e := e) hse)] exact e.continuousOn_invFun.isOpen_inter_preimage e.open_target hs /-- The image of the restriction of an open set to the source is open. -/ theorem isOpen_image_source_inter {s : Set X} (hs : IsOpen s) : IsOpen (e '' (e.source ∩ s)) := e.isOpen_image_of_subset_source (e.open_source.inter hs) inter_subset_left /-- The inverse of an open partial homeomorphism `e` is an open map on `e.target`. -/ lemma isOpen_image_symm_of_subset_target {t : Set Y} (ht : IsOpen t) (hte : t ⊆ e.target) : IsOpen (e.symm '' t) := isOpen_image_of_subset_source e.symm ht (e.symm_source ▸ hte) lemma isOpen_symm_image_iff_of_subset_target {t : Set Y} (hs : t ⊆ e.target) : IsOpen (e.symm '' t) ↔ IsOpen t := by refine ⟨fun h ↦ ?_, fun h ↦ e.symm.isOpen_image_of_subset_source h hs⟩ have hs' : e.symm '' t ⊆ e.source := by rw [e.symm_image_eq_source_inter_preimage hs] apply Set.inter_subset_left rw [← e.image_symm_image_of_subset_target hs] exact e.isOpen_image_of_subset_source h hs' theorem isOpen_image_iff_of_subset_source {s : Set X} (hs : s ⊆ e.source) : IsOpen (e '' s) ↔ IsOpen s := by rw [← e.symm.isOpen_symm_image_iff_of_subset_target hs, e.symm_symm] section IsImage /-! ### `OpenPartialHomeomorph.IsImage` relation We say that `t : Set Y` is an image of `s : Set X` under an open partial homeomorphism `e` if any of the following equivalent conditions hold: `e '' (e.source ∩ s) = e.target ∩ t`; * `e.source ∩ e ⁻¹ t = e.source ∩ s`; * `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition). This definition is a restatement of `PartialEquiv.IsImage` for open partial homeomorphisms. In this section we transfer API about `PartialEquiv.IsImage` to open partial homeomorphisms and add a few `OpenPartialHomeomorph`-specific lemmas like `OpenPartialHomeomorph.IsImage.closure`. -/ /-- We say that `t : Set Y` is an image of `s : Set X` under an open partial homeomorphism `e` if any of the following equivalent conditions hold: * `e '' (e.source ∩ s) = e.target ∩ t`; * `e.source ∩ e ⁻¹ t = e.source ∩ s`; * `∀ x ∈ e.source, e x ∈ t ↔ x ∈ s` (this one is used in the definition). -/ def IsImage (s : Set X) (t : Set Y) : Prop := ∀ ⦃x⦄, x ∈ e.source → (e x ∈ t ↔ x ∈ s) namespace IsImage variable {e} {s : Set X} {t : Set Y} {x : X} {y : Y} theorem toPartialEquiv (h : e.IsImage s t) : e.toPartialEquiv.IsImage s t := h theorem apply_mem_iff (h : e.IsImage s t) (hx : x ∈ e.source) : e x ∈ t ↔ x ∈ s := h hx protected theorem symm (h : e.IsImage s t) : e.symm.IsImage t s := h.toPartialEquiv.symm theorem symm_apply_mem_iff (h : e.IsImage s t) (hy : y ∈ e.target) : e.symm y ∈ s ↔ y ∈ t := h.symm hy @[simp] theorem symm_iff : e.symm.IsImage t s ↔ e.IsImage s t := ⟨fun h => h.symm, fun h => h.symm⟩ protected theorem mapsTo (h : e.IsImage s t) : MapsTo e (e.source ∩ s) (e.target ∩ t) := h.toPartialEquiv.mapsTo theorem symm_mapsTo (h : e.IsImage s t) : MapsTo e.symm (e.target ∩ t) (e.source ∩ s) := h.symm.mapsTo theorem image_eq (h : e.IsImage s t) : e '' (e.source ∩ s) = e.target ∩ t := h.toPartialEquiv.image_eq theorem symm_image_eq (h : e.IsImage s t) : e.symm '' (e.target ∩ t) = e.source ∩ s := h.symm.image_eq theorem iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s := PartialEquiv.IsImage.iff_preimage_eq alias ⟨preimage_eq, of_preimage_eq⟩ := iff_preimage_eq theorem iff_symm_preimage_eq : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' s = e.target ∩ t := symm_iff.symm.trans iff_preimage_eq alias ⟨symm_preimage_eq, of_symm_preimage_eq⟩ := iff_symm_preimage_eq theorem iff_symm_preimage_eq' : e.IsImage s t ↔ e.target ∩ e.symm ⁻¹' (e.source ∩ s) = e.target ∩ t := by rw [iff_symm_preimage_eq, ← image_source_inter_eq, ← image_source_inter_eq'] alias ⟨symm_preimage_eq', of_symm_preimage_eq'⟩ := iff_symm_preimage_eq' theorem iff_preimage_eq' : e.IsImage s t ↔ e.source ∩ e ⁻¹' (e.target ∩ t) = e.source ∩ s := symm_iff.symm.trans iff_symm_preimage_eq' alias ⟨preimage_eq', of_preimage_eq'⟩ := iff_preimage_eq' theorem of_image_eq (h : e '' (e.source ∩ s) = e.target ∩ t) : e.IsImage s t := PartialEquiv.IsImage.of_image_eq h theorem of_symm_image_eq (h : e.symm '' (e.target ∩ t) = e.source ∩ s) : e.IsImage s t := PartialEquiv.IsImage.of_symm_image_eq h protected theorem compl (h : e.IsImage s t) : e.IsImage sᶜ tᶜ := fun _ hx => (h hx).not protected theorem inter {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') : e.IsImage (s ∩ s') (t ∩ t') := fun _ hx => (h hx).and (h' hx) protected theorem union {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') : e.IsImage (s ∪ s') (t ∪ t') := fun _ hx => (h hx).or (h' hx) protected theorem diff {s' t'} (h : e.IsImage s t) (h' : e.IsImage s' t') : e.IsImage (s \ s') (t \ t') := h.inter h'.compl theorem leftInvOn_piecewise {e' : OpenPartialHomeomorph X Y} [∀ i, Decidable (i ∈ s)] [∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) : LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) := h.toPartialEquiv.leftInvOn_piecewise h' theorem inter_eq_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t) (h' : e'.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) : e.target ∩ t = e'.target ∩ t := h.toPartialEquiv.inter_eq_of_inter_eq_of_eqOn h' hs Heq theorem symm_eqOn_of_inter_eq_of_eqOn {e' : OpenPartialHomeomorph X Y} (h : e.IsImage s t) (hs : e.source ∩ s = e'.source ∩ s) (Heq : EqOn e e' (e.source ∩ s)) : EqOn e.symm e'.symm (e.target ∩ t) := h.toPartialEquiv.symm_eq_on_of_inter_eq_of_eqOn hs Heq theorem map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x := by rw [e.map_nhdsWithin_eq hx, h.image_eq, e.nhdsWithin_target_inter (e.map_source hx)] protected theorem closure (h : e.IsImage s t) : e.IsImage (closure s) (closure t) := fun x hx => by simp only [mem_closure_iff_nhdsWithin_neBot, ← h.map_nhdsWithin_eq hx, map_neBot_iff] protected theorem interior (h : e.IsImage s t) : e.IsImage (interior s) (interior t) := by simpa only [closure_compl, compl_compl] using h.compl.closure.compl protected theorem frontier (h : e.IsImage s t) : e.IsImage (frontier s) (frontier t) := h.closure.diff h.interior theorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t) := ⟨fun hs => h.symm_preimage_eq' ▸ e.symm.isOpen_inter_preimage hs, fun hs => h.preimage_eq' ▸ e.isOpen_inter_preimage hs⟩ /-- Restrict an `OpenPartialHomeomorph` to a pair of corresponding open sets. -/ @[simps toPartialEquiv] def restr (h : e.IsImage s t) (hs : IsOpen (e.source ∩ s)) : OpenPartialHomeomorph X Y where toPartialEquiv := h.toPartialEquiv.restr open_source := hs open_target := h.isOpen_iff.1 hs continuousOn_toFun := e.continuousOn.mono inter_subset_left continuousOn_invFun := e.symm.continuousOn.mono inter_subset_left end IsImage theorem isImage_source_target : e.IsImage e.source e.target := e.toPartialEquiv.isImage_source_target theorem isImage_source_target_of_disjoint (e' : OpenPartialHomeomorph X Y) (hs : Disjoint e.source e'.source) (ht : Disjoint e.target e'.target) : e.IsImage e'.source e'.target := e.toPartialEquiv.isImage_source_target_of_disjoint e'.toPartialEquiv hs ht /-- Preimage of interior or interior of preimage coincide for open partial homeomorphisms, when restricted to the source. -/ theorem preimage_interior (s : Set Y) : e.source ∩ e ⁻¹' interior s = e.source ∩ interior (e ⁻¹' s) := (IsImage.of_preimage_eq rfl).interior.preimage_eq theorem preimage_closure (s : Set Y) : e.source ∩ e ⁻¹' closure s = e.source ∩ closure (e ⁻¹' s) := (IsImage.of_preimage_eq rfl).closure.preimage_eq theorem preimage_frontier (s : Set Y) : e.source ∩ e ⁻¹' frontier s = e.source ∩ frontier (e ⁻¹' s) := (IsImage.of_preimage_eq rfl).frontier.preimage_eq end IsImage /-- A `PartialEquiv` with continuous open forward map and open source is a `OpenPartialHomeomorph`. -/ def ofContinuousOpenRestrict (e : PartialEquiv X Y) (hc : ContinuousOn e e.source) (ho : IsOpenMap (e.source.restrict e)) (hs : IsOpen e.source) : OpenPartialHomeomorph X Y where toPartialEquiv := e open_source := hs open_target := by simpa only [range_restrict, e.image_source_eq_target] using ho.isOpen_range continuousOn_toFun := hc continuousOn_invFun := e.image_source_eq_target ▸ ho.continuousOn_image_of_leftInvOn e.leftInvOn /-- A `PartialEquiv` with continuous open forward map and open source is a `OpenPartialHomeomorph`. -/ def ofContinuousOpen (e : PartialEquiv X Y) (hc : ContinuousOn e e.source) (ho : IsOpenMap e) (hs : IsOpen e.source) : OpenPartialHomeomorph X Y := ofContinuousOpenRestrict e hc (ho.restrict hs) hs /-- Restricting an open partial homeomorphism `e` to `e.source ∩ s` when `s` is open. This is sometimes hard to use because of the openness assumption, but it has the advantage that when it can be used then its `PartialEquiv` is defeq to `PartialEquiv.restr`. -/ protected def restrOpen (s : Set X) (hs : IsOpen s) : OpenPartialHomeomorph X Y := (@IsImage.of_symm_preimage_eq X Y _ _ e s (e.symm ⁻¹' s) rfl).restr (IsOpen.inter e.open_source hs) @[simp, mfld_simps] theorem restrOpen_toPartialEquiv (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).toPartialEquiv = e.toPartialEquiv.restr s := rfl -- Already simp via `PartialEquiv` theorem restrOpen_source (s : Set X) (hs : IsOpen s) : (e.restrOpen s hs).source = e.source ∩ s := rfl /-- Restricting an open partial homeomorphism `e` to `e.source ∩ interior s`. We use the interior to make sure that the restriction is well defined whatever the set s, since open partial homeomorphisms are by definition defined on open sets. In applications where `s` is open, this coincides with the restriction of partial equivalences. -/ @[simps! (attr := mfld_simps) -fullyApplied apply symm_apply, simps! (attr := grind =) -isSimp source target] protected def restr (s : Set X) : OpenPartialHomeomorph X Y := e.restrOpen (interior s) isOpen_interior @[simp, mfld_simps] theorem restr_toPartialEquiv (s : Set X) : (e.restr s).toPartialEquiv = e.toPartialEquiv.restr (interior s) := rfl theorem restr_source' (s : Set X) (hs : IsOpen s) : (e.restr s).source = e.source ∩ s := by grind theorem restr_toPartialEquiv' (s : Set X) (hs : IsOpen s) : (e.restr s).toPartialEquiv = e.toPartialEquiv.restr s := by rw [e.restr_toPartialEquiv, hs.interior_eq] theorem restr_eq_of_source_subset {e : OpenPartialHomeomorph X Y} {s : Set X} (h : e.source ⊆ s) : e.restr s = e := toPartialEquiv_injective <\| PartialEquiv.restr_eq_of_source_subset <\| interior_maximal h e.open_source @[simp, mfld_simps] theorem restr_univ {e : OpenPartialHomeomorph X Y} : e.restr univ = e := restr_eq_of_source_subset (subset_univ _) @[simp, grind =] theorem restr_source_inter (s : Set X) : e.restr (e.source ∩ s) = e.restr s := by refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_ simp [e.open_source.interior_eq, ← inter_assoc] /-- The identity on the whole space as an open partial homeomorphism. -/ @[simps! (attr := mfld_simps) -fullyApplied apply, simps! -isSimp source target] protected def refl (X : Type) [TopologicalSpace X] : OpenPartialHomeomorph X X := (Homeomorph.refl X).toOpenPartialHomeomorph @[simp, mfld_simps] theorem refl_partialEquiv : (OpenPartialHomeomorph.refl X).toPartialEquiv = PartialEquiv.refl X := rfl @[simp, mfld_simps] theorem refl_symm : (OpenPartialHomeomorph.refl X).symm = OpenPartialHomeomorph.refl X := rfl /-! const: `PartialEquiv.const` as an open partial homeomorphism -/ section const variable {a : X} {b : Y} /-- This is `PartialEquiv.single` as an open partial homeomorphism: a constant map, whose source and target are necessarily singleton sets. -/ def const (ha : IsOpen {a}) (hb : IsOpen {b}) : OpenPartialHomeomorph X Y where toPartialEquiv := PartialEquiv.single a b open_source := ha open_target := hb continuousOn_toFun := by simp continuousOn_invFun := by simp @[simp, mfld_simps] lemma const_apply (ha : IsOpen {a}) (hb : IsOpen {b}) (x : X) : (const ha hb) x = b := rfl @[simp, mfld_simps] lemma const_source (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).source = {a} := rfl @[simp, mfld_simps] lemma const_target (ha : IsOpen {a}) (hb : IsOpen {b}) : (const ha hb).target = {b} := rfl end const /-! ofSet: the identity on a set `s` -/ section ofSet variable {s : Set X} (hs : IsOpen s) /-- The identity partial equivalence on a set `s` -/ @[simps! (attr := mfld_simps) -fullyApplied apply, simps! -isSimp source target] def ofSet (s : Set X) (hs : IsOpen s) : OpenPartialHomeomorph X X where toPartialEquiv := PartialEquiv.ofSet s open_source := hs open_target := hs continuousOn_toFun := continuous_id.continuousOn continuousOn_invFun := continuous_id.continuousOn @[simp, mfld_simps] theorem ofSet_toPartialEquiv : (ofSet s hs).toPartialEquiv = PartialEquiv.ofSet s := rfl @[simp, mfld_simps] theorem ofSet_symm : (ofSet s hs).symm = ofSet s hs := rfl @[simp, mfld_simps] theorem ofSet_univ_eq_refl : ofSet univ isOpen_univ = OpenPartialHomeomorph.refl X := by ext <;> simp end ofSet /-! `trans`: composition of two open partial homeomorphisms -/ section trans variable (e' : OpenPartialHomeomorph Y Z) /-- Composition of two open partial homeomorphisms when the target of the first and the source of the second coincide. -/ @[simps! apply symm_apply toPartialEquiv, simps! -isSimp source target] protected def trans' (h : e.target = e'.source) : OpenPartialHomeomorph X Z where toPartialEquiv := PartialEquiv.trans' e.toPartialEquiv e'.toPartialEquiv h open_source := e.open_source open_target := e'.open_target continuousOn_toFun := e'.continuousOn.comp e.continuousOn <\| h ▸ e.mapsTo continuousOn_invFun := e.continuousOn_symm.comp e'.continuousOn_symm <\| h.symm ▸ e'.symm_mapsTo /-- Composing two open partial homeomorphisms, by restricting to the maximal domain where their composition is well defined. Within the `Manifold` namespace, there is the notation `e ≫ₕ f` for this. -/ @[trans] protected def trans : OpenPartialHomeomorph X Z := OpenPartialHomeomorph.trans' (e.symm.restrOpen e'.source e'.open_source).symm (e'.restrOpen e.target e.open_target) (by simp [inter_comm]) @[simp, mfld_simps] theorem trans_toPartialEquiv : (e.trans e').toPartialEquiv = e.toPartialEquiv.trans e'.toPartialEquiv := rfl @[simp, mfld_simps] theorem coe_trans : (e.trans e' : X → Z) = e' ∘ e := rfl @[simp, mfld_simps] theorem coe_trans_symm : ((e.trans e').symm : Z → X) = e.symm ∘ e'.symm := rfl theorem trans_apply {x : X} : (e.trans e') x = e' (e x) := rfl theorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm := rfl /- This could be considered as a simp lemma, but there are many situations where it makes something simple into something more complicated. -/ theorem trans_source : (e.trans e').source = e.source ∩ e ⁻¹' e'.source := PartialEquiv.trans_source e.toPartialEquiv e'.toPartialEquiv theorem trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) := PartialEquiv.trans_source' e.toPartialEquiv e'.toPartialEquiv theorem trans_source'' : (e.trans e').source = e.symm '' (e.target ∩ e'.source) := PartialEquiv.trans_source'' e.toPartialEquiv e'.toPartialEquiv theorem image_trans_source : e '' (e.trans e').source = e.target ∩ e'.source := PartialEquiv.image_trans_source e.toPartialEquiv e'.toPartialEquiv theorem trans_target : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' e.target := rfl theorem trans_target' : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' (e'.source ∩ e.target) := trans_source' e'.symm e.symm theorem trans_target'' : (e.trans e').target = e' '' (e'.source ∩ e.target) := trans_source'' e'.symm e.symm theorem inv_image_trans_target : e'.symm '' (e.trans e').target = e'.source ∩ e.target := image_trans_source e'.symm e.symm theorem trans_assoc (e'' : OpenPartialHomeomorph Z Z') : (e.trans e').trans e'' = e.trans (e'.trans e'') := toPartialEquiv_injective <\| e.1.trans_assoc _ _ @[simp, mfld_simps] theorem trans_refl : e.trans (OpenPartialHomeomorph.refl Y) = e := toPartialEquiv_injective e.1.trans_refl @[simp, mfld_simps] theorem refl_trans : (OpenPartialHomeomorph.refl X).trans e = e := toPartialEquiv_injective e.1.refl_trans theorem trans_ofSet {s : Set Y} (hs : IsOpen s) : e.trans (ofSet s hs) = e.restr (e ⁻¹' s) := OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <\| by rw [trans_source, restr_source, ofSet_source, ← preimage_interior, hs.interior_eq] theorem trans_of_set' {s : Set Y} (hs : IsOpen s) : e.trans (ofSet s hs) = e.restr (e.source ∩ e ⁻¹' s) := by rw [trans_ofSet, restr_source_inter] theorem ofSet_trans {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr s := OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) <\| by simp [hs.interior_eq, inter_comm] theorem ofSet_trans' {s : Set X} (hs : IsOpen s) : (ofSet s hs).trans e = e.restr (e.source ∩ s) := by rw [ofSet_trans, restr_source_inter] @[simp, mfld_simps] theorem ofSet_trans_ofSet {s : Set X} (hs : IsOpen s) {s' : Set X} (hs' : IsOpen s') : (ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') (IsOpen.inter hs hs') := by rw [(ofSet s hs).trans_ofSet hs'] ext <;> simp [hs'.interior_eq] theorem restr_trans (s : Set X) : (e.restr s).trans e' = (e.trans e').restr s := toPartialEquiv_injective <\| PartialEquiv.restr_trans e.toPartialEquiv e'.toPartialEquiv (interior s) end trans /-! `EqOnSource`: equivalence on their source -/ section EqOnSource /-- `EqOnSource e e'` means that `e` and `e'` have the same source, and coincide there. They should really be considered the same partial equivalence. -/ def EqOnSource (e e' : OpenPartialHomeomorph X Y) : Prop := e.source = e'.source ∧ EqOn e e' e.source theorem eqOnSource_iff (e e' : OpenPartialHomeomorph X Y) : EqOnSource e e' ↔ PartialEquiv.EqOnSource e.toPartialEquiv e'.toPartialEquiv := Iff.rfl /-- `EqOnSource` is an equivalence relation. -/ instance eqOnSourceSetoid : Setoid (OpenPartialHomeomorph X Y) := { PartialEquiv.eqOnSourceSetoid.comap toPartialEquiv with r := EqOnSource } theorem eqOnSource_refl : e ≈ e := Setoid.refl _ /-- If two open partial homeomorphisms are equivalent, so are their inverses. -/ theorem EqOnSource.symm' {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : e.symm ≈ e'.symm := PartialEquiv.EqOnSource.symm' h /-- Two equivalent open partial homeomorphisms have the same source. -/ theorem EqOnSource.source_eq {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : e.source = e'.source := h.1 /-- Two equivalent open partial homeomorphisms have the same target. -/ theorem EqOnSource.target_eq {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : e.target = e'.target := h.symm'.1 /-- Two equivalent open partial homeomorphisms have coinciding `toFun` on the source -/ theorem EqOnSource.eqOn {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : EqOn e e' e.source := h.2 /-- Two equivalent open partial homeomorphisms have coinciding `invFun` on the target -/ theorem EqOnSource.symm_eqOn_target {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') : EqOn e.symm e'.symm e.target := h.symm'.2 /-- Composition of open partial homeomorphisms respects equivalence. -/ theorem EqOnSource.trans' {e e' : OpenPartialHomeomorph X Y} {f f' : OpenPartialHomeomorph Y Z} (he : e ≈ e') (hf : f ≈ f') : e.trans f ≈ e'.trans f' := PartialEquiv.EqOnSource.trans' he hf /-- Restriction of open partial homeomorphisms respects equivalence -/ theorem EqOnSource.restr {e e' : OpenPartialHomeomorph X Y} (he : e ≈ e') (s : Set X) : e.restr s ≈ e'.restr s := PartialEquiv.EqOnSource.restr he _ /-- Two equivalent open partial homeomorphisms are equal when the source and target are `univ`. -/ theorem Set.EqOn.restr_eqOn_source {e e' : OpenPartialHomeomorph X Y} (h : EqOn e e' (e.source ∩ e'.source)) : e.restr e'.source ≈ e'.restr e.source := by constructor · rw [e'.restr_source' _ e.open_source] rw [e.restr_source' _ e'.open_source] exact Set.inter_comm _ _ · rw [e.restr_source' _ e'.open_source] refine (EqOn.trans ?_ h).trans ?_ <;> simp only [mfld_simps, eqOn_refl] /-- Composition of an open partial homeomorphism and its inverse is equivalent to the restriction of the identity to the source -/ theorem self_trans_symm : e.trans e.symm ≈ OpenPartialHomeomorph.ofSet e.source e.open_source := PartialEquiv.self_trans_symm _ theorem symm_trans_self : e.symm.trans e ≈ OpenPartialHomeomorph.ofSet e.target e.open_target := e.symm.self_trans_symm theorem eq_of_eqOnSource_univ {e e' : OpenPartialHomeomorph X Y} (h : e ≈ e') (s : e.source = univ) (t : e.target = univ) : e = e' := toPartialEquiv_injective <\| PartialEquiv.eq_of_eqOnSource_univ _ _ h s t variable {s : Set X} theorem restr_symm_trans {e' : OpenPartialHomeomorph X Y} (hs : IsOpen s) (hs' : IsOpen (e '' s)) (hs'' : s ⊆ e.source) : (e.restr s).symm.trans e' ≈ (e.symm.trans e').restr (e '' s) := by refine ⟨?_, ?_⟩ · simp only [trans_toPartialEquiv, symm_toPartialEquiv, restr_toPartialEquiv, PartialEquiv.trans_source, PartialEquiv.symm_source, PartialEquiv.restr_target, coe_coe_symm, PartialEquiv.restr_coe_symm, PartialEquiv.restr_source] rw [interior_eq_iff_isOpen.mpr hs', interior_eq_iff_isOpen.mpr hs] -- Get rid of the middle term, which is merely distracting. rw [inter_assoc, inter_assoc, inter_comm _ (e '' s), ← inter_assoc, ← inter_assoc] congr 1 -- Now, just a bunch of rewrites: should this be a separate lemma? rw [← image_source_inter_eq', ← image_source_eq_target] refine image_inter_on ?_ intro x hx y hy h rw [← left_inv e hy, ← left_inv e (hs'' hx), h] · simp_rw [coe_trans, restr_symm_apply, restr_apply, coe_trans] intro x hx simp theorem symm_trans_restr (e' : OpenPartialHomeomorph X Y) (hs : IsOpen s) : e'.symm.trans (e.restr s) ≈ (e'.symm.trans e).restr (e'.target ∩ e'.symm ⁻¹' s) := by have ht : IsOpen (e'.target ∩ e'.symm ⁻¹' s) := by rw [← image_source_inter_eq'] exact isOpen_image_source_inter e' hs refine ⟨?_, ?_⟩ · simp only [trans_toPartialEquiv, symm_toPartialEquiv, restr_toPartialEquiv, PartialEquiv.trans_source, PartialEquiv.symm_source, coe_coe_symm, PartialEquiv.restr_source, preimage_inter] -- Shuffle the intersections, pull e'.target into the interior and use interior_inter. rw [interior_eq_iff_isOpen.mpr hs, ← inter_assoc, inter_comm e'.target, inter_assoc, inter_assoc] congr 1 nth_rw 2 [← interior_eq_iff_isOpen.mpr e'.open_target] rw [← interior_inter, ← inter_assoc, inter_self, interior_eq_iff_isOpen.mpr ht] · simp [Set.eqOn_refl] lemma restr_eqOnSource_restr {s' : Set X} (hss' : e.source ∩ interior s = e.source ∩ interior s') : e.restr s ≈ e.restr s' := by constructor · simpa [e.restr_source] · simp [Set.eqOn_refl] lemma restr_inter_source : e.restr (e.source ∩ s) ≈ e.restr s := e.restr_eqOnSource_restr (by simp [interior_eq_iff_isOpen.mpr e.open_source]) end EqOnSource /-! product of two open partial homeomorphisms -/ section Prod /-- The product of two open partial homeomorphisms, as an open partial homeomorphism on the product space. -/ @[simps! (attr := mfld_simps) -fullyApplied toPartialEquiv apply, simps! -isSimp source target symm_apply] def prod (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') : OpenPartialHomeomorph (X × Y) (X' × Y') where open_source := eX.open_source.prod eY.open_source open_target := eX.open_target.prod eY.open_target continuousOn_toFun := eX.continuousOn.prodMap eY.continuousOn continuousOn_invFun := eX.continuousOn_symm.prodMap eY.continuousOn_symm toPartialEquiv := eX.toPartialEquiv.prod eY.toPartialEquiv @[simp, mfld_simps] theorem prod_symm (eX : OpenPartialHomeomorph X X') (eY : OpenPartialHomeomorph Y Y') : (eX.prod eY).symm = eX.symm.prod eY.symm := rfl @[simp] theorem refl_prod_refl : (OpenPartialHomeomorph.refl X).prod (OpenPartialHomeomorph.refl Y) = OpenPartialHomeomorph.refl (X × Y) := OpenPartialHomeomorph.ext _ _ (fun _ => rfl) (fun _ => rfl) univ_prod_univ @[simp, mfld_simps] theorem prod_trans (e : OpenPartialHomeomorph X Y) (f : OpenPartialHomeomorph Y Z) (e' : OpenPartialHomeomorph X' Y') (f' : OpenPartialHomeomorph Y' Z') : (e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') := toPartialEquiv_injective <\| e.1.prod_trans .. theorem prod_eq_prod_of_nonempty {eX eX' : OpenPartialHomeomorph X X'} {eY eY' : OpenPartialHomeomorph Y Y'} (h : (eX.prod eY).source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by obtain ⟨⟨x, y⟩, -⟩ := id h haveI : Nonempty X := ⟨x⟩ haveI : Nonempty X' := ⟨eX x⟩ haveI : Nonempty Y := ⟨y⟩ haveI : Nonempty Y' := ⟨eY y⟩ simp_rw [OpenPartialHomeomorph.ext_iff, prod_apply, prod_symm_apply, prod_source, Prod.ext_iff, Set.prod_eq_prod_iff_of_nonempty h, forall_and, Prod.forall, forall_const, and_assoc, and_left_comm] theorem prod_eq_prod_of_nonempty' {eX eX' : OpenPartialHomeomorph X X'} {eY eY' : OpenPartialHomeomorph Y Y'} (h : (eX'.prod eY').source.Nonempty) : eX.prod eY = eX'.prod eY' ↔ eX = eX' ∧ eY = eY' := by rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ eY'] end Prod /-! finite product of partial homeomorphisms -/ section Pi variable {ι : Type} [Finite ι] {X Y : ι → Type} [∀ i, TopologicalSpace (X i)] [∀ i, TopologicalSpace (Y i)] (ei : ∀ i, OpenPartialHomeomorph (X i) (Y i)) /-- The product of a finite family of `OpenPartialHomeomorph`s. -/ @[simps! toPartialEquiv apply symm_apply source target] def pi : OpenPartialHomeomorph (∀ i, X i) (∀ i, Y i) where toPartialEquiv := PartialEquiv.pi fun i => (ei i).toPartialEquiv open_source := isOpen_set_pi finite_univ fun i _ => (ei i).open_source open_target := isOpen_set_pi finite_univ fun i _ => (ei i).open_target continuousOn_toFun := continuousOn_pi.2 fun i => (ei i).continuousOn.comp (continuous_apply _).continuousOn fun _f hf => hf i trivial continuousOn_invFun := continuousOn_pi.2 fun i => (ei i).continuousOn_symm.comp (continuous_apply _).continuousOn fun _f hf => hf i trivial end Pi /-! combining two partial homeomorphisms using `Set.piecewise` -/ section Piecewise /-- Combine two `OpenPartialHomeomorph`s using `Set.piecewise`. The source of the new `OpenPartialHomeomorph` is `s.ite e.source e'.source = e.source ∩ s ∪ e'.source \ s`, and similarly for target. The function sends `e.source ∩ s` to `e.target ∩ t` using `e` and `e'.source \ s` to `e'.target \ t` using `e'`, and similarly for the inverse function. To ensure the maps `toFun` and `invFun` are inverse of each other on the new `source` and `target`, the definition assumes that the sets `s` and `t` are related both by `e.is_image` and `e'.is_image`. To ensure that the new maps are continuous on `source`/`target`, it also assumes that `e.source` and `e'.source` meet `frontier s` on the same set and `e x = e' x` on this intersection. -/ @[simps! -fullyApplied toPartialEquiv apply] def piecewise (e e' : OpenPartialHomeomorph X Y) (s : Set X) (t : Set Y) [∀ x, Decidable (x ∈ s)] [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : EqOn e e' (e.source ∩ frontier s)) : OpenPartialHomeomorph X Y where toPartialEquiv := e.toPartialEquiv.piecewise e'.toPartialEquiv s t H H' open_source := e.open_source.ite e'.open_source Hs open_target := e.open_target.ite e'.open_target <\| H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq continuousOn_toFun := continuousOn_piecewise_ite e.continuousOn e'.continuousOn Hs Heq continuousOn_invFun := continuousOn_piecewise_ite e.continuousOn_symm e'.continuousOn_symm (H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq) (H.frontier.symm_eqOn_of_inter_eq_of_eqOn Hs Heq) @[simp] theorem symm_piecewise (e e' : OpenPartialHomeomorph X Y) {s : Set X} {t : Set Y} [∀ x, Decidable (x ∈ s)] [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : EqOn e e' (e.source ∩ frontier s)) : (e.piecewise e' s t H H' Hs Heq).symm = e.symm.piecewise e'.symm t s H.symm H'.symm (H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq) (H.frontier.symm_eqOn_of_inter_eq_of_eqOn Hs Heq) := rfl /-- Combine two `OpenPartialHomeomorph`s with disjoint sources and disjoint targets. We reuse `OpenPartialHomeomorph.piecewise` then override `toPartialEquiv` to `PartialEquiv.disjointUnion`. This way we have better definitional equalities for `source` and `target`. -/ def disjointUnion (e e' : OpenPartialHomeomorph X Y) [∀ x, Decidable (x ∈ e.source)] [∀ y, Decidable (y ∈ e.target)] (Hs : Disjoint e.source e'.source) (Ht : Disjoint e.target e'.target) : OpenPartialHomeomorph X Y := (e.piecewise e' e.source e.target e.isImage_source_target (e'.isImage_source_target_of_disjoint e Hs.symm Ht.symm) (by rw [e.open_source.inter_frontier_eq, (Hs.symm.frontier_right e'.open_source).inter_eq]) (by rw [e.open_source.inter_frontier_eq] exact eqOn_empty _ _)).replaceEquiv (e.toPartialEquiv.disjointUnion e'.toPartialEquiv Hs Ht) (PartialEquiv.disjointUnion_eq_piecewise _ _ _ _).symm end Piecewise section Continuity /-- Continuity within a set at a point can be read under right composition with a local homeomorphism, if the point is in its target -/ theorem continuousWithinAt_iff_continuousWithinAt_comp_right {f : Y → Z} {s : Set Y} {x : Y} (h : x ∈ e.target) : ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ e) (e ⁻¹' s) (e.symm x) := by simp_rw [ContinuousWithinAt, ← @tendsto_map'_iff _ _ _ _ e, e.map_nhdsWithin_preimage_eq (e.map_target h), (· ∘ ·), e.right_inv h] /-- Continuity at a point can be read under right composition with an open partial homeomorphism, if the point is in its target -/ theorem continuousAt_iff_continuousAt_comp_right {f : Y → Z} {x : Y} (h : x ∈ e.target) : ContinuousAt f x ↔ ContinuousAt (f ∘ e) (e.symm x) := by rw [← continuousWithinAt_univ, e.continuousWithinAt_iff_continuousWithinAt_comp_right h, preimage_univ, continuousWithinAt_univ] /-- A function is continuous on a set if and only if its composition with an open partial homeomorphism on the right is continuous on the corresponding set. -/ theorem continuousOn_iff_continuousOn_comp_right {f : Y → Z} {s : Set Y} (h : s ⊆ e.target) : ContinuousOn f s ↔ ContinuousOn (f ∘ e) (e.source ∩ e ⁻¹' s) := by simp only [← e.symm_image_eq_source_inter_preimage h, ContinuousOn, forall_mem_image] refine forall₂_congr fun x hx => ?_ rw [e.continuousWithinAt_iff_continuousWithinAt_comp_right (h hx), e.symm_image_eq_source_inter_preimage h, inter_comm, continuousWithinAt_inter] exact IsOpen.mem_nhds e.open_source (e.map_target (h hx)) /-- Continuity within a set at a point can be read under left composition with a local homeomorphism if a neighborhood of the initial point is sent to the source of the local homeomorphism -/ theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : Set Z} {x : Z} (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) : ContinuousWithinAt f s x ↔ ContinuousWithinAt (e ∘ f) s x := by refine ⟨(e.continuousAt hx).comp_continuousWithinAt, fun fe_cont => ?_⟩ rw [← continuousWithinAt_inter' h] at fe_cont ⊢ have : ContinuousWithinAt (e.symm ∘ e ∘ f) (s ∩ f ⁻¹' e.source) x := haveI : ContinuousWithinAt e.symm univ (e (f x)) := (e.continuousAt_symm (e.map_source hx)).continuousWithinAt ContinuousWithinAt.comp this fe_cont (subset_univ _) exact this.congr (fun y hy => by simp [e.left_inv hy.2]) (by simp [e.left_inv hx]) /-- Continuity at a point can be read under left composition with an open partial homeomorphism if a neighborhood of the initial point is sent to the source of the partial homeomorphism -/ theorem continuousAt_iff_continuousAt_comp_left {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x) : ContinuousAt f x ↔ ContinuousAt (e ∘ f) x := by have hx : f x ∈ e.source := (mem_of_mem_nhds h :) have h' : f ⁻¹' e.source ∈ 𝓝[univ] x := by rwa [nhdsWithin_univ] rw [← continuousWithinAt_univ, ← continuousWithinAt_univ, e.continuousWithinAt_iff_continuousWithinAt_comp_left hx h'] /-- A function is continuous on a set if and only if its composition with an open partial homeomorphism on the left is continuous on the corresponding set. -/ theorem continuousOn_iff_continuousOn_comp_left {f : Z → X} {s : Set Z} (h : s ⊆ f ⁻¹' e.source) : ContinuousOn f s ↔ ContinuousOn (e ∘ f) s := forall₂_congr fun _x hx => e.continuousWithinAt_iff_continuousWithinAt_comp_left (h hx) (mem_of_superset self_mem_nhdsWithin h) /-- A function is continuous if and only if its composition with an open partial homeomorphism on the left is continuous and its image is contained in the source. -/ theorem continuous_iff_continuous_comp_left {f : Z → X} (h : f ⁻¹' e.source = univ) : Continuous f ↔ Continuous (e ∘ f) := by simp only [← continuousOn_univ] exact e.continuousOn_iff_continuousOn_comp_left (Eq.symm h).subset end Continuity /-- The homeomorphism obtained by restricting an `OpenPartialHomeomorph` to a subset of the source. -/ @[simps] def homeomorphOfImageSubsetSource {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : e '' s = t) : s ≃ₜ t := have h₁ : MapsTo e s t := mapsTo_iff_image_subset.2 ht.subset have h₂ : t ⊆ e.target := ht ▸ e.image_source_eq_target ▸ image_mono hs have h₃ : MapsTo e.symm t s := ht ▸ forall_mem_image.2 fun _x hx => (e.left_inv (hs hx)).symm ▸ hx { toFun := MapsTo.restrict e s t h₁ invFun := MapsTo.restrict e.symm t s h₃ left_inv := fun a => Subtype.ext (e.left_inv (hs a.2)) right_inv := fun b => Subtype.eq <\| e.right_inv (h₂ b.2) continuous_toFun := (e.continuousOn.mono hs).mapsToRestrict h₁ continuous_invFun := (e.continuousOn_symm.mono h₂).mapsToRestrict h₃ } /-- An open partial homeomorphism defines a homeomorphism between its source and target. -/ @[simps!] def toHomeomorphSourceTarget : e.source ≃ₜ e.target := e.homeomorphOfImageSubsetSource subset_rfl e.image_source_eq_target theorem secondCountableTopology_source [SecondCountableTopology Y] : SecondCountableTopology e.source := e.toHomeomorphSourceTarget.secondCountableTopology theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) : 𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source := by lift x to e.source using hx rw [← e.open_source.nhdsWithin_eq x.2, ← map_nhds_subtype_val, ← map_comap_setCoe_val, e.toHomeomorphSourceTarget.nhds_eq_comap, nhds_subtype_eq_comap] simp only [Function.comp_def, toHomeomorphSourceTarget_apply_coe, comap_comap] /-- If an open partial homeomorphism has source and target equal to univ, then it induces a homeomorphism between the whole spaces, expressed in this definition. -/ @[simps (attr := mfld_simps) -fullyApplied apply symm_apply] -- TODO: add a `PartialEquiv` version def toHomeomorphOfSourceEqUnivTargetEqUniv (h : e.source = (univ : Set X)) (h' : e.target = univ) : X ≃ₜ Y where toFun := e invFun := e.symm left_inv x := e.left_inv <\| by rw [h] exact mem_univ _ right_inv x := e.right_inv <\| by rw [h'] exact mem_univ _ continuous_toFun := by simpa only [continuousOn_univ, h] using e.continuousOn continuous_invFun := by simpa only [continuousOn_univ, h'] using e.continuousOn_symm theorem isOpenEmbedding_restrict : IsOpenEmbedding (e.source.restrict e) := by refine .of_continuous_injective_isOpenMap (e.continuousOn.comp_continuous continuous_subtype_val Subtype.prop) e.injOn.injective fun V hV ↦ ?_ rw [Set.restrict_eq, Set.image_comp] exact e.isOpen_image_of_subset_source (e.open_source.isOpenMap_subtype_val V hV) fun _ ⟨x, _, h⟩ ↦ h ▸ x.2 /-- An open partial homeomorphism whose source is all of `X` defines an open embedding of `X` into `Y`. The converse is also true; see `IsOpenEmbedding.toOpenPartialHomeomorph`. -/ theorem to_isOpenEmbedding (h : e.source = Set.univ) : IsOpenEmbedding e := e.isOpenEmbedding_restrict.comp ((Homeomorph.setCongr h).trans <\| Homeomorph.Set.univ X).symm.isOpenEmbedding end OpenPartialHomeomorph namespace Homeomorph variable (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) /- Register as simp lemmas that the fields of an open partial homeomorphism built from a homeomorphism correspond to the fields of the original homeomorphism. -/ @[simp, mfld_simps] theorem refl_toOpenPartialHomeomorph : (Homeomorph.refl X).toOpenPartialHomeomorph = OpenPartialHomeomorph.refl X := rfl @[deprecated (since := "2025-08-29")] alias refl_toPartialHomeomorph := refl_toOpenPartialHomeomorph @[simp, mfld_simps] theorem symm_toOpenPartialHomeomorph : e.symm.toOpenPartialHomeomorph = e.toOpenPartialHomeomorph.symm := rfl @[deprecated (since := "2025-08-29")] alias symm_toPartialHomeomorph := symm_toOpenPartialHomeomorph @[simp, mfld_simps] theorem trans_toOpenPartialHomeomorph : (e.trans e').toOpenPartialHomeomorph = e.toOpenPartialHomeomorph.trans e'.toOpenPartialHomeomorph := OpenPartialHomeomorph.toPartialEquiv_injective <\| Equiv.trans_toPartialEquiv _ _ @[deprecated (since := "2025-08-29")] alias trans_toPartialHomeomorph := trans_toOpenPartialHomeomorph /-- Precompose an open partial homeomorphism with a homeomorphism. We modify the source and target to have better definitional behavior. -/ @[simps! -fullyApplied] def transOpenPartialHomeomorph (e : X ≃ₜ Y) (f' : OpenPartialHomeomorph Y Z) : OpenPartialHomeomorph X Z where toPartialEquiv := e.toEquiv.transPartialEquiv f'.toPartialEquiv open_source := f'.open_source.preimage e.continuous open_target := f'.open_target continuousOn_toFun := f'.continuousOn.comp e.continuous.continuousOn fun _ => id continuousOn_invFun := e.symm.continuous.comp_continuousOn f'.symm.continuousOn @[deprecated (since := "2025-08-29")] alias transPartialHomeomorph := transOpenPartialHomeomorph theorem transOpenPartialHomeomorph_eq_trans (e : X ≃ₜ Y) (f' : OpenPartialHomeomorph Y Z) : e.transOpenPartialHomeomorph f' = e.toOpenPartialHomeomorph.trans f' := OpenPartialHomeomorph.toPartialEquiv_injective <\| Equiv.transPartialEquiv_eq_trans _ _ @[deprecated (since := "2025-08-29")] alias transPartialHomeomorph_eq_trans := transOpenPartialHomeomorph_eq_trans @[simp, mfld_simps] theorem transOpenPartialHomeomorph_trans (e : X ≃ₜ Y) (f : OpenPartialHomeomorph Y Z) (f' : OpenPartialHomeomorph Z Z') : (e.transOpenPartialHomeomorph f).trans f' = e.transOpenPartialHomeomorph (f.trans f') := by simp only [transOpenPartialHomeomorph_eq_trans, OpenPartialHomeomorph.trans_assoc] @[deprecated (since := "2025-08-29")] alias transPartialHomeomorph_trans := transOpenPartialHomeomorph_trans @[simp, mfld_simps] theorem trans_transOpenPartialHomeomorph (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) (f'' : OpenPartialHomeomorph Z Z') : (e.trans e').transOpenPartialHomeomorph f'' = e.transOpenPartialHomeomorph (e'.transOpenPartialHomeomorph f'') := by simp only [transOpenPartialHomeomorph_eq_trans, OpenPartialHomeomorph.trans_assoc, trans_toOpenPartialHomeomorph] @[deprecated (since := "2025-08-29")] alias trans_transPartialHomeomorph := trans_transOpenPartialHomeomorph end Homeomorph namespace Topology.IsOpenEmbedding variable (f : X → Y) (h : IsOpenEmbedding f) /-- An open embedding of `X` into `Y`, with `X` nonempty, defines an open partial homeomorphism whose source is all of `X`. The converse is also true; see `OpenPartialHomeomorph.to_isOpenEmbedding`. -/ @[simps! (attr := mfld_simps) -fullyApplied apply source target] noncomputable def toOpenPartialHomeomorph [Nonempty X] : OpenPartialHomeomorph X Y := OpenPartialHomeomorph.ofContinuousOpen (h.isEmbedding.injective.injOn.toPartialEquiv f univ) h.continuous.continuousOn h.isOpenMap isOpen_univ @[deprecated (since := "2025-08-29")] alias toPartialHomeomorph := toOpenPartialHomeomorph variable [Nonempty X] lemma toOpenPartialHomeomorph_left_inv {x : X} : (h.toOpenPartialHomeomorph f).symm (f x) = x := by rw [← congr_fun (h.toOpenPartialHomeomorph_apply f), OpenPartialHomeomorph.left_inv] exact Set.mem_univ _ @[deprecated (since := "2025-08-29")] alias toPartialHomeomorph_left_inv := toOpenPartialHomeomorph_left_inv lemma toOpenPartialHomeomorph_right_inv {x : Y} (hx : x ∈ Set.range f) : f ((h.toOpenPartialHomeomorph f).symm x) = x := by rw [← congr_fun (h.toOpenPartialHomeomorph_apply f), OpenPartialHomeomorph.right_inv] rwa [toOpenPartialHomeomorph_target] @[deprecated (since := "2025-08-29")] alias toPartialHomeomorph_right_inv := toOpenPartialHomeomorph_right_inv end Topology.IsOpenEmbedding /-! inclusion of an open set in a topological space -/ namespace TopologicalSpace.Opens /- `Nonempty s` is not a type class argument because `s`, being a subset, rarely comes with a type class instance. Then we'd have to manually provide the instance every time we use the following lemmas, tediously using `haveI := ...` or `@foobar _ _ _ ...`. -/ variable (s : Opens X) (hs : Nonempty s) /-- The inclusion of an open subset `s` of a space `X` into `X` is an open partial homeomorphism from the subtype `s` to `X`. -/ noncomputable def openPartialHomeomorphSubtypeCoe : OpenPartialHomeomorph s X := IsOpenEmbedding.toOpenPartialHomeomorph _ s.2.isOpenEmbedding_subtypeVal @[deprecated (since := "2025-08-29")] alias partialHomeomorphSubtypeCoe := openPartialHomeomorphSubtypeCoe @[simp, mfld_simps] theorem openPartialHomeomorphSubtypeCoe_coe : (s.openPartialHomeomorphSubtypeCoe hs : s → X) = (↑) := rfl @[deprecated (since := "2025-08-29")] alias partialHomeomorphSubtypeCoe_coe := openPartialHomeomorphSubtypeCoe_coe @[simp, mfld_simps] theorem openPartialHomeomorphSubtypeCoe_source : (s.openPartialHomeomorphSubtypeCoe hs).source = Set.univ := rfl @[deprecated (since := "2025-08-29")] alias partialHomeomorphSubtypeCoe_source := openPartialHomeomorphSubtypeCoe_source @[simp, mfld_simps] theorem openPartialHomeomorphSubtypeCoe_target : (s.openPartialHomeomorphSubtypeCoe hs).target = s := by simp only [openPartialHomeomorphSubtypeCoe, Subtype.range_coe_subtype, mfld_simps] rfl @[deprecated (since := "2025-08-29")] alias partialHomeomorphSubtypeCoe_target := openPartialHomeomorphSubtypeCoe_target end TopologicalSpace.Opens namespace OpenPartialHomeomorph /- post-compose with an open partial homeomorphism -/ section transHomeomorph /-- Postcompose an open partial homeomorphism with a homeomorphism. We modify the source and target to have better definitional behavior. -/ @[simps! -fullyApplied] def transHomeomorph (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) : OpenPartialHomeomorph X Z where toPartialEquiv := e.toPartialEquiv.transEquiv f'.toEquiv open_source := e.open_source open_target := e.open_target.preimage f'.symm.continuous continuousOn_toFun := f'.continuous.comp_continuousOn e.continuousOn continuousOn_invFun := e.symm.continuousOn.comp f'.symm.continuous.continuousOn fun _ => id theorem transHomeomorph_eq_trans (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) : e.transHomeomorph f' = e.trans f'.toOpenPartialHomeomorph := toPartialEquiv_injective <\| PartialEquiv.transEquiv_eq_trans _ _ @[simp, mfld_simps] theorem transHomeomorph_transHomeomorph (e : OpenPartialHomeomorph X Y) (f' : Y ≃ₜ Z) (f'' : Z ≃ₜ Z') : (e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'') := by simp only [transHomeomorph_eq_trans, trans_assoc, Homeomorph.trans_toOpenPartialHomeomorph] @[simp, mfld_simps] theorem trans_transHomeomorph (e : OpenPartialHomeomorph X Y) (e' : OpenPartialHomeomorph Y Z) (f'' : Z ≃ₜ Z') : (e.trans e').transHomeomorph f'' = e.trans (e'.transHomeomorph f'') := by simp only [transHomeomorph_eq_trans, trans_assoc] end transHomeomorph /-! `subtypeRestr`: restriction to a subtype -/ section subtypeRestr open TopologicalSpace variable (e : OpenPartialHomeomorph X Y) variable {s : Opens X} (hs : Nonempty s) /-- The restriction of an open partial homeomorphism `e` to an open subset `s` of the domain type produces an open partial homeomorphism whose domain is the subtype `s`. -/ noncomputable def subtypeRestr : OpenPartialHomeomorph s Y := (s.openPartialHomeomorphSubtypeCoe hs).trans e theorem subtypeRestr_def : e.subtypeRestr hs = (s.openPartialHomeomorphSubtypeCoe hs).trans e := rfl @[simp, mfld_simps] theorem subtypeRestr_coe : ((e.subtypeRestr hs : OpenPartialHomeomorph s Y) : s → Y) = Set.restrict ↑s (e : X → Y) := rfl @[simp, mfld_simps] theorem subtypeRestr_source : (e.subtypeRestr hs).source = (↑) ⁻¹' e.source := by simp only [subtypeRestr_def, mfld_simps] theorem map_subtype_source {x : s} (hxe : (x : X) ∈ e.source) : e x ∈ (e.subtypeRestr hs).target := by refine ⟨e.map_source hxe, ?_⟩ rw [s.openPartialHomeomorphSubtypeCoe_target, mem_preimage, e.leftInvOn hxe] exact x.prop /-- This lemma characterizes the transition functions of an open subset in terms of the transition functions of the original space. -/ theorem subtypeRestr_symm_trans_subtypeRestr (f f' : OpenPartialHomeomorph X Y) : (f.subtypeRestr hs).symm.trans (f'.subtypeRestr hs) ≈ (f.symm.trans f').restr (f.target ∩ f.symm ⁻¹' s) := by simp only [subtypeRestr_def, trans_symm_eq_symm_trans_symm] have openness₁ : IsOpen (f.target ∩ f.symm ⁻¹' s) := f.isOpen_inter_preimage_symm s.2 rw [← ofSet_trans _ openness₁, ← trans_assoc, ← trans_assoc] refine EqOnSource.trans' ?_ (eqOnSource_refl _) -- f' has been eliminated !!! have set_identity : f.symm.source ∩ (f.target ∩ f.symm ⁻¹' s) = f.symm.source ∩ f.symm ⁻¹' s := by mfld_set_tac have openness₂ : IsOpen (s : Set X) := s.2 rw [ofSet_trans', set_identity, ← trans_of_set' _ openness₂, trans_assoc] refine EqOnSource.trans' (eqOnSource_refl _) ?_ -- f has been eliminated !!! refine Setoid.trans (symm_trans_self (s.openPartialHomeomorphSubtypeCoe hs)) ?_ simp only [mfld_simps, Setoid.refl] theorem subtypeRestr_symm_eqOn {U : Opens X} (hU : Nonempty U) : EqOn e.symm (Subtype.val ∘ (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target := by intro y hy rw [eq_comm, eq_symm_apply _ _ hy.1] · change restrict _ e _ = _ rw [← subtypeRestr_coe _ hU, (e.subtypeRestr hU).right_inv hy] · have := map_target _ hy; rwa [subtypeRestr_source] at this theorem subtypeRestr_symm_eqOn_of_le {U V : Opens X} (hU : Nonempty U) (hV : Nonempty V) (hUV : U ≤ V) : EqOn (e.subtypeRestr hV).symm (Set.inclusion hUV ∘ (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target := by set i := Set.inclusion hUV intro y hy dsimp [OpenPartialHomeomorph.subtypeRestr_def] at hy ⊢ have hyV : e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target := by rw [Opens.openPartialHomeomorphSubtypeCoe_target] at hy ⊢ exact hUV hy.2 refine (V.openPartialHomeomorphSubtypeCoe hV).injOn ?_ trivial ?_ · rw [← OpenPartialHomeomorph.symm_target] apply OpenPartialHomeomorph.map_source rw [OpenPartialHomeomorph.symm_source] exact hyV · rw [(V.openPartialHomeomorphSubtypeCoe hV).right_inv hyV] change _ = U.openPartialHomeomorphSubtypeCoe hU _ rw [(U.openPartialHomeomorphSubtypeCoe hU).right_inv hy.2] end subtypeRestr variable {X X' Z : Type} [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Z] [Nonempty Z] {f : X → X'} /-- Extend an open partial homeomorphism `e : X → Z` to `X' → Z`, using an open embedding `ι : X → X'`. On `ι(X)`, the extension is specified by `e`; its value elsewhere is arbitrary (and uninteresting). -/ noncomputable def lift_openEmbedding (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : OpenPartialHomeomorph X' Z where toFun := extend f e (fun _ ↦ (Classical.arbitrary Z)) invFun := f ∘ e.invFun source := f '' e.source target := e.target map_source' := by rintro x ⟨x₀, hx₀, hxx₀⟩ rw [← hxx₀, hf.injective.extend_apply e] exact e.map_source' hx₀ map_target' z hz := mem_image_of_mem f (e.map_target' hz) left_inv' := by intro x ⟨x₀, hx₀, hxx₀⟩ rw [← hxx₀, hf.injective.extend_apply e, comp_apply] congr exact e.left_inv' hx₀ right_inv' z hz := by simpa only [comp_apply, hf.injective.extend_apply e] using e.right_inv' hz open_source := hf.isOpenMap _ e.open_source open_target := e.open_target continuousOn_toFun := by by_cases Nonempty X; swap · intro x hx; simp_all set F := (extend f e (fun _ ↦ (Classical.arbitrary Z))) with F_eq have heq : EqOn F (e ∘ (hf.toOpenPartialHomeomorph).symm) (f '' e.source) := by intro x ⟨x₀, hx₀, hxx₀⟩ rw [← hxx₀, F_eq, hf.injective.extend_apply e, comp_apply, hf.toOpenPartialHomeomorph_left_inv] have : ContinuousOn (e ∘ (hf.toOpenPartialHomeomorph).symm) (f '' e.source) := by apply e.continuousOn_toFun.comp; swap · intro x' ⟨x, hx, hx'x⟩ rw [← hx'x, hf.toOpenPartialHomeomorph_left_inv]; exact hx have : ContinuousOn (hf.toOpenPartialHomeomorph).symm (f '' univ) := (hf.toOpenPartialHomeomorph).continuousOn_invFun exact this.mono <\| image_mono <\| subset_univ _ exact ContinuousOn.congr this heq continuousOn_invFun := hf.continuous.comp_continuousOn e.continuousOn_invFun @[simp, mfld_simps] lemma lift_openEmbedding_toFun (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf) = extend f e (fun _ ↦ (Classical.arbitrary Z)) := rfl lemma lift_openEmbedding_apply (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) {x : X} : (lift_openEmbedding e hf) (f x) = e x := by simp_rw [e.lift_openEmbedding_toFun] apply hf.injective.extend_apply @[simp, mfld_simps] lemma lift_openEmbedding_source (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf).source = f '' e.source := rfl @[simp, mfld_simps] lemma lift_openEmbedding_target (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf).target = e.target := rfl @[simp, mfld_simps] lemma lift_openEmbedding_symm (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf).symm = f ∘ e.symm := rfl @[simp, mfld_simps] lemma lift_openEmbedding_symm_source (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf).symm.source = e.target := rfl @[simp, mfld_simps] lemma lift_openEmbedding_symm_target (e : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf).symm.target = f '' e.source := by rw [OpenPartialHomeomorph.symm_target, e.lift_openEmbedding_source] lemma lift_openEmbedding_trans_apply (e e' : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) (z : Z) : (e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) z = (e.symm.trans e') z := by simp [hf.injective.extend_apply e'] @[simp, mfld_simps] lemma lift_openEmbedding_trans (e e' : OpenPartialHomeomorph X Z) (hf : IsOpenEmbedding f) : (e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) = e.symm.trans e' := by ext z · exact e.lift_openEmbedding_trans_apply e' hf z · simp [hf.injective.extend_apply e] · simp_rw [OpenPartialHomeomorph.trans_source, e.lift_openEmbedding_symm_source, e.symm_source, e.lift_openEmbedding_symm, e'.lift_openEmbedding_source] refine ⟨fun ⟨hx, ⟨y, hy, hxy⟩⟩ ↦ ⟨hx, ?_⟩, fun ⟨hx, hx'⟩ ↦ ⟨hx, mem_image_of_mem f hx'⟩⟩ rw [mem_preimage]; rw [comp_apply] at hxy exact (hf.injective hxy) ▸ hy end OpenPartialHomeomorph set_option linter.style.longFile 1700
.lake/packages/mathlib/Mathlib/Topology/QuasiSeparated.lean	import Mathlib.Topology.Compactness.Bases import Mathlib.Topology.NoetherianSpace /-! # Quasi-separated spaces A topological space is quasi-separated if the intersections of any pairs of compact open subsets are still compact. Notable examples include spectral spaces, Noetherian spaces, and Hausdorff spaces. A non-example is the interval `[0, 1]` with doubled origin: the two copies of `[0, 1]` are compact open subsets, but their intersection `(0, 1]` is not. ## Main results - `IsQuasiSeparated`: A subset `s` of a topological space is quasi-separated if the intersections of any pairs of compact open subsets of `s` are still compact. - `QuasiSeparatedSpace`: A topological space is quasi-separated if the intersections of any pairs of compact open subsets are still compact. - `QuasiSeparatedSpace.of_isOpenEmbedding`: If `f : α → β` is an open embedding, and `β` is a quasi-separated space, then so is `α`. -/ open Set TopologicalSpace Topology variable {α β : Type} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} /-- A subset `s` of a topological space is quasi-separated if the intersections of any pairs of compact open subsets of `s` are still compact. Note that this is equivalent to `s` being a `QuasiSeparatedSpace` only when `s` is open. -/ def IsQuasiSeparated (s : Set α) : Prop := ∀ U V : Set α, U ⊆ s → IsOpen U → IsCompact U → V ⊆ s → IsOpen V → IsCompact V → IsCompact (U ∩ V) /-- A topological space is quasi-separated if the intersections of any pairs of compact open subsets are still compact. -/ @[mk_iff] class QuasiSeparatedSpace (α : Type) [TopologicalSpace α] : Prop where /-- The intersection of two open compact subsets of a quasi-separated space is compact. -/ inter_isCompact : ∀ U V : Set α, IsOpen U → IsCompact U → IsOpen V → IsCompact V → IsCompact (U ∩ V) theorem isQuasiSeparated_univ_iff {α : Type} [TopologicalSpace α] : IsQuasiSeparated (Set.univ : Set α) ↔ QuasiSeparatedSpace α := by rw [quasiSeparatedSpace_iff] simp [IsQuasiSeparated] theorem isQuasiSeparated_univ {α : Type} [TopologicalSpace α] [QuasiSeparatedSpace α] : IsQuasiSeparated (Set.univ : Set α) := isQuasiSeparated_univ_iff.mpr inferInstance theorem IsQuasiSeparated.image_of_isEmbedding {s : Set α} (H : IsQuasiSeparated s) (h : IsEmbedding f) : IsQuasiSeparated (f '' s) := by intro U V hU hU' hU'' hV hV' hV'' convert (H (f ⁻¹' U) (f ⁻¹' V) ?_ (h.continuous.1 _ hU') ?_ ?_ (h.continuous.1 _ hV') ?_).image h.continuous · symm rw [← Set.preimage_inter, Set.image_preimage_eq_inter_range, Set.inter_eq_left] exact Set.inter_subset_left.trans (hU.trans (Set.image_subset_range _ _)) · intro x hx rw [← h.injective.injOn.mem_image_iff (Set.subset_univ _) trivial] exact hU hx · rw [h.isCompact_iff] convert hU'' rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left] exact hU.trans (Set.image_subset_range _ _) · intro x hx rw [← h.injective.injOn.mem_image_iff (Set.subset_univ _) trivial] exact hV hx · rw [h.isCompact_iff] convert hV'' rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left] exact hV.trans (Set.image_subset_range _ _) theorem Topology.IsOpenEmbedding.isQuasiSeparated_iff (h : IsOpenEmbedding f) {s : Set α} : IsQuasiSeparated s ↔ IsQuasiSeparated (f '' s) := by refine ⟨fun hs => hs.image_of_isEmbedding h.isEmbedding, ?_⟩ intro H U V hU hU' hU'' hV hV' hV'' rw [h.isEmbedding.isCompact_iff, Set.image_inter h.injective] exact H (f '' U) (f '' V) (image_mono hU) (h.isOpenMap _ hU') (hU''.image h.continuous) (image_mono hV) (h.isOpenMap _ hV') (hV''.image h.continuous) theorem isQuasiSeparated_iff_quasiSeparatedSpace (s : Set α) (hs : IsOpen s) : IsQuasiSeparated s ↔ QuasiSeparatedSpace s := by rw [← isQuasiSeparated_univ_iff] convert (hs.isOpenEmbedding_subtypeVal.isQuasiSeparated_iff (s := Set.univ)).symm simp theorem IsQuasiSeparated.of_subset {s t : Set α} (ht : IsQuasiSeparated t) (h : s ⊆ t) : IsQuasiSeparated s := by intro U V hU hU' hU'' hV hV' hV'' exact ht U V (hU.trans h) hU' hU'' (hV.trans h) hV' hV'' instance (priority := 100) T2Space.to_quasiSeparatedSpace [T2Space α] : QuasiSeparatedSpace α := ⟨fun _ _ _ hU' _ hV' => hU'.inter hV'⟩ instance (priority := 100) NoetherianSpace.to_quasiSeparatedSpace [NoetherianSpace α] : QuasiSeparatedSpace α := ⟨fun _ _ _ _ _ _ => NoetherianSpace.isCompact _⟩ lemma QuasiSeparatedSpace.of_isTopologicalBasis {ι : Type*} {b : ι → Set α} (basis : IsTopologicalBasis (range b)) (isCompact_inter : ∀ i j, IsCompact (b i ∩ b j)) : QuasiSeparatedSpace α where inter_isCompact U V hUopen hUcomp hVopen hVcomp := by have aux := isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis b basis fun i ↦ by simpa using isCompact_inter i i obtain ⟨s, hs, rfl⟩ := (aux _).1 ⟨hUcomp, hUopen⟩ obtain ⟨t, ht, rfl⟩ := (aux _).1 ⟨hVcomp, hVopen⟩ rw [iUnion₂_inter_iUnion₂] exact hs.isCompact_biUnion fun i hi ↦ ht.isCompact_biUnion fun j hj ↦ isCompact_inter .. section QuasiSeparatedSpace variable [QuasiSeparatedSpace α] {U V : Set α} lemma IsQuasiSeparated.of_quasiSeparatedSpace (s : Set α) : IsQuasiSeparated s := isQuasiSeparated_univ.of_subset (Set.subset_univ _) lemma QuasiSeparatedSpace.of_isOpenEmbedding {f : β → α} (h : IsOpenEmbedding f) : QuasiSeparatedSpace β := isQuasiSeparated_univ_iff.mp (h.isQuasiSeparated_iff.mpr <\| .of_quasiSeparatedSpace _) lemma IsCompact.inter_of_isOpen (hUcomp : IsCompact U) (hVcomp : IsCompact V) (hUopen : IsOpen U) (hVopen : IsOpen V) : IsCompact (U ∩ V) := QuasiSeparatedSpace.inter_isCompact _ _ hUopen hUcomp hVopen hVcomp end QuasiSeparatedSpace lemma quasiSeparatedSpace_congr (e : α ≃ₜ β) : QuasiSeparatedSpace α ↔ QuasiSeparatedSpace β where mp _ := .of_isOpenEmbedding e.symm.isOpenEmbedding mpr _ := .of_isOpenEmbedding e.isOpenEmbedding
.lake/packages/mathlib/Mathlib/Topology/LocalAtTarget.lean	import Mathlib.Topology.Homeomorph.Lemmas import Mathlib.Topology.Sets.OpenCover import Mathlib.Topology.LocallyClosed import Mathlib.Topology.Maps.Proper.Basic /-! # Properties of maps that are local at the target or at the source. We show that the following properties of continuous maps are local at the target : - `IsInducing` - `IsOpenMap` - `IsClosedMap` - `IsEmbedding` - `IsOpenEmbedding` - `IsClosedEmbedding` - `GeneralizingMap` We show that the following properties of continuous maps are local at the source: - `IsOpenMap` - `GeneralizingMap` -/ open Filter Set TopologicalSpace Topology variable {α β : Type} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} variable {ι : Type} {U : ι → Opens β} theorem Set.restrictPreimage_isInducing (s : Set β) (h : IsInducing f) : IsInducing (s.restrictPreimage f) := by simp_rw [← IsInducing.subtypeVal.of_comp_iff, isInducing_iff_nhds, restrictPreimage, MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h ⊢ intro a rw [← h, ← IsInducing.subtypeVal.nhds_eq_comap] alias Topology.IsInducing.restrictPreimage := Set.restrictPreimage_isInducing theorem Set.restrictPreimage_isEmbedding (s : Set β) (h : IsEmbedding f) : IsEmbedding (s.restrictPreimage f) := ⟨h.1.restrictPreimage s, h.2.restrictPreimage s⟩ alias Topology.IsEmbedding.restrictPreimage := Set.restrictPreimage_isEmbedding theorem Set.restrictPreimage_isOpenEmbedding (s : Set β) (h : IsOpenEmbedding f) : IsOpenEmbedding (s.restrictPreimage f) := ⟨h.1.restrictPreimage s, (s.range_restrictPreimage f).symm ▸ continuous_subtype_val.isOpen_preimage _ h.isOpen_range⟩ alias Topology.IsOpenEmbedding.restrictPreimage := Set.restrictPreimage_isOpenEmbedding theorem Set.restrictPreimage_isClosedEmbedding (s : Set β) (h : IsClosedEmbedding f) : IsClosedEmbedding (s.restrictPreimage f) := ⟨h.1.restrictPreimage s, (s.range_restrictPreimage f).symm ▸ IsInducing.subtypeVal.isClosed_preimage _ h.isClosed_range⟩ alias Topology.IsClosedEmbedding.restrictPreimage := Set.restrictPreimage_isClosedEmbedding theorem IsClosedMap.restrictPreimage (H : IsClosedMap f) (s : Set β) : IsClosedMap (s.restrictPreimage f) := by intro t suffices ∀ u, IsClosed u → Subtype.val ⁻¹' u = t → ∃ v, IsClosed v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by simpa [isClosed_induced_iff] exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩ theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) : IsOpenMap (s.restrictPreimage f) := by intro t suffices ∀ u, IsOpen u → Subtype.val ⁻¹' u = t → ∃ v, IsOpen v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by simpa [isOpen_induced_iff] exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩ lemma GeneralizingMap.restrictPreimage (H : GeneralizingMap f) (s : Set β) : GeneralizingMap (s.restrictPreimage f) := by intro x y h obtain ⟨a, ha, hy⟩ := H (h.map <\| continuous_subtype_val (p := s)) use ⟨a, by simp [hy]⟩ simp [hy, subtype_specializes_iff, ha] lemma IsProperMap.restrictPreimage (H : IsProperMap f) (s : Set β) : IsProperMap (s.restrictPreimage f) := by rw [isProperMap_iff_isClosedMap_and_compact_fibers] refine ⟨H.continuous.restrictPreimage, H.isClosedMap.restrictPreimage _, fun y ↦ ?_⟩ rw [IsEmbedding.subtypeVal.isCompact_iff, image_val_preimage_restrictPreimage, image_singleton] exact H.isCompact_preimage isCompact_singleton namespace TopologicalSpace.IsOpenCover section LocalAtTarget variable {U : ι → Opens β} {s : Set β} (hU : IsOpenCover U) include hU theorem isOpen_iff_inter : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) := by constructor · exact fun H i ↦ H.inter (U i).isOpen · intro H simpa [← inter_iUnion, hU.iSup_set_eq_univ] using isOpen_iUnion H theorem isOpen_iff_coe_preimage : IsOpen s ↔ ∀ i, IsOpen ((↑) ⁻¹' s : Set (U i)) := by simp [hU.isOpen_iff_inter (s := s), (U _).2.isOpenEmbedding_subtypeVal.isOpen_iff_image_isOpen, image_preimage_eq_inter_range] theorem isClosed_iff_coe_preimage {s : Set β} : IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) := by simpa using hU.isOpen_iff_coe_preimage (s := sᶜ) theorem isLocallyClosed_iff_coe_preimage {s : Set β} : IsLocallyClosed s ↔ ∀ i, IsLocallyClosed ((↑) ⁻¹' s : Set (U i)) := by have (i : _) : coborder ((↑) ⁻¹' s : Set (U i)) = Subtype.val ⁻¹' coborder s := (U i).isOpen.isOpenEmbedding_subtypeVal.coborder_preimage _ simp [isLocallyClosed_iff_isOpen_coborder, hU.isOpen_iff_coe_preimage, this] theorem isOpenMap_iff_restrictPreimage : IsOpenMap f ↔ ∀ i, IsOpenMap ((U i).1.restrictPreimage f) := by refine ⟨fun h i ↦ h.restrictPreimage _, fun H s hs ↦ ?_⟩ rw [hU.isOpen_iff_coe_preimage] intro i convert H i _ (hs.preimage continuous_subtype_val) ext ⟨x, hx⟩ suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by simpa [← Subtype.coe_inj] exact ⟨fun ⟨a, b, c⟩ ↦ ⟨a, b, c.symm ▸ hx, c⟩, by tauto⟩ theorem isClosedMap_iff_restrictPreimage : IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) := by refine ⟨fun h i => h.restrictPreimage _, fun H s hs ↦ ?_⟩ rw [hU.isClosed_iff_coe_preimage] intro i convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩ ext ⟨x, hx⟩ suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by simpa [← Subtype.coe_inj] exact ⟨fun ⟨a, b, c⟩ => ⟨a, b, c.symm ▸ hx, c⟩, by tauto⟩ theorem isInducing_iff_restrictPreimage (h : Continuous f) : IsInducing f ↔ ∀ i, IsInducing ((U i).1.restrictPreimage f) := by simp_rw [← IsInducing.subtypeVal.of_comp_iff, isInducing_iff_nhds, restrictPreimage, MapsTo.coe_restrict, restrict_eq, ← Filter.comap_comap] constructor · intro H i x rw [Function.comp_apply, ← H, ← IsInducing.subtypeVal.nhds_eq_comap] · intro H x obtain ⟨i, hi⟩ := Opens.mem_iSup.mp (show f x ∈ iSup U by simp [hU.iSup_eq_top]) simpa [← ((h.1 _ (U i).2).isOpenEmbedding_subtypeVal).map_nhds_eq ⟨x, hi⟩, H i ⟨x, hi⟩, subtype_coe_map_comap] using preimage_mem_comap ((U i).2.mem_nhds hi) theorem isEmbedding_iff_restrictPreimage (h : Continuous f) : IsEmbedding f ↔ ∀ i, IsEmbedding ((U i).1.restrictPreimage f) := by simpa [isEmbedding_iff, forall_and] using and_congr (hU.isInducing_iff_restrictPreimage h) (injective_iff_injective_of_iUnion_eq_univ hU.iSup_set_eq_univ) theorem isOpenEmbedding_iff_restrictPreimage (h : Continuous f) : IsOpenEmbedding f ↔ ∀ i, IsOpenEmbedding ((U i).1.restrictPreimage f) := by simp_rw [isOpenEmbedding_iff, forall_and] apply and_congr · exact hU.isEmbedding_iff_restrictPreimage h · simp_rw [range_restrictPreimage] exact hU.isOpen_iff_coe_preimage theorem isClosedEmbedding_iff_restrictPreimage (h : Continuous f) : IsClosedEmbedding f ↔ ∀ i, IsClosedEmbedding ((U i).1.restrictPreimage f) := by simp_rw [isClosedEmbedding_iff, forall_and] apply and_congr · exact hU.isEmbedding_iff_restrictPreimage h · simp_rw [range_restrictPreimage] exact hU.isClosed_iff_coe_preimage omit [TopologicalSpace α] in theorem denseRange_iff_restrictPreimage : DenseRange f ↔ ∀ i, DenseRange ((U i).1.restrictPreimage f) := by simp_rw [denseRange_iff_closure_range, Set.range_restrictPreimage, ← (U _).2.isOpenEmbedding_subtypeVal.isOpenMap.preimage_closure_eq_closure_preimage continuous_subtype_val] simp only [Opens.carrier_eq_coe, SetLike.coe_sort_coe, preimage_eq_univ_iff, Subtype.range_coe_subtype, SetLike.mem_coe] rw [← iUnion_subset_iff, ← Set.univ_subset_iff, iff_iff_eq] congr 1 exact hU.iSup_set_eq_univ.symm lemma generalizingMap_iff_restrictPreimage : GeneralizingMap f ↔ ∀ i, GeneralizingMap ((U i).1.restrictPreimage f) := by refine ⟨fun hf ↦ fun i ↦ hf.restrictPreimage _, fun hf ↦ fun x y h ↦ ?_⟩ obtain ⟨i, hx⟩ := hU.exists_mem (f x) have h : (⟨y, (U i).2.stableUnderGeneralization h hx⟩ : U i) ⤳ (U i).1.restrictPreimage f ⟨x, hx⟩ := by rwa [subtype_specializes_iff] obtain ⟨a, ha, heq⟩ := hf i h refine ⟨a, ?_, congr(($heq).val)⟩ rwa [subtype_specializes_iff] at ha end LocalAtTarget section LocalAtSource variable {U : ι → Opens α} (hU : IsOpenCover U) include hU lemma isOpenMap_iff_comp : IsOpenMap f ↔ ∀ i, IsOpenMap (f ∘ ((↑) : U i → α)) := by refine ⟨fun hf ↦ fun i ↦ hf.comp (U i).isOpenEmbedding'.isOpenMap, fun hf ↦ ?_⟩ intro V hV convert isOpen_iUnion (fun i ↦ hf i _ <\| isOpen_induced hV) simp_rw [Set.image_comp, Set.image_preimage_eq_inter_range, ← Set.image_iUnion, Subtype.range_coe_subtype, SetLike.setOf_mem_eq, hU.iUnion_inter] lemma generalizingMap_iff_comp : GeneralizingMap f ↔ ∀ i, GeneralizingMap (f ∘ ((↑) : U i → α)) := by refine ⟨fun hf ↦ fun i ↦ ((U i).isOpenEmbedding'.generalizingMap (U i).isOpenEmbedding'.isOpen_range.stableUnderGeneralization).comp hf, fun hf ↦ fun x y h ↦ ?_⟩ obtain ⟨i, hi⟩ := hU.exists_mem x replace h : y ⤳ (f ∘ ((↑) : U i → α)) ⟨x, hi⟩ := h obtain ⟨a, ha, rfl⟩ := hf i h use a.val simp [ha.map (U i).isOpenEmbedding'.continuous] end LocalAtSource end TopologicalSpace.IsOpenCover /-- Given a continuous map `f : X → Y` between topological spaces. Suppose we have an open cover `U i` of the range of `f`, and a family of continuous maps `V i → X` whose images are a cover of `X` that is coarser than the pullback of `U` under `f`. To check that `f` is an embedding it suffices to check that `V i → Y` is an embedding for all `i`. -/ -- TODO : the lemma name does not match the content (there is no hypothesis `iSup_eq_top`!) theorem isEmbedding_of_iSup_eq_top_of_preimage_subset_range {X Y} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : Continuous f) {ι : Type} (U : ι → Opens Y) (hU : Set.range f ⊆ (iSup U :)) (V : ι → Type) [∀ i, TopologicalSpace (V i)] (iV : ∀ i, V i → X) (hiV : ∀ i, Continuous (iV i)) (hV : ∀ i, f ⁻¹' U i ⊆ Set.range (iV i)) (hV' : ∀ i, IsEmbedding (f ∘ iV i)) : IsEmbedding f := by wlog hU' : iSup U = ⊤ · let f₀ : X → Set.range f := fun x ↦ ⟨f x, ⟨x, rfl⟩⟩ suffices IsEmbedding f₀ from IsEmbedding.subtypeVal.comp this have hU'' : (⨆ i, (U i).comap ⟨Subtype.val, continuous_subtype_val⟩ : Opens (Set.range f)) = ⊤ := by rw [← top_le_iff] simpa [Set.range_subset_iff, SetLike.le_def] using hU refine this _ ?_ _ ?_ V iV hiV ?_ ?_ hU'' · fun_prop · rw [hU'']; simp · exact hV · exact fun i ↦ IsEmbedding.of_comp (by fun_prop) continuous_subtype_val (hV' i) rw [(IsOpenCover.mk hU').isEmbedding_iff_restrictPreimage h] intro i let f' := (Subtype.val ∘ (f ⁻¹' U i).restrictPreimage (iV i)) have : IsEmbedding f' := IsEmbedding.subtypeVal.comp ((IsEmbedding.of_comp (hiV i) h (hV' _)).restrictPreimage _) have hf' : Set.range f' = f ⁻¹' U i := by simpa [f', Set.range_comp, Set.range_restrictPreimage] using hV i let e := this.toHomeomorph.trans (Homeomorph.setCongr hf') refine IsEmbedding.of_comp (by fun_prop) continuous_subtype_val ?_ convert ((hV' i).comp IsEmbedding.subtypeVal).comp e.symm.isEmbedding ext x obtain ⟨x, rfl⟩ := e.surjective x simp rfl
.lake/packages/mathlib/Mathlib/Topology/PartitionOfUnity.lean	import Mathlib.Algebra.BigOperators.Finprod import Mathlib.Topology.ContinuousMap.Algebra import Mathlib.Topology.Compactness.Paracompact import Mathlib.Topology.ShrinkingLemma import Mathlib.Topology.UrysohnsLemma import Mathlib.Topology.ContinuousMap.Ordered /-! # Continuous partition of unity In this file we define `PartitionOfUnity (ι X : Type) [TopologicalSpace X] (s : Set X := univ)` to be a continuous partition of unity on `s` indexed by `ι`. More precisely, `f : PartitionOfUnity ι X s` is a collection of continuous functions `f i : C(X, ℝ)`, `i : ι`, such that the supports of `f i` form a locally finite family of sets; * each `f i` is nonnegative; * `∑ᶠ i, f i x = 1` for all `x ∈ s`; * `∑ᶠ i, f i x ≤ 1` for all `x : X`. In the case `s = univ` the last assumption follows from the previous one but it is convenient to have this assumption in the case `s ≠ univ`. We also define a bump function covering, `BumpCovering (ι X : Type) [TopologicalSpace X] (s : Set X := univ)`, to be a collection of functions `f i : C(X, ℝ)`, `i : ι`, such that the supports of `f i` form a locally finite family of sets; * each `f i` is nonnegative; * for each `x ∈ s` there exists `i : ι` such that `f i y = 1` in a neighborhood of `x`. The term is motivated by the smooth case. If `f` is a bump function covering indexed by a linearly ordered type, then `g i x = f i x * ∏ᶠ j < i, (1 - f j x)` is a partition of unity, see `BumpCovering.toPartitionOfUnity`. Note that only finitely many terms `1 - f j x` are not equal to one, so this product is well-defined. Note that `g i x = ∏ᶠ j ≤ i, (1 - f j x) - ∏ᶠ j < i, (1 - f j x)`, so most terms in the sum `∑ᶠ i, g i x` cancel, and we get `∑ᶠ i, g i x = 1 - ∏ᶠ i, (1 - f i x)`, and the latter product equals zero because one of `f i x` is equal to one. We say that a partition of unity or a bump function covering `f` is subordinate to a family of sets `U i`, `i : ι`, if the closure of the support of each `f i` is included in `U i`. We use Urysohn's Lemma to prove that a locally finite open covering of a normal topological space admits a subordinate bump function covering (hence, a subordinate partition of unity), see `BumpCovering.exists_isSubordinate_of_locallyFinite`. If `X` is a paracompact space, then any open covering admits a locally finite refinement, hence it admits a subordinate bump function covering and a subordinate partition of unity, see `BumpCovering.exists_isSubordinate`. We also provide two slightly more general versions of these lemmas, `BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop` and `BumpCovering.exists_isSubordinate_of_prop`, to be used later in the construction of a smooth partition of unity. ## Implementation notes Most (if not all) books only define a partition of unity of the whole space. However, quite a few proofs only deal with `f i` such that `tsupport (f i)` meets a specific closed subset, and it is easier to formalize these proofs if we don't have other functions right away. We use `WellOrderingRel j i` instead of `j < i` in the definition of `BumpCovering.toPartitionOfUnity` to avoid a `[LinearOrder ι]` assumption. While `WellOrderingRel j i` is a well order, not only a strict linear order, we never use this property. ## Tags partition of unity, bump function, Urysohn's lemma, normal space, paracompact space -/ universe u v open Function Set Filter Topology noncomputable section /-- A continuous partition of unity on a set `s : Set X` is a collection of continuous functions `f i` such that * the supports of `f i` form a locally finite family of sets, i.e., for every point `x : X` there exists a neighborhood `U ∋ x` such that all but finitely many functions `f i` are zero on `U`; * the functions `f i` are nonnegative; * the sum `∑ᶠ i, f i x` is equal to one for every `x ∈ s` and is less than or equal to one otherwise. If `X` is a normal paracompact space, then `PartitionOfUnity.exists_isSubordinate` guarantees that for every open covering `U : Set (Set X)` of `s` there exists a partition of unity that is subordinate to `U`. -/ structure PartitionOfUnity (ι X : Type) [TopologicalSpace X] (s : Set X := univ) where /-- The collection of continuous functions underlying this partition of unity -/ toFun : ι → C(X, ℝ) /-- the supports of the underlying functions are a locally finite family of sets -/ locallyFinite' : LocallyFinite fun i => support (toFun i) /-- the functions are non-negative -/ nonneg' : 0 ≤ toFun /-- the functions sum up to one on `s` -/ sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1 /-- the functions sum up to at most one, globally -/ sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1 /-- A `BumpCovering ι X s` is an indexed family of functions `f i`, `i : ι`, such that the supports of `f i` form a locally finite family of sets, i.e., for every point `x : X` there exists a neighborhood `U ∋ x` such that all but finitely many functions `f i` are zero on `U`; * for all `i`, `x` we have `0 ≤ f i x ≤ 1`; * each point `x ∈ s` belongs to the interior of `{x \| f i x = 1}` for some `i`. One of the main use cases for a `BumpCovering` is to define a `PartitionOfUnity`, see `BumpCovering.toPartitionOfUnity`, but some proofs can directly use a `BumpCovering` instead of a `PartitionOfUnity`. If `X` is a normal paracompact space, then `BumpCovering.exists_isSubordinate` guarantees that for every open covering `U : Set (Set X)` of `s` there exists a `BumpCovering` of `s` that is subordinate to `U`. -/ structure BumpCovering (ι X : Type) [TopologicalSpace X] (s : Set X := univ) where /-- The collections of continuous functions underlying this bump covering -/ toFun : ι → C(X, ℝ) /-- the supports of the underlying functions are a locally finite family of sets -/ locallyFinite' : LocallyFinite fun i => support (toFun i) /-- the functions are non-negative -/ nonneg' : 0 ≤ toFun /-- the functions are each at most one -/ le_one' : toFun ≤ 1 /-- Each point `x ∈ s` belongs to the interior of `{x \| f i x = 1}` for some `i`. -/ eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1 variable {ι : Type u} {X : Type v} [TopologicalSpace X] namespace PartitionOfUnity variable {E : Type} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {s : Set X} (f : PartitionOfUnity ι X s) instance : FunLike (PartitionOfUnity ι X s) ι C(X, ℝ) where coe := toFun coe_injective' f g h := by cases f; cases g; congr protected theorem locallyFinite : LocallyFinite fun i => support (f i) := f.locallyFinite' theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) := f.locallyFinite.closure theorem nonneg (i : ι) (x : X) : 0 ≤ f i x := f.nonneg' i x theorem sum_eq_one {x : X} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 := f.sum_eq_one' x hx /-- If `f` is a partition of unity on `s`, then for every `x ∈ s` there exists an index `i` such that `0 < f i x`. -/ theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x := by have H := f.sum_eq_one hx contrapose! H simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one theorem sum_le_one (x : X) : ∑ᶠ i, f i x ≤ 1 := f.sum_le_one' x theorem sum_nonneg (x : X) : 0 ≤ ∑ᶠ i, f i x := finsum_nonneg fun i => f.nonneg i x theorem le_one (i : ι) (x : X) : f i x ≤ 1 := (single_le_finsum i (f.locallyFinite.point_finite x) fun j => f.nonneg j x).trans (f.sum_le_one x) section finsupport variable {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) /-- The support of a partition of unity at a point `x₀` as a `Finset`. This is the set of `i : ι` such that `x₀ ∈ support f i`, i.e. `f i ≠ x₀`. -/ def finsupport : Finset ι := (ρ.locallyFinite.point_finite x₀).toFinset @[simp] theorem mem_finsupport (x₀ : X) {i} : i ∈ ρ.finsupport x₀ ↔ i ∈ support fun i ↦ ρ i x₀ := by simp only [finsupport, mem_support, Finite.mem_toFinset, mem_setOf_eq] @[simp] theorem coe_finsupport (x₀ : X) : (ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ := by ext rw [Finset.mem_coe, mem_finsupport] variable {x₀ : X} theorem sum_finsupport (hx₀ : x₀ ∈ s) : ∑ i ∈ ρ.finsupport x₀, ρ i x₀ = 1 := by rw [← ρ.sum_eq_one hx₀, finsum_eq_sum_of_support_subset _ (ρ.coe_finsupport x₀).superset] theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport x₀ ⊆ I) : ∑ i ∈ I, ρ i x₀ = 1 := by classical rw [← Finset.sum_sdiff hI, ρ.sum_finsupport hx₀] suffices ∑ i ∈ I \ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ I \ ρ.finsupport x₀, 0 by rw [this, add_eq_right, Finset.sum_const_zero] apply Finset.sum_congr rfl rintro x hx simp only [Finset.mem_sdiff, ρ.mem_finsupport, mem_support, Classical.not_not] at hx exact hx.2 theorem sum_finsupport_smul_eq_finsum {M : Type} [AddCommMonoid M] [Module ℝ M] (φ : ι → X → M) : ∑ i ∈ ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ := by apply (finsum_eq_sum_of_support_subset _ _).symm have : (fun i ↦ (ρ i) x₀ • φ i x₀) = (fun i ↦ (ρ i) x₀) • (fun i ↦ φ i x₀) := funext fun _ => (Pi.smul_apply' _ _ _).symm rw [ρ.coe_finsupport x₀, this, support_smul] exact inter_subset_left end finsupport section fintsupport -- partitions of unity have locally finite `tsupport` variable {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) /-- The `tsupport`s of a partition of unity are locally finite. -/ theorem finite_tsupport : {i \| x₀ ∈ tsupport (ρ i)}.Finite := by rcases ρ.locallyFinite x₀ with ⟨t, t_in, ht⟩ apply ht.subset rintro i hi simp only [inter_comm] exact mem_closure_iff_nhds.mp hi t t_in /-- The tsupport of a partition of unity at a point `x₀` as a `Finset`. This is the set of `i : ι` such that `x₀ ∈ tsupport f i`. -/ def fintsupport (x₀ : X) : Finset ι := (ρ.finite_tsupport x₀).toFinset theorem mem_fintsupport_iff (i : ι) : i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport (ρ i) := Finite.mem_toFinset _ theorem eventually_fintsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀ := by apply (ρ.locallyFinite.closure.eventually_subset (fun _ ↦ isClosed_closure) x₀).mono intro y hy z hz rw [PartitionOfUnity.mem_fintsupport_iff] at exact hy hz theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ := fun i hi ↦ by rw [ρ.mem_fintsupport_iff] apply subset_closure exact (ρ.mem_finsupport x₀).mp hi theorem eventually_finsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀ := (ρ.eventually_fintsupport_subset x₀).mono fun y hy ↦ (ρ.finsupport_subset_fintsupport y).trans hy end fintsupport /-- If `f` is a partition of unity on `s : Set X` and `g : X → E` is continuous at every point of the topological support of some `f i`, then `fun x ↦ f i x • g x` is continuous on the whole space. -/ theorem continuous_smul {g : X → E} {i : ι} (hg : ∀ x ∈ tsupport (f i), ContinuousAt g x) : Continuous fun x => f i x • g x := continuous_of_tsupport fun x hx => ((f i).continuousAt x).smul <\| hg x <\| tsupport_smul_subset_left _ _ hx /-- If `f` is a partition of unity on a set `s : Set X` and `g : ι → X → E` is a family of functions such that each `g i` is continuous at every point of the topological support of `f i`, then the sum `fun x ↦ ∑ᶠ i, f i x • g i x` is continuous on the whole space. -/ theorem continuous_finsum_smul [ContinuousAdd E] {g : ι → X → E} (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContinuousAt (g i) x) : Continuous fun x => ∑ᶠ i, f i x • g i x := (continuous_finsum fun i => f.continuous_smul (hg i)) <\| f.locallyFinite.subset fun _ => support_smul_subset_left _ _ /-- A partition of unity `f i` is subordinate to a family of sets `U i` indexed by the same type if for each `i` the closure of the support of `f i` is a subset of `U i`. -/ def IsSubordinate (U : ι → Set X) : Prop := ∀ i, tsupport (f i) ⊆ U i variable {f} theorem exists_finset_nhds' {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) : ∃ I : Finset ι, (∀ᶠ x in 𝓝[s] x₀, ∑ i ∈ I, ρ i x = 1) ∧ ∀ᶠ x in 𝓝 x₀, support (ρ · x) ⊆ I := by rcases ρ.locallyFinite.exists_finset_support x₀ with ⟨I, hI⟩ refine ⟨I, eventually_nhdsWithin_iff.mpr (hI.mono fun x hx x_in ↦ ?_), hI⟩ have : ∑ᶠ i : ι, ρ i x = ∑ i ∈ I, ρ i x := finsum_eq_sum_of_support_subset _ hx rwa [eq_comm, ρ.sum_eq_one x_in] at this @[deprecated (since := "2025-05-22")] alias exists_finset_nhd' := exists_finset_nhds' theorem exists_finset_nhds (ρ : PartitionOfUnity ι X univ) (x₀ : X) : ∃ I : Finset ι, ∀ᶠ x in 𝓝 x₀, ∑ i ∈ I, ρ i x = 1 ∧ support (ρ · x) ⊆ I := by rcases ρ.exists_finset_nhds' x₀ with ⟨I, H⟩ use I rwa [nhdsWithin_univ, ← eventually_and] at H @[deprecated (since := "2025-05-22")] alias exists_finset_nhd := exists_finset_nhds theorem exists_finset_nhds_support_subset {U : ι → Set X} (hso : f.IsSubordinate U) (ho : ∀ i, IsOpen (U i)) (x : X) : ∃ is : Finset ι, ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (support (f · z)) ⊆ is := f.locallyFinite.exists_finset_nhds_support_subset hso ho x @[deprecated (since := "2025-05-22")] alias exists_finset_nhd_support_subset := exists_finset_nhds_support_subset /-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and `g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum `fun x ↦ ∑ᶠ i, f i x • g i x` is a continuous function. -/ theorem IsSubordinate.continuous_finsum_smul [ContinuousAdd E] {U : ι → Set X} (ho : ∀ i, IsOpen (U i)) (hf : f.IsSubordinate U) {g : ι → X → E} (hg : ∀ i, ContinuousOn (g i) (U i)) : Continuous fun x => ∑ᶠ i, f i x • g i x := f.continuous_finsum_smul fun i _ hx => (hg i).continuousAt <\| (ho i).mem_nhds <\| hf i hx end PartitionOfUnity namespace BumpCovering variable {s : Set X} (f : BumpCovering ι X s) instance : FunLike (BumpCovering ι X s) ι C(X, ℝ) where coe := toFun coe_injective' f g h := by cases f; cases g; congr @[simp] lemma toFun_eq_coe : f.toFun = f := rfl protected theorem locallyFinite : LocallyFinite fun i => support (f i) := f.locallyFinite' theorem locallyFinite_tsupport : LocallyFinite fun i => tsupport (f i) := f.locallyFinite.closure protected theorem point_finite (x : X) : { i \| f i x ≠ 0 }.Finite := f.locallyFinite.point_finite x theorem nonneg (i : ι) (x : X) : 0 ≤ f i x := f.nonneg' i x theorem le_one (i : ι) (x : X) : f i x ≤ 1 := f.le_one' i x open Classical in /-- A `BumpCovering` that consists of a single function, uniformly equal to one, defined as an example for `Inhabited` instance. -/ protected def single (i : ι) (s : Set X) : BumpCovering ι X s where toFun := Pi.single i 1 locallyFinite' x := by refine ⟨univ, univ_mem, (finite_singleton i).subset ?_⟩ rintro j ⟨x, hx, -⟩ contrapose! hx rw [mem_singleton_iff] at hx simp [hx] nonneg' := le_update_iff.2 ⟨fun _ => zero_le_one, fun _ _ => le_rfl⟩ le_one' := update_le_iff.2 ⟨le_rfl, fun _ _ _ => zero_le_one⟩ eventuallyEq_one' x _ := ⟨i, by rw [Pi.single_eq_same, ContinuousMap.coe_one]⟩ open Classical in @[simp] theorem coe_single (i : ι) (s : Set X) : ⇑(BumpCovering.single i s) = Pi.single i 1 := by rfl instance [Inhabited ι] : Inhabited (BumpCovering ι X s) := ⟨BumpCovering.single default s⟩ /-- A collection of bump functions `f i` is subordinate to a family of sets `U i` indexed by the same type if for each `i` the closure of the support of `f i` is a subset of `U i`. -/ def IsSubordinate (f : BumpCovering ι X s) (U : ι → Set X) : Prop := ∀ i, tsupport (f i) ⊆ U i theorem IsSubordinate.mono {f : BumpCovering ι X s} {U V : ι → Set X} (hU : f.IsSubordinate U) (hV : ∀ i, U i ⊆ V i) : f.IsSubordinate V := fun i => Subset.trans (hU i) (hV i) /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a closed set `s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. If `X` is a paracompact space, then the assumption `hf : LocallyFinite U` can be omitted, see `BumpCovering.exists_isSubordinate`. This version assumes that `p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a `BumpCovering` such that each function of the covering satisfies `p`. -/ theorem exists_isSubordinate_of_locallyFinite_of_prop [NormalSpace X] (p : (X → ℝ) → Prop) (h01 : ∀ s t, IsClosed s → IsClosed t → Disjoint s t → ∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U := by rcases exists_subset_iUnion_closure_subset hs ho (fun x _ => hf.point_finite x) hU with ⟨V, hsV, hVo, hVU⟩ have hVU' : ∀ i, V i ⊆ U i := fun i => Subset.trans subset_closure (hVU i) rcases exists_subset_iUnion_closure_subset hs hVo (fun x _ => (hf.subset hVU').point_finite x) hsV with ⟨W, hsW, hWo, hWV⟩ choose f hfp hf0 hf1 hf01 using fun i => h01 _ _ (isClosed_compl_iff.2 <\| hVo i) isClosed_closure (disjoint_right.2 fun x hx => Classical.not_not.2 (hWV i hx)) have hsupp : ∀ i, support (f i) ⊆ V i := fun i => support_subset_iff'.2 (hf0 i) refine ⟨⟨f, hf.subset fun i => Subset.trans (hsupp i) (hVU' i), fun i x => (hf01 i x).1, fun i x => (hf01 i x).2, fun x hx => ?_⟩, hfp, fun i => Subset.trans (closure_mono (hsupp i)) (hVU i)⟩ rcases mem_iUnion.1 (hsW hx) with ⟨i, hi⟩ exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩ /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a closed set `s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. If `X` is a paracompact space, then the assumption `hf : LocallyFinite U` can be omitted, see `BumpCovering.exists_isSubordinate`. -/ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, f.IsSubordinate U := let ⟨f, _, hfU⟩ := exists_isSubordinate_of_locallyFinite_of_prop (fun _ => True) (fun _ _ hs ht hd => (exists_continuous_zero_one_of_isClosed hs ht hd).imp fun _ hf => ⟨trivial, hf⟩) hs U ho hf hU ⟨f, hfU⟩ /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set `s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. This version assumes that `p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a `BumpCovering` such that each function of the covering satisfies `p`. -/ theorem exists_isSubordinate_of_prop [NormalSpace X] [ParacompactSpace X] (p : (X → ℝ) → Prop) (h01 : ∀ s t, IsClosed s → IsClosed t → Disjoint s t → ∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U := by rcases precise_refinement_set hs _ ho hU with ⟨V, hVo, hsV, hVf, hVU⟩ rcases exists_isSubordinate_of_locallyFinite_of_prop p h01 hs V hVo hVf hsV with ⟨f, hfp, hf⟩ exact ⟨f, hfp, hf.mono hVU⟩ /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set `s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. -/ theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, f.IsSubordinate U := by rcases precise_refinement_set hs _ ho hU with ⟨V, hVo, hsV, hVf, hVU⟩ rcases exists_isSubordinate_of_locallyFinite hs V hVo hVf hsV with ⟨f, hf⟩ exact ⟨f, hf.mono hVU⟩ /-- If `X` is a locally compact T2 topological space and `U i`, `i : ι`, is a locally finite open covering of a compact set `s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. If `X` is a paracompact space, then the assumption `hf : LocallyFinite U` can be omitted, see `BumpCovering.exists_isSubordinate`. This version assumes that `p : (X → ℝ) → Prop` is a predicate that satisfies Urysohn's lemma, and provides a `BumpCovering` such that each function of the covering satisfies `p`. -/ theorem exists_isSubordinate_of_locallyFinite_of_prop_t2space [LocallyCompactSpace X] [T2Space X] (p : (X → ℝ) → Prop) (h01 : ∀ s t, IsClosed s → IsCompact t → Disjoint s t → ∃ f : C(X, ℝ), p f ∧ EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1) (hs : IsCompact s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, (∀ i, p (f i)) ∧ f.IsSubordinate U ∧ ∀ i, HasCompactSupport (f i) := by rcases exists_subset_iUnion_closure_subset_t2space hs ho (fun x _ => hf.point_finite x) hU with ⟨V, hsV, hVo, hVU, hcp⟩ have hVU' i : V i ⊆ U i := subset_closure.trans (hVU i) rcases exists_subset_iUnion_closure_subset_t2space hs hVo (fun x _ => (hf.subset hVU').point_finite x) hsV with ⟨W, hsW, hWo, hWV, hWc⟩ choose f hfp hf0 hf1 hf01 using fun i => h01 _ _ (isClosed_compl_iff.2 <\| hVo i) (hWc i) (disjoint_right.2 fun x hx => Classical.not_not.2 (hWV i hx)) have hsupp i : support (f i) ⊆ V i := support_subset_iff'.2 (hf0 i) refine ⟨⟨f, hf.subset fun i => Subset.trans (hsupp i) (hVU' i), fun i x => (hf01 i x).1, fun i x => (hf01 i x).2, fun x hx => ?_⟩, hfp, fun i => Subset.trans (closure_mono (hsupp i)) (hVU i), fun i => IsCompact.of_isClosed_subset (hcp i) isClosed_closure <\| closure_mono (hsupp i)⟩ rcases mem_iUnion.1 (hsW hx) with ⟨i, hi⟩ exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩ /-- If `X` is a normal topological space and `U i`, `i : ι`, is a locally finite open covering of a closed set `s`, then there exists a `BumpCovering ι X s` that is subordinate to `U`. If `X` is a paracompact space, then the assumption `hf : LocallyFinite U` can be omitted, see `BumpCovering.exists_isSubordinate`. -/ theorem exists_isSubordinate_hasCompactSupport_of_locallyFinite_t2space [LocallyCompactSpace X] [T2Space X] (hs : IsCompact s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, f.IsSubordinate U ∧ ∀ i, HasCompactSupport (f i) := by -- need to switch 0 and 1 in `exists_continuous_zero_one_of_isCompact` simpa using exists_isSubordinate_of_locallyFinite_of_prop_t2space (fun _ => True) (fun _ _ ht hs hd => (exists_continuous_zero_one_of_isCompact' hs ht hd.symm).imp fun _ hf => ⟨trivial, hf⟩) hs U ho hf hU /-- Index of a bump function such that `fs i =ᶠ[𝓝 x] 1`. -/ def ind (x : X) (hx : x ∈ s) : ι := (f.eventuallyEq_one' x hx).choose theorem eventuallyEq_one (x : X) (hx : x ∈ s) : f (f.ind x hx) =ᶠ[𝓝 x] 1 := (f.eventuallyEq_one' x hx).choose_spec theorem ind_apply (x : X) (hx : x ∈ s) : f (f.ind x hx) x = 1 := (f.eventuallyEq_one x hx).eq_of_nhds /-- Partition of unity defined by a `BumpCovering`. We use this auxiliary definition to prove some properties of the new family of functions before bundling it into a `PartitionOfUnity`. Do not use this definition, use `BumpCovering.toPartitionOfUnity` instead. The partition of unity is given by the formula `g i x = f i x * ∏ᶠ j < i, (1 - f j x)`. In other words, `g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x)`, so `∑ᶠ i, g i x = 1 - ∏ᶠ j, (1 - f j x)`. If `x ∈ s`, then one of `f j x` equals one, hence the product of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`. In order to avoid an assumption `LinearOrder ι`, we use `WellOrderingRel` instead of `(<)`. -/ def toPOUFun (i : ι) (x : X) : ℝ := f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), (1 - f j x) theorem toPOUFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPOUFun i x = 0 := by rw [toPOUFun, h, zero_mul] theorem support_toPOUFun_subset (i : ι) : support (f.toPOUFun i) ⊆ support (f i) := fun _ => mt <\| f.toPOUFun_zero_of_zero open Classical in theorem toPOUFun_eq_mul_prod (i : ι) (x : X) (t : Finset ι) (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) : f.toPOUFun i x = f i x * ∏ j ∈ t with WellOrderingRel j i, (1 - f j x) := by refine congr_arg _ (finprod_cond_eq_prod_of_cond_iff _ fun {j} hj => ?_) rw [Ne, sub_eq_self] at hj rw [Finset.mem_filter, Iff.comm, and_iff_right_iff_imp] exact flip (ht j) hj theorem sum_toPOUFun_eq (x : X) : ∑ᶠ i, f.toPOUFun i x = 1 - ∏ᶠ i, (1 - f i x) := by set s := (f.point_finite x).toFinset have hs : (s : Set ι) = { i \| f i x ≠ 0 } := Finite.coe_toFinset _ have A : (support fun i => toPOUFun f i x) ⊆ s := by rw [hs] exact fun i hi => f.support_toPOUFun_subset i hi have B : (mulSupport fun i => 1 - f i x) ⊆ s := by rw [hs, mulSupport_one_sub] exact fun i => id classical letI : LinearOrder ι := linearOrderOfSTO WellOrderingRel rw [finsum_eq_sum_of_support_subset _ A, finprod_eq_prod_of_mulSupport_subset _ B, Finset.prod_one_sub_ordered, sub_sub_cancel] refine Finset.sum_congr rfl fun i _ => ?_ convert f.toPOUFun_eq_mul_prod _ _ _ fun j _ hj => _ rwa [Finite.mem_toFinset] open Classical in theorem exists_finset_toPOUFun_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι, f.toPOUFun i =ᶠ[𝓝 x] f i * ∏ j ∈ t with WellOrderingRel j i, (1 - f j) := by rcases f.locallyFinite x with ⟨U, hU, hf⟩ use hf.toFinset filter_upwards [hU] with y hyU simp only [ContinuousMap.coe_prod, Pi.mul_apply, Finset.prod_apply] apply toPOUFun_eq_mul_prod intro j _ hj exact hf.mem_toFinset.2 ⟨y, ⟨hj, hyU⟩⟩ theorem continuous_toPOUFun (i : ι) : Continuous (f.toPOUFun i) := by refine (map_continuous <\| f i).mul <\| continuous_finprod_cond (fun j _ => by fun_prop) ?_ simp only [mulSupport_one_sub] exact f.locallyFinite /-- The partition of unity defined by a `BumpCovering`. The partition of unity is given by the formula `g i x = f i x * ∏ᶠ j < i, (1 - f j x)`. In other words, `g i x = ∏ᶠ j < i, (1 - f j x) - ∏ᶠ j ≤ i, (1 - f j x)`, so `∑ᶠ i, g i x = 1 - ∏ᶠ j, (1 - f j x)`. If `x ∈ s`, then one of `f j x` equals one, hence the product of `1 - f j x` vanishes, and `∑ᶠ i, g i x = 1`. In order to avoid an assumption `LinearOrder ι`, we use `WellOrderingRel` instead of `(<)`. -/ def toPartitionOfUnity : PartitionOfUnity ι X s where toFun i := ⟨f.toPOUFun i, f.continuous_toPOUFun i⟩ locallyFinite' := f.locallyFinite.subset f.support_toPOUFun_subset nonneg' i x := mul_nonneg (f.nonneg i x) (finprod_cond_nonneg fun j _ => sub_nonneg.2 <\| f.le_one j x) sum_eq_one' x hx := by simp only [ContinuousMap.coe_mk, sum_toPOUFun_eq, sub_eq_self] apply finprod_eq_zero (fun i => 1 - f i x) (f.ind x hx) · simp only [f.ind_apply x hx, sub_self] · rw [mulSupport_one_sub] exact f.point_finite x sum_le_one' x := by simp only [ContinuousMap.coe_mk, sum_toPOUFun_eq, sub_le_self_iff] exact finprod_nonneg fun i => sub_nonneg.2 <\| f.le_one i x theorem toPartitionOfUnity_apply (i : ι) (x : X) : f.toPartitionOfUnity i x = f i x * ∏ᶠ (j) (_ : WellOrderingRel j i), (1 - f j x) := rfl open Classical in theorem toPartitionOfUnity_eq_mul_prod (i : ι) (x : X) (t : Finset ι) (ht : ∀ j, WellOrderingRel j i → f j x ≠ 0 → j ∈ t) : f.toPartitionOfUnity i x = f i x * ∏ j ∈ t with WellOrderingRel j i, (1 - f j x) := f.toPOUFun_eq_mul_prod i x t ht open Classical in theorem exists_finset_toPartitionOfUnity_eventuallyEq (i : ι) (x : X) : ∃ t : Finset ι, f.toPartitionOfUnity i =ᶠ[𝓝 x] f i * ∏ j ∈ t with WellOrderingRel j i, (1 - f j) := f.exists_finset_toPOUFun_eventuallyEq i x theorem toPartitionOfUnity_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPartitionOfUnity i x = 0 := f.toPOUFun_zero_of_zero h theorem support_toPartitionOfUnity_subset (i : ι) : support (f.toPartitionOfUnity i) ⊆ support (f i) := f.support_toPOUFun_subset i theorem sum_toPartitionOfUnity_eq (x : X) : ∑ᶠ i, f.toPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - f i x) := f.sum_toPOUFun_eq x theorem IsSubordinate.toPartitionOfUnity {f : BumpCovering ι X s} {U : ι → Set X} (h : f.IsSubordinate U) : f.toPartitionOfUnity.IsSubordinate U := fun i => Subset.trans (closure_mono <\| f.support_toPartitionOfUnity_subset i) (h i) end BumpCovering namespace PartitionOfUnity variable {s : Set X} instance [Inhabited ι] : Inhabited (PartitionOfUnity ι X s) := ⟨BumpCovering.toPartitionOfUnity default⟩ /-- If `X` is a normal topological space and `U` is a locally finite open covering of a closed set `s`, then there exists a `PartitionOfUnity ι X s` that is subordinate to `U`. If `X` is a paracompact space, then the assumption `hf : LocallyFinite U` can be omitted, see `BumpCovering.exists_isSubordinate`. -/ theorem exists_isSubordinate_of_locallyFinite [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f : PartitionOfUnity ι X s, f.IsSubordinate U := let ⟨f, hf⟩ := BumpCovering.exists_isSubordinate_of_locallyFinite hs U ho hf hU ⟨f.toPartitionOfUnity, hf.toPartitionOfUnity⟩ /-- If `X` is a paracompact normal topological space and `U` is an open covering of a closed set `s`, then there exists a `PartitionOfUnity ι X s` that is subordinate to `U`. -/ theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) : ∃ f : PartitionOfUnity ι X s, f.IsSubordinate U := let ⟨f, hf⟩ := BumpCovering.exists_isSubordinate hs U ho hU ⟨f.toPartitionOfUnity, hf.toPartitionOfUnity⟩ /-- If `X` is a locally compact T2 topological space and `U` is a locally finite open covering of a compact set `s`, then there exists a `PartitionOfUnity ι X s` that is subordinate to `U`. -/ theorem exists_isSubordinate_of_locallyFinite_t2space [LocallyCompactSpace X] [T2Space X] (hs : IsCompact s) (U : ι → Set X) (ho : ∀ i, IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f : PartitionOfUnity ι X s, f.IsSubordinate U ∧ ∀ i, HasCompactSupport (f i) := let ⟨f, hfsub, hfcp⟩ := BumpCovering.exists_isSubordinate_hasCompactSupport_of_locallyFinite_t2space hs U ho hf hU ⟨f.toPartitionOfUnity, hfsub.toPartitionOfUnity, fun i => IsCompact.of_isClosed_subset (hfcp i) isClosed_closure <\| closure_mono (f.support_toPartitionOfUnity_subset i)⟩ end PartitionOfUnity /-- A variation of Urysohn's lemma. In a locally compact T2 space `X`, for a compact set `t` and a finite family of open sets `{s i}_i` such that `t ⊆ ⋃ i, s i`, there is a family of compactly supported continuous functions `{f i}_i` supported in `s i`, `∑ i, f i x = 1` on `t` and `0 ≤ f i x ≤ 1`. -/ theorem exists_continuous_sum_one_of_isOpen_isCompact [T2Space X] [LocallyCompactSpace X] {n : ℕ} {t : Set X} {s : Fin n → Set X} (hs : ∀ (i : Fin n), IsOpen (s i)) (htcp : IsCompact t) (hst : t ⊆ ⋃ i, s i) : ∃ f : Fin n → C(X, ℝ), (∀ (i : Fin n), tsupport (f i) ⊆ s i) ∧ EqOn (∑ i, f i) 1 t ∧ (∀ (i : Fin n), ∀ (x : X), f i x ∈ Icc (0 : ℝ) 1) ∧ (∀ (i : Fin n), HasCompactSupport (f i)) := by obtain ⟨f, hfsub, hfcp⟩ := PartitionOfUnity.exists_isSubordinate_of_locallyFinite_t2space htcp s hs (locallyFinite_of_finite _) hst use f refine ⟨fun i ↦ hfsub i, ?_, ?_, fun i => hfcp i⟩ · intro x hx simp only [Finset.sum_apply, Pi.one_apply] have h := f.sum_eq_one' x hx rw [finsum_eq_sum (fun i => (f.toFun i) x) (Finite.subset finite_univ (subset_univ (support fun i ↦ (f.toFun i) x)))] at h rwa [Fintype.sum_subset (by simp)] at h intro i x exact ⟨f.nonneg i x, PartitionOfUnity.le_one f i x⟩
.lake/packages/mathlib/Mathlib/Topology/DiscreteSubset.lean	import Mathlib.Tactic.TautoSet import Mathlib.Topology.Constructions import Mathlib.Data.Set.Subset import Mathlib.Topology.Separation.Basic /-! # Discrete subsets of topological spaces This file contains various additional properties of discrete subsets of topological spaces. ## Discreteness and compact sets Given a topological space `X` together with a subset `s ⊆ X`, there are two distinct concepts of "discreteness" which may hold. These are: (i) Every point of `s` is isolated (i.e., the subset topology induced on `s` is the discrete topology). (ii) Every compact subset of `X` meets `s` only finitely often (i.e., the inclusion map `s → X` tends to the cocompact filter along the cofinite filter on `s`). When `s` is closed, the two conditions are equivalent provided `X` is locally compact and T1, see `IsClosed.tendsto_coe_cofinite_iff`. ### Main statements * `tendsto_cofinite_cocompact_iff`: * `IsClosed.tendsto_coe_cofinite_iff`: ## Co-discrete open sets We define the filter `Filter.codiscreteWithin S`, which is the supremum of all `𝓝[S \ {x}] x`. This is the filter of all open codiscrete sets within S. We also define `Filter.codiscrete` as `Filter.codiscreteWithin univ`, which is the filter of all open codiscrete sets in the space. -/ open Set Filter Function Topology variable {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} theorem discreteTopology_subtype_iff {S : Set Y} : DiscreteTopology S ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥ := by simp_rw [discreteTopology_iff_nhds_ne, SetCoe.forall', nhds_ne_subtype_eq_bot_iff] lemma discreteTopology_subtype_iff' {S : Set Y} : DiscreteTopology S ↔ ∀ y ∈ S, ∃ U : Set Y, IsOpen U ∧ U ∩ S = {y} := by simp [discreteTopology_iff_isOpen_singleton, isOpen_induced_iff, Set.ext_iff] grind section cofinite_cocompact omit [TopologicalSpace X] in lemma tendsto_cofinite_cocompact_iff : Tendsto f cofinite (cocompact _) ↔ ∀ K, IsCompact K → Set.Finite (f ⁻¹' K) := by rw [hasBasis_cocompact.tendsto_right_iff] refine forall₂_congr (fun K _ ↦ ?_) simp only [mem_compl_iff, eventually_cofinite, not_not, preimage] lemma Continuous.discrete_of_tendsto_cofinite_cocompact [T1Space X] [WeaklyLocallyCompactSpace Y] (hf' : Continuous f) (hf : Tendsto f cofinite (cocompact _)) : DiscreteTopology X := by refine discreteTopology_iff_isOpen_singleton.mpr (fun x ↦ ?_) obtain ⟨K : Set Y, hK : IsCompact K, hK' : K ∈ 𝓝 (f x)⟩ := exists_compact_mem_nhds (f x) obtain ⟨U : Set Y, hU₁ : U ⊆ K, hU₂ : IsOpen U, hU₃ : f x ∈ U⟩ := mem_nhds_iff.mp hK' have hU₄ : Set.Finite (f⁻¹' U) := Finite.subset (tendsto_cofinite_cocompact_iff.mp hf K hK) (preimage_mono hU₁) exact isOpen_singleton_of_finite_mem_nhds _ ((hU₂.preimage hf').mem_nhds hU₃) hU₄ lemma tendsto_cofinite_cocompact_of_discrete [DiscreteTopology X] (hf : Tendsto f (cocompact _) (cocompact _)) : Tendsto f cofinite (cocompact _) := by convert hf rw [cocompact_eq_cofinite X] lemma IsClosed.tendsto_coe_cofinite_of_discreteTopology {s : Set X} (hs : IsClosed s) (_hs' : DiscreteTopology s) : Tendsto ((↑) : s → X) cofinite (cocompact _) := tendsto_cofinite_cocompact_of_discrete hs.isClosedEmbedding_subtypeVal.tendsto_cocompact lemma IsClosed.tendsto_coe_cofinite_iff [T1Space X] [WeaklyLocallyCompactSpace X] {s : Set X} (hs : IsClosed s) : Tendsto ((↑) : s → X) cofinite (cocompact _) ↔ DiscreteTopology s := ⟨continuous_subtype_val.discrete_of_tendsto_cofinite_cocompact, fun _ ↦ hs.tendsto_coe_cofinite_of_discreteTopology inferInstance⟩ end cofinite_cocompact section codiscrete_filter /-- Criterion for a subset `S ⊆ X` to be closed and discrete in terms of the punctured neighbourhood filter at an arbitrary point of `X`. (Compare `discreteTopology_subtype_iff`.) -/ theorem isClosed_and_discrete_iff {S : Set X} : IsClosed S ∧ DiscreteTopology S ↔ ∀ x, Disjoint (𝓝[≠] x) (𝓟 S) := by rw [discreteTopology_subtype_iff, isClosed_iff_clusterPt, ← forall_and] congrm (∀ x, ?_) rw [← not_imp_not, clusterPt_iff_not_disjoint, not_not, ← disjoint_iff] constructor <;> intro H · by_cases hx : x ∈ S exacts [H.2 hx, (H.1 hx).mono_left nhdsWithin_le_nhds] · refine ⟨fun hx ↦ ?_, fun _ ↦ H⟩ simpa [disjoint_iff, nhdsWithin, inf_assoc, hx] using H /-- The filter of sets with no accumulation points inside a set `S : Set X`, implemented as the supremum over all punctured neighborhoods within `S`. -/ def Filter.codiscreteWithin (S : Set X) : Filter X := ⨆ x ∈ S, 𝓝[S \ {x}] x lemma mem_codiscreteWithin {S T : Set X} : S ∈ codiscreteWithin T ↔ ∀ x ∈ T, Disjoint (𝓝[≠] x) (𝓟 (T \ S)) := by simp only [codiscreteWithin, mem_iSup, mem_nhdsWithin, disjoint_principal_right, subset_def, mem_diff, mem_inter_iff, mem_compl_iff] congr! 7 with x - u y tauto /-- A set `s` is codiscrete within `U` iff `s ∪ Uᶜ` is a punctured neighborhood of every point in `U`. -/ theorem mem_codiscreteWithin_iff_forall_mem_nhdsNE {S T : Set X} : S ∈ codiscreteWithin T ↔ ∀ x ∈ T, S ∪ Tᶜ ∈ 𝓝[≠] x := by simp_rw [mem_codiscreteWithin, disjoint_principal_right, Set.compl_diff] lemma mem_codiscreteWithin_accPt {S T : Set X} : S ∈ codiscreteWithin T ↔ ∀ x ∈ T, ¬AccPt x (𝓟 (T \ S)) := by simp only [mem_codiscreteWithin, disjoint_iff, AccPt, not_neBot] /-- Any set is codiscrete within itself. -/ @[simp] theorem Filter.self_mem_codiscreteWithin (U : Set X) : U ∈ Filter.codiscreteWithin U := by simp [mem_codiscreteWithin] /-- If a set is codiscrete within `U`, then it is codiscrete within any subset of `U`. -/ lemma Filter.codiscreteWithin.mono {U₁ U : Set X} (hU : U₁ ⊆ U) : codiscreteWithin U₁ ≤ codiscreteWithin U := by refine (biSup_mono hU).trans <\| iSup₂_mono fun _ _ ↦ ?_ gcongr /-- If `s` is codiscrete within `U`, then `sᶜ ∩ U` has discrete topology. -/ theorem discreteTopology_of_codiscreteWithin {U s : Set X} (h : s ∈ Filter.codiscreteWithin U) : DiscreteTopology ((sᶜ ∩ U) : Set X) := by rw [(by simp : ((sᶜ ∩ U) : Set X) = ((s ∪ Uᶜ)ᶜ : Set X)), discreteTopology_subtype_iff] simp_rw [mem_codiscreteWithin, Filter.disjoint_principal_right] at h intro x hx rw [← Filter.mem_iff_inf_principal_compl, ← Set.compl_diff] simp_all only [h x, Set.compl_union, compl_compl, Set.mem_inter_iff, Set.mem_compl_iff] /-- Helper lemma for `codiscreteWithin_iff_locallyFiniteComplementWithin`: A set `s` is `codiscreteWithin U` iff every point `z ∈ U` has a punctured neighborhood that does not intersect `U \ s`. -/ lemma codiscreteWithin_iff_locallyEmptyComplementWithin {s U : Set X} : s ∈ codiscreteWithin U ↔ ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \ s) = ∅ := by simp only [mem_codiscreteWithin, disjoint_principal_right] refine ⟨fun h z hz ↦ ⟨(U \ s)ᶜ, h z hz, by simp⟩, fun h z hz ↦ ?_⟩ rw [← exists_mem_subset_iff] obtain ⟨t, h₁t, h₂t⟩ := h z hz use t, h₁t, (disjoint_iff_inter_eq_empty.mpr h₂t).subset_compl_right /-- If `U` is closed and `s` is codiscrete within `U`, then `U \ s` is closed. -/ theorem isClosed_sdiff_of_codiscreteWithin {s U : Set X} (hs : s ∈ codiscreteWithin U) (hU : IsClosed U) : IsClosed (U \ s) := by rw [← isOpen_compl_iff, isOpen_iff_eventually] intro x hx by_cases h₁x : x ∈ U · rw [mem_codiscreteWithin] at hs filter_upwards [eventually_nhdsWithin_iff.1 (disjoint_principal_right.1 (hs x h₁x))] intro a ha by_cases h₂a : a = x · tauto_set · specialize ha h₂a tauto_set · rw [eventually_iff_exists_mem] use Uᶜ, hU.compl_mem_nhds h₁x intro y hy tauto_set /-- In a T1Space, punctured neighborhoods are stable under removing finite sets of points. -/ theorem nhdsNE_of_nhdsNE_sdiff_finite {X : Type} [TopologicalSpace X] [T1Space X] {x : X} {U s : Set X} (hU : U ∈ 𝓝[≠] x) (hs : Finite s) : U \ s ∈ 𝓝[≠] x := by rw [mem_nhdsWithin] at hU ⊢ obtain ⟨t, ht, h₁ts, h₂ts⟩ := hU use t \ (s \ {x}) constructor · rw [← isClosed_compl_iff, compl_diff] exact s.toFinite.diff.isClosed.union (isClosed_compl_iff.2 ht) · tauto_set @[deprecated (since := "2025-05-22")] alias nhdNE_of_nhdNE_sdiff_finite := nhdsNE_of_nhdsNE_sdiff_finite /-- In a T1Space, a set `s` is codiscreteWithin `U` iff it has locally finite complement within `U`. More precisely: `s` is codiscreteWithin `U` iff every point `z ∈ U` has a punctured neighborhood intersect `U \ s` in only finitely many points. -/ theorem codiscreteWithin_iff_locallyFiniteComplementWithin [T1Space X] {s U : Set X} : s ∈ codiscreteWithin U ↔ ∀ z ∈ U, ∃ t ∈ 𝓝 z, Set.Finite (t ∩ (U \ s)) := by rw [codiscreteWithin_iff_locallyEmptyComplementWithin] constructor · intro h z h₁z obtain ⟨t, h₁t, h₂t⟩ := h z h₁z use insert z t, insert_mem_nhds_iff.mpr h₁t by_cases hz : z ∈ U \ s · rw [inter_comm, inter_insert_of_mem hz, inter_comm, h₂t] simp · rw [inter_comm, inter_insert_of_notMem hz, inter_comm, h₂t] simp · intro h z h₁z obtain ⟨t, h₁t, h₂t⟩ := h z h₁z use t \ (t ∩ (U \ s)), nhdsNE_of_nhdsNE_sdiff_finite (mem_nhdsWithin_of_mem_nhds h₁t) h₂t simp /-- In any topological space, the open sets with discrete complement form a filter, defined as the supremum of all punctured neighborhoods. See `Filter.mem_codiscrete'` for the equivalence. -/ def Filter.codiscrete (X : Type) [TopologicalSpace X] : Filter X := codiscreteWithin Set.univ lemma mem_codiscrete {S : Set X} : S ∈ codiscrete X ↔ ∀ x, Disjoint (𝓝[≠] x) (𝓟 Sᶜ) := by simp [codiscrete, mem_codiscreteWithin, compl_eq_univ_diff] lemma mem_codiscrete_accPt {S : Set X} : S ∈ codiscrete X ↔ ∀ x, ¬AccPt x (𝓟 Sᶜ) := by simp only [mem_codiscrete, disjoint_iff, AccPt, not_neBot] lemma mem_codiscrete' {S : Set X} : S ∈ codiscrete X ↔ IsOpen S ∧ DiscreteTopology ↑Sᶜ := by rw [mem_codiscrete, ← isClosed_compl_iff, isClosed_and_discrete_iff] lemma mem_codiscrete_subtype_iff_mem_codiscreteWithin {S : Set X} {U : Set S} : U ∈ codiscrete S ↔ (↑) '' U ∈ codiscreteWithin S := by simp only [mem_codiscrete, disjoint_principal_right, compl_compl, Subtype.forall, mem_codiscreteWithin] congr! with x hx constructor · rw [nhdsWithin_subtype, mem_comap] rintro ⟨t, ht1, ht2⟩ rw [mem_nhdsWithin] at ht1 ⊢ obtain ⟨u, hu1, hu2, hu3⟩ := ht1 refine ⟨u, hu1, hu2, fun v hv ↦ ?_⟩ simpa using fun hv2 ↦ ⟨hv2, ht2 <\| hu3 <\| by simpa [hv2]⟩ · suffices Tendsto (↑) (𝓝[≠] (⟨x, hx⟩ : S)) (𝓝[≠] x) by convert tendsto_def.mp this _; ext; simp exact tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ continuous_subtype_val.continuousWithinAt <\| eventually_mem_nhdsWithin.mono (by simp) end codiscrete_filter section discrete_union lemma compl_mem_codiscrete_iff {S : Set X} : Sᶜ ∈ codiscrete X ↔ IsClosed S ∧ DiscreteTopology ↑S := by rw [mem_codiscrete, compl_compl, isClosed_and_discrete_iff] /-- The union of two discrete closed subsets is discrete. -/ theorem discreteTopology_union {S T : Set X} (hs : DiscreteTopology S) (ht : DiscreteTopology T) (hs' : IsClosed S) (ht' : IsClosed T) : DiscreteTopology ↑(S ∪ T) := by suffices (S ∪ T)ᶜ ∈ codiscrete X from compl_mem_codiscrete_iff.mp this \|>.2 have hS : Sᶜ ∈ codiscrete X := by simpa [compl_mem_codiscrete_iff] using ⟨hs', hs⟩ have hT : Tᶜ ∈ codiscrete X := by simpa [compl_mem_codiscrete_iff] using ⟨ht', ht⟩ simpa using inter_mem hS hT /-- The union of finitely many discrete closed subsets is discrete. -/ theorem discreteTopology_biUnion_finset {ι : Type} {I : Finset ι} {s : ι → Set X} (hs : ∀ i ∈ I, DiscreteTopology (s i)) (hs' : ∀ i ∈ I, IsClosed (s i)) : DiscreteTopology (⋃ i ∈ I, s i) := by suffices (⋃ i ∈ I, s i)ᶜ ∈ codiscrete X from (compl_mem_codiscrete_iff.mp this).2 simpa [biInter_finset_mem I] using fun i hi ↦ compl_mem_codiscrete_iff.mpr ⟨hs' i hi, hs i hi⟩ /-- The union of finitely many discrete closed subsets is discrete. -/ theorem discreteTopology_iUnion_fintype {ι : Type*} [Fintype ι] {s : ι → Set X} (hs : ∀ i, DiscreteTopology (s i)) (hs' : ∀ i, IsClosed (s i)) : DiscreteTopology (⋃ i, s i) := by convert discreteTopology_biUnion_finset (I := .univ) (fun i _ ↦ hs i) (fun i _ ↦ hs' i) <;> simp end discrete_union
.lake/packages/mathlib/Mathlib/Topology/Sequences.lean	import Mathlib.Topology.Defs.Sequences import Mathlib.Topology.UniformSpace.Cauchy /-! # Sequences in topological spaces In this file we prove theorems about relations between closure/compactness/continuity etc. and their sequential counterparts. ## Main definitions The following notions are defined in `Topology/Defs/Sequences`. We build theory about these definitions here, so we remind the definitions. ### Set operation * `seqClosure s`: sequential closure of a set, the set of limits of sequences of points of `s`; ### Predicates * `IsSeqClosed s`: predicate saying that a set is sequentially closed, i.e., `seqClosure s ⊆ s`; * `SeqContinuous f`: predicate saying that a function is sequentially continuous, i.e., for any sequence `u : ℕ → X` that converges to a point `x`, the sequence `f ∘ u` converges to `f x`; * `IsSeqCompact s`: predicate saying that a set is sequentially compact, i.e., every sequence taking values in `s` has a converging subsequence. ### Type classes * `FrechetUrysohnSpace X`: a typeclass saying that a topological space is a Fréchet-Urysohn space, i.e., the sequential closure of any set is equal to its closure. * `SequentialSpace X`: a typeclass saying that a topological space is a sequential space, i.e., any sequentially closed set in this space is closed. This condition is weaker than being a Fréchet-Urysohn space. * `SeqCompactSpace X`: a typeclass saying that a topological space is sequentially compact, i.e., every sequence in `X` has a converging subsequence. ## Main results * `seqClosure_subset_closure`: closure of a set includes its sequential closure; * `IsClosed.isSeqClosed`: a closed set is sequentially closed; * `IsSeqClosed.seqClosure_eq`: sequential closure of a sequentially closed set `s` is equal to `s`; * `seqClosure_eq_closure`: in a Fréchet-Urysohn space, the sequential closure of a set is equal to its closure; * `tendsto_nhds_iff_seq_tendsto`, `FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto`: a topological space is a Fréchet-Urysohn space if and only if sequential convergence implies convergence; * `FirstCountableTopology.frechetUrysohnSpace`: every topological space with first countable topology is a Fréchet-Urysohn space; * `FrechetUrysohnSpace.to_sequentialSpace`: every Fréchet-Urysohn space is a sequential space; * `IsSeqCompact.isCompact`: a sequentially compact set in a uniform space with countably generated uniformity is compact. ## Tags sequentially closed, sequentially compact, sequential space -/ open Bornology Filter Function Set TopologicalSpace Topology open scoped Uniformity variable {X Y : Type} /-! ### Sequential closures, sequential continuity, and sequential spaces. -/ section TopologicalSpace variable [TopologicalSpace X] [TopologicalSpace Y] theorem subset_seqClosure {s : Set X} : s ⊆ seqClosure s := fun p hp => ⟨const ℕ p, fun _ => hp, tendsto_const_nhds⟩ /-- The sequential closure of a set is contained in the closure of that set. The converse is not true. -/ theorem seqClosure_subset_closure {s : Set X} : seqClosure s ⊆ closure s := fun _p ⟨_x, xM, xp⟩ => mem_closure_of_tendsto xp (univ_mem' xM) /-- The sequential closure of a sequentially closed set is the set itself. -/ theorem IsSeqClosed.seqClosure_eq {s : Set X} (hs : IsSeqClosed s) : seqClosure s = s := Subset.antisymm (fun _p ⟨_x, hx, hp⟩ => hs hx hp) subset_seqClosure /-- If a set is equal to its sequential closure, then it is sequentially closed. -/ theorem isSeqClosed_of_seqClosure_eq {s : Set X} (hs : seqClosure s = s) : IsSeqClosed s := fun x _p hxs hxp => hs ▸ ⟨x, hxs, hxp⟩ /-- A set is sequentially closed iff it is equal to its sequential closure. -/ theorem isSeqClosed_iff {s : Set X} : IsSeqClosed s ↔ seqClosure s = s := ⟨IsSeqClosed.seqClosure_eq, isSeqClosed_of_seqClosure_eq⟩ /-- A set is sequentially closed if it is closed. -/ protected theorem IsClosed.isSeqClosed {s : Set X} (hc : IsClosed s) : IsSeqClosed s := fun _u _x hu hx => hc.mem_of_tendsto hx (Eventually.of_forall hu) theorem seqClosure_eq_closure [FrechetUrysohnSpace X] (s : Set X) : seqClosure s = closure s := seqClosure_subset_closure.antisymm <\| FrechetUrysohnSpace.closure_subset_seqClosure s /-- In a Fréchet-Urysohn space, a point belongs to the closure of a set iff it is a limit of a sequence taking values in this set. -/ theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) := by rw [← seqClosure_eq_closure] rfl /-- If the domain of a function `f : α → β` is a Fréchet-Urysohn space, then convergence is equivalent to sequential convergence. See also `Filter.tendsto_iff_seq_tendsto` for a version that works for any pair of filters assuming that the filter in the domain is countably generated. This property is equivalent to the definition of `FrechetUrysohnSpace`, see `FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto`. -/ theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} : Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) := by refine ⟨fun hf u hu => hf.comp hu, fun h => ((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 ?_⟩ rintro s ⟨hbs, hsc⟩ refine ⟨closure (f ⁻¹' s), ⟨mt ?_ hbs, isClosed_closure⟩, fun x => mt fun hx => subset_closure hx⟩ rw [← seqClosure_eq_closure] rintro ⟨u, hus, hu⟩ exact hsc.mem_of_tendsto (h u hu) (Eventually.of_forall hus) /-- An alternative construction for `FrechetUrysohnSpace`: if sequential convergence implies convergence, then the space is a Fréchet-Urysohn space. -/ theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto (h : ∀ (f : X → Prop) (a : X), (∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) : FrechetUrysohnSpace X := by refine ⟨fun s x hcx => ?_⟩ by_cases hx : x ∈ s · exact subset_seqClosure hx · obtain ⟨u, hux, hus⟩ : ∃ u : ℕ → X, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ x in atTop, u x ∈ s := by simpa only [ContinuousAt, hx, tendsto_nhds_true, (· ∘ ·), ← not_frequently, exists_prop, ← mem_closure_iff_frequently, hcx, imp_false, not_forall, not_not, not_false_eq_true, not_true_eq_false] using h (· ∉ s) x rcases extraction_of_frequently_atTop hus with ⟨φ, φ_mono, hφ⟩ exact ⟨u ∘ φ, hφ, hux.comp φ_mono.tendsto_atTop⟩ -- see Note [lower instance priority] /-- Every first-countable space is a Fréchet-Urysohn space. -/ instance (priority := 100) FirstCountableTopology.frechetUrysohnSpace [FirstCountableTopology X] : FrechetUrysohnSpace X := FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto fun _ _ => tendsto_iff_seq_tendsto.2 -- see Note [lower instance priority] /-- Every Fréchet-Urysohn space is a sequential space. -/ instance (priority := 100) FrechetUrysohnSpace.to_sequentialSpace [FrechetUrysohnSpace X] : SequentialSpace X := ⟨fun s hs => by rw [← closure_eq_iff_isClosed, ← seqClosure_eq_closure, hs.seqClosure_eq]⟩ theorem Topology.IsInducing.frechetUrysohnSpace [FrechetUrysohnSpace Y] {f : X → Y} (hf : IsInducing f) : FrechetUrysohnSpace X := by refine ⟨fun s x hx ↦ ?_⟩ rw [hf.closure_eq_preimage_closure_image, mem_preimage, mem_closure_iff_seq_limit] at hx rcases hx with ⟨u, hus, hu⟩ choose v hv hvu using hus refine ⟨v, hv, ?_⟩ simpa only [hf.tendsto_nhds_iff, Function.comp_def, hvu] /-- Subtype of a Fréchet-Urysohn space is a Fréchet-Urysohn space. -/ instance Subtype.instFrechetUrysohnSpace [FrechetUrysohnSpace X] {p : X → Prop} : FrechetUrysohnSpace (Subtype p) := IsInducing.subtypeVal.frechetUrysohnSpace /-- In a sequential space, a set is closed iff it's sequentially closed. -/ theorem isSeqClosed_iff_isClosed [SequentialSpace X] {M : Set X} : IsSeqClosed M ↔ IsClosed M := ⟨IsSeqClosed.isClosed, IsClosed.isSeqClosed⟩ /-- The preimage of a sequentially closed set under a sequentially continuous map is sequentially closed. -/ theorem IsSeqClosed.preimage {f : X → Y} {s : Set Y} (hs : IsSeqClosed s) (hf : SeqContinuous f) : IsSeqClosed (f ⁻¹' s) := fun _x _p hx hp => hs hx (hf hp) -- A continuous function is sequentially continuous. protected theorem Continuous.seqContinuous {f : X → Y} (hf : Continuous f) : SeqContinuous f := fun _x p hx => (hf.tendsto p).comp hx /-- A sequentially continuous function defined on a sequential space is continuous. -/ protected theorem SeqContinuous.continuous [SequentialSpace X] {f : X → Y} (hf : SeqContinuous f) : Continuous f := continuous_iff_isClosed.mpr fun _s hs => (hs.isSeqClosed.preimage hf).isClosed /-- If the domain of a function is a sequential space, then continuity of this function is equivalent to its sequential continuity. -/ theorem continuous_iff_seqContinuous [SequentialSpace X] {f : X → Y} : Continuous f ↔ SeqContinuous f := ⟨Continuous.seqContinuous, SeqContinuous.continuous⟩ theorem SequentialSpace.coinduced [SequentialSpace X] {Y} (f : X → Y) : @SequentialSpace Y (.coinduced f ‹_›) := letI : TopologicalSpace Y := .coinduced f ‹_› ⟨fun _ hs ↦ isClosed_coinduced.2 (hs.preimage continuous_coinduced_rng.seqContinuous).isClosed⟩ protected theorem SequentialSpace.iSup {X} {ι : Sort} {t : ι → TopologicalSpace X} (h : ∀ i, @SequentialSpace X (t i)) : @SequentialSpace X (⨆ i, t i) := by letI : TopologicalSpace X := ⨆ i, t i refine ⟨fun s hs ↦ isClosed_iSup_iff.2 fun i ↦ ?_⟩ letI := t i exact IsSeqClosed.isClosed fun u x hus hux ↦ hs hus <\| hux.mono_right <\| nhds_mono <\| le_iSup _ _ protected theorem SequentialSpace.sup {X} {t₁ t₂ : TopologicalSpace X} (h₁ : @SequentialSpace X t₁) (h₂ : @SequentialSpace X t₂) : @SequentialSpace X (t₁ ⊔ t₂) := by rw [sup_eq_iSup] exact .iSup <\| Bool.forall_bool.2 ⟨h₂, h₁⟩ lemma Topology.IsQuotientMap.sequentialSpace [SequentialSpace X] {f : X → Y} (hf : IsQuotientMap f) : SequentialSpace Y := hf.2.symm ▸ .coinduced f /-- The quotient of a sequential space is a sequential space. -/ instance Quotient.instSequentialSpace [SequentialSpace X] {s : Setoid X} : SequentialSpace (Quotient s) := isQuotientMap_quot_mk.sequentialSpace /-- The sum (disjoint union) of two sequential spaces is a sequential space. -/ instance Sum.instSequentialSpace [SequentialSpace X] [SequentialSpace Y] : SequentialSpace (X ⊕ Y) := .sup (.coinduced Sum.inl) (.coinduced Sum.inr) /-- The disjoint union of an indexed family of sequential spaces is a sequential space. -/ instance Sigma.instSequentialSpace {ι : Type} {X : ι → Type} [∀ i, TopologicalSpace (X i)] [∀ i, SequentialSpace (X i)] : SequentialSpace (Σ i, X i) := .iSup fun _ ↦ .coinduced _ end TopologicalSpace section SeqCompact open TopologicalSpace FirstCountableTopology variable [TopologicalSpace X] theorem IsSeqCompact.subseq_of_frequently_in {s : Set X} (hs : IsSeqCompact s) {x : ℕ → X} (hx : ∃ᶠ n in atTop, x n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) := let ⟨ψ, hψ, huψ⟩ := extraction_of_frequently_atTop hx let ⟨a, a_in, φ, hφ, h⟩ := hs huψ ⟨a, a_in, ψ ∘ φ, hψ.comp hφ, h⟩ theorem SeqCompactSpace.tendsto_subseq [SeqCompactSpace X] (x : ℕ → X) : ∃ (a : X) (φ : ℕ → ℕ), StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) := let ⟨a, _, φ, mono, h⟩ := isSeqCompact_univ fun n => mem_univ (x n) ⟨a, φ, mono, h⟩ section FirstCountableTopology variable [FirstCountableTopology X] open FirstCountableTopology protected theorem IsCompact.isSeqCompact {s : Set X} (hs : IsCompact s) : IsSeqCompact s := fun _x x_in => let ⟨a, a_in, ha⟩ := hs (tendsto_principal.mpr (Eventually.of_forall x_in)) ⟨a, a_in, tendsto_subseq ha⟩ theorem IsCompact.tendsto_subseq' {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∃ᶠ n in atTop, x n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) := hs.isSeqCompact.subseq_of_frequently_in hx theorem IsCompact.tendsto_subseq {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∀ n, x n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) := hs.isSeqCompact hx -- see Note [lower instance priority] instance (priority := 100) FirstCountableTopology.seq_compact_of_compact [CompactSpace X] : SeqCompactSpace X := ⟨isCompact_univ.isSeqCompact⟩ theorem CompactSpace.tendsto_subseq [CompactSpace X] (x : ℕ → X) : ∃ (a : _) (φ : ℕ → ℕ), StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) := SeqCompactSpace.tendsto_subseq x end FirstCountableTopology section Image variable [TopologicalSpace Y] {f : X → Y} /-- Sequential compactness of sets is preserved under sequentially continuous functions. -/ theorem IsSeqCompact.image (f_cont : SeqContinuous f) {K : Set X} (K_cpt : IsSeqCompact K) : IsSeqCompact (f '' K) := by intro ys ys_in_fK choose xs xs_in_K fxs_eq_ys using ys_in_fK obtain ⟨a, a_in_K, phi, phi_mono, xs_phi_lim⟩ := K_cpt xs_in_K refine ⟨f a, mem_image_of_mem f a_in_K, phi, phi_mono, ?_⟩ exact (f_cont xs_phi_lim).congr fun x ↦ fxs_eq_ys (phi x) /-- The range of sequentially continuous function on a sequentially compact space is sequentially compact. -/ theorem IsSeqCompact.range [SeqCompactSpace X] (f_cont : SeqContinuous f) : IsSeqCompact (Set.range f) := by simpa using isSeqCompact_univ.image f_cont end Image end SeqCompact section UniformSpaceSeqCompact open uniformity open UniformSpace Prod variable [UniformSpace X] {s : Set X} theorem IsSeqCompact.exists_tendsto_of_frequently_mem (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∃ᶠ n in atTop, u n ∈ s) (huc : CauchySeq u) : ∃ x ∈ s, Tendsto u atTop (𝓝 x) := let ⟨x, hxs, _φ, φ_mono, hx⟩ := hs.subseq_of_frequently_in hu ⟨x, hxs, tendsto_nhds_of_cauchySeq_of_subseq huc φ_mono.tendsto_atTop hx⟩ theorem IsSeqCompact.exists_tendsto (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) (huc : CauchySeq u) : ∃ x ∈ s, Tendsto u atTop (𝓝 x) := hs.exists_tendsto_of_frequently_mem (Frequently.of_forall hu) huc /-- A sequentially compact set in a uniform space is totally bounded. -/ protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBounded s := by intro V V_in unfold IsSeqCompact at h contrapose! h obtain ⟨u, u_in, hu⟩ : ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ ∀ n m, m < n → u m ∉ ball (u n) V := by simp only [not_subset, mem_iUnion₂, not_exists, exists_prop] at h simpa only [forall_and, forall_mem_image, not_and] using seq_of_forall_finite_exists h refine ⟨u, u_in, fun x _ φ hφ huφ => ?_⟩ obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V := huφ.cauchySeq.mem_entourage V_in exact hu (φ <\| N + 1) (φ N) (hφ <\| Nat.lt_add_one N) (hN (N + 1) N N.le_succ le_rfl) variable [IsCountablyGenerated (𝓤 X)] /-- A sequentially compact set in a uniform space with countably generated uniformity filter is complete. -/ protected theorem IsSeqCompact.isComplete (hs : IsSeqCompact s) : IsComplete s := fun l hl hls => by have := hl.1 rcases exists_antitone_basis (𝓤 X) with ⟨V, hV⟩ choose W hW hWV using fun n => comp_mem_uniformity_sets (hV.mem n) have hWV' : ∀ n, W n ⊆ V n := fun n ⟨x, y⟩ hx => @hWV n (x, y) ⟨x, refl_mem_uniformity <\| hW _, hx⟩ obtain ⟨t, ht_anti, htl, htW, hts⟩ : ∃ t : ℕ → Set X, Antitone t ∧ (∀ n, t n ∈ l) ∧ (∀ n, t n ×ˢ t n ⊆ W n) ∧ ∀ n, t n ⊆ s := by have : ∀ n, ∃ t ∈ l, t ×ˢ t ⊆ W n ∧ t ⊆ s := by rw [le_principal_iff] at hls have : ∀ n, W n ∩ s ×ˢ s ∈ l ×ˢ l := fun n => inter_mem (hl.2 (hW n)) (prod_mem_prod hls hls) simpa only [l.basis_sets.prod_self.mem_iff, true_imp_iff, subset_inter_iff, prod_self_subset_prod_self, and_assoc] using this choose t htl htW hts using this have : ∀ n : ℕ, ⋂ k ≤ n, t k ⊆ t n := fun n => by apply iInter₂_subset; rfl exact ⟨fun n => ⋂ k ≤ n, t k, fun m n h => biInter_subset_biInter_left fun k (hk : k ≤ m) => hk.trans h, fun n => (biInter_mem (finite_le_nat n)).2 fun k _ => htl k, fun n => (prod_mono (this n) (this n)).trans (htW n), fun n => (this n).trans (hts n)⟩ choose u hu using fun n => Filter.nonempty_of_mem (htl n) have huc : CauchySeq u := hV.toHasBasis.cauchySeq_iff.2 fun N _ => ⟨N, fun m hm n hn => hWV' _ <\| @htW N (_, _) ⟨ht_anti hm (hu _), ht_anti hn (hu _)⟩⟩ rcases hs.exists_tendsto (fun n => hts n (hu n)) huc with ⟨x, hxs, hx⟩ refine ⟨x, hxs, (nhds_basis_uniformity' hV.toHasBasis).ge_iff.2 fun N _ => ?_⟩ obtain ⟨n, hNn, hn⟩ : ∃ n, N ≤ n ∧ u n ∈ ball x (W N) := ((eventually_ge_atTop N).and (hx <\| ball_mem_nhds x (hW N))).exists refine mem_of_superset (htl n) fun y hy => hWV N ⟨u n, hn, htW N ?_⟩ exact ⟨ht_anti hNn (hu n), ht_anti hNn hy⟩ /-- If `𝓤 β` is countably generated, then any sequentially compact set is compact. -/ protected theorem IsSeqCompact.isCompact (hs : IsSeqCompact s) : IsCompact s := isCompact_iff_totallyBounded_isComplete.2 ⟨hs.totallyBounded, hs.isComplete⟩ /-- A version of Bolzano-Weierstrass: in a uniform space with countably generated uniformity filter (e.g., in a metric space), a set is compact if and only if it is sequentially compact. -/ protected theorem UniformSpace.isCompact_iff_isSeqCompact : IsCompact s ↔ IsSeqCompact s := ⟨fun H => H.isSeqCompact, fun H => H.isCompact⟩ theorem UniformSpace.compactSpace_iff_seqCompactSpace : CompactSpace X ↔ SeqCompactSpace X := by simp only [← isCompact_univ_iff, seqCompactSpace_iff, UniformSpace.isCompact_iff_isSeqCompact] end UniformSpaceSeqCompact
.lake/packages/mathlib/Mathlib/Topology/IsClosedRestrict.lean	import Mathlib.Topology.Maps.Proper.Basic /-! # Restriction of a closed compact set in a product space to a set of coordinates We show that the image of a compact closed set `s` in a product `Π i : ι, α i` by the restriction to a subset of coordinates `S : Set ι` is a closed set. The idea of the proof is to use `isClosedMap_snd_of_compactSpace`, which is the fact that if `X` is a compact topological space, then `Prod.snd : X × Y → Y` is a closed map. We remark that `s` is included in the set `Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s)`, and we build a homeomorphism `Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s) ≃ₜ Sᶜ.restrict '' s × Π i : S, α i`. `Sᶜ.restrict '' s` is a compact space since `s` is compact, and the lemma applies, with `X = Sᶜ.restrict '' s` and `Y = Π i : S, α i`. -/ open Set variable {ι : Type} {α : ι → Type} {s : Set (Π i, α i)} {i : ι} {S : Set ι} namespace Topology open Classical in /-- Given a set in a product space `s : Set (Π j, α j)` and a set of coordinates `S : Set ι`, `Sᶜ.restrict '' s × (Π i : S, α i)` is the set of functions that coincide with an element of `s` on `Sᶜ` and are arbitrary on `S`. `reorderRestrictProd` sends a term of that type to `Π j, α j` by looking for the value at `j` in one part of the product or the other depending on whether `j` is in `S` or not. -/ noncomputable def reorderRestrictProd (S : Set ι) (s : Set (Π j, α j)) (p : Sᶜ.restrict '' s × (Π i : S, α i)) : Π j, α j := fun j ↦ if h : j ∈ S then (p.2 : Π j : ↑(S : Set ι), α j) ⟨j, h⟩ else (p.1 : Π j : ↑(Sᶜ : Set ι), α j) ⟨j, h⟩ @[simp] lemma reorderRestrictProd_of_mem (p : Sᶜ.restrict '' s × (Π i : S, α i)) (j : S) : reorderRestrictProd S s p j = (p.2 : Π j : ↑(S : Set ι), α j) j := by have hj : ↑j ∈ S := j.prop simp [reorderRestrictProd, hj] @[simp] lemma reorderRestrictProd_of_compl (p : Sᶜ.restrict '' s × (Π i : S, α i)) (j : (Sᶜ : Set ι)) : reorderRestrictProd S s p j = (p.1 : Π j : ↑(Sᶜ : Set ι), α j) j := by have hj : ↑j ∉ S := j.prop simp [reorderRestrictProd, hj] @[simp] lemma restrict_compl_reorderRestrictProd (p : Sᶜ.restrict '' s × (Π i : S, α i)) : Sᶜ.restrict (reorderRestrictProd S s p) = p.1 := by ext; simp lemma continuous_reorderRestrictProd [∀ i, TopologicalSpace (α i)] : Continuous (reorderRestrictProd S s) := by refine continuous_pi fun j ↦ ?_ simp only [reorderRestrictProd] split_ifs with h · fun_prop · exact ((continuous_apply _).comp continuous_subtype_val).comp continuous_fst lemma reorderRestrictProd_mem_preimage_image_restrict (p : Sᶜ.restrict '' s × (Π i : S, α i)) : reorderRestrictProd S s p ∈ Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s) := by obtain ⟨y, hy_mem_s, hy_eq⟩ := p.1.2 exact ⟨y, hy_mem_s, hy_eq.trans (restrict_compl_reorderRestrictProd p).symm⟩ @[simp] lemma reorderRestrictProd_restrict_compl (x : Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s)) : reorderRestrictProd S s ⟨⟨Sᶜ.restrict x, x.2⟩, fun i ↦ (x : Π j, α j) i⟩ = (x : Π j, α j) := by ext; simp [reorderRestrictProd] /-- Homeomorphism between the set of functions that coincide with a given set of functions away from a given set `S`, and dependent functions away from `S` times any value on `S`. -/ noncomputable def _root_.Homeomorph.preimageImageRestrict (α : ι → Type*) [∀ i, TopologicalSpace (α i)] (S : Set ι) (s : Set (Π j, α j)) : Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s) ≃ₜ Sᶜ.restrict '' s × (Π i : S, α i) where toFun x := ⟨⟨Sᶜ.restrict x, x.2⟩, fun i ↦ (x : Π j, α j) i⟩ invFun p := ⟨reorderRestrictProd S s p, reorderRestrictProd_mem_preimage_image_restrict p⟩ left_inv x := by ext; simp right_inv p := by ext <;> simp continuous_toFun := by refine Continuous.prodMk ?_ ?_ · exact ((Pi.continuous_restrict _).comp continuous_subtype_val).subtype_mk _ · rw [continuous_pi_iff] exact fun _ ↦ (continuous_apply _).comp continuous_subtype_val continuous_invFun := continuous_reorderRestrictProd.subtype_mk _ /-- The image by `preimageImageRestrict α S s` of `s` seen as a set of `Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s)` is a set of `Sᶜ.restrict '' s × (Π i : S, α i)`, and the image of that set by `Prod.snd` is `S.restrict '' s`. Used in `IsCompact.isClosed_image_restrict` to prove that the restriction of a compact closed set in a product space to a set of coordinates is closed. -/ lemma image_snd_preimageImageRestrict [∀ i, TopologicalSpace (α i)] : Prod.snd '' (Homeomorph.preimageImageRestrict α S s '' ((fun (x : Sᶜ.restrict ⁻¹' (Sᶜ.restrict '' s)) ↦ (x : Π j, α j)) ⁻¹' s)) = S.restrict '' s := by ext x simp only [Homeomorph.preimageImageRestrict, Homeomorph.homeomorph_mk_coe, Equiv.coe_fn_mk, mem_image, mem_preimage, Subtype.exists, exists_and_left, Prod.exists, Prod.mk.injEq, exists_and_right, exists_eq_right, Subtype.mk.injEq, exists_prop] constructor · rintro ⟨y, _, z, hz_mem, _, hzx⟩ exact ⟨z, hz_mem, hzx⟩ · rintro ⟨z, hz_mem, hzx⟩ exact ⟨Sᶜ.restrict z, mem_image_of_mem Sᶜ.restrict hz_mem, z, hz_mem, ⟨⟨⟨z, hz_mem, rfl⟩, rfl⟩, hzx⟩⟩ end Topology section IsClosed variable [∀ i, TopologicalSpace (α i)] /-- The restriction of a compact closed set in a product space to a set of coordinates is closed. -/ theorem IsCompact.isClosed_image_restrict (S : Set ι) (hs_compact : IsCompact s) (hs_closed : IsClosed s) : IsClosed (S.restrict '' s) := by rw [← Topology.image_snd_preimageImageRestrict] have : CompactSpace (Sᶜ.restrict '' s) := isCompact_iff_compactSpace.mp (hs_compact.image (Pi.continuous_restrict _)) refine isClosedMap_snd_of_compactSpace _ ?_ rw [Homeomorph.isClosed_image] exact hs_closed.preimage continuous_subtype_val lemma isClosedMap_restrict_of_compactSpace [∀ i, CompactSpace (α i)] : IsClosedMap (S.restrict : (Π i, α i) → _) := fun s hs ↦ by classical have : S.restrict (π := α) = Prod.fst ∘ (Homeomorph.piEquivPiSubtypeProd S α) := rfl rw [this, image_comp] exact isClosedMap_fst_of_compactSpace _ <\| (Homeomorph.isClosed_image _).mpr hs lemma IsClosed.isClosed_image_eval (i : ι) (hs_compact : IsCompact s) (hs_closed : IsClosed s) : IsClosed ((fun x ↦ x i) '' s) := by suffices IsClosed (Set.restrict {i} '' s) by have : Homeomorph.piUnique _ ∘ Set.restrict {i} = fun (x : Π j, α j) ↦ x i := rfl rwa [← this, image_comp, Homeomorph.isClosed_image (Homeomorph.piUnique _)] exact hs_compact.isClosed_image_restrict {i} hs_closed end IsClosed
.lake/packages/mathlib/Mathlib/Topology/Constructible.lean	import Mathlib.Order.BooleanSubalgebra import Mathlib.Topology.Compactness.Bases import Mathlib.Topology.LocalAtTarget import Mathlib.Topology.QuasiSeparated import Mathlib.Topology.Spectral.Hom import Mathlib.Topology.Spectral.Prespectral /-! # Constructible sets This file defines constructible sets, which are morally sets in a topological space which we can make out of finite unions and intersections of open and closed sets. Precisely, constructible sets are the Boolean subalgebra generated by open retrocompact sets, where a set is retrocompact if its intersection with every compact open set is compact. In a locally Noetherian space, all sets are retrocompact, in which case this Boolean subalgebra is simply the one generated by the open sets. Constructible sets are useful because the image of a constructible set under a finitely presented morphism of schemes is a constructible set (and this is not true at the level of varieties). ## Main declarations * `IsRetrocompact`: Predicate for a set to be retrocompact, namely to have its intersection with every compact open be compact. * `IsConstructible`: Predicate for a set to be constructible, namely to belong to the Boolean subalgebra generated by open retrocompact sets. * `IsLocallyConstructible`: Predicate for a set to be locally constructible, namely to be partitionable along an open cover such that each of its parts is constructible in the respective open subspace. -/ open Set TopologicalSpace Topology open scoped Set.Notation variable {ι : Sort} {X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s t U : Set X} {a : X} /-! ### retrocompact sets -/ /-- A retrocompact set is a set whose intersection with every compact open is compact. -/ @[stacks 005A] def IsRetrocompact (s : Set X) : Prop := ∀ ⦃U⦄, IsCompact U → IsOpen U → IsCompact (s ∩ U) @[simp] lemma IsRetrocompact.empty : IsRetrocompact (∅ : Set X) := by simp [IsRetrocompact] @[simp] lemma IsRetrocompact.univ : IsRetrocompact (univ : Set X) := by simp +contextual [IsRetrocompact] @[simp] lemma IsRetrocompact.singleton : IsRetrocompact {a} := fun _ _ _ ↦ Subsingleton.singleton_inter.isCompact lemma IsRetrocompact.union (hs : IsRetrocompact s) (ht : IsRetrocompact t) : IsRetrocompact (s ∪ t : Set X) := fun _U hUcomp hUopen ↦ union_inter_distrib_right .. ▸ (hs hUcomp hUopen).union (ht hUcomp hUopen) private lemma supClosed_isRetrocompact : SupClosed {s : Set X \| IsRetrocompact s} := fun _s hs _t ht ↦ hs.union ht lemma IsRetrocompact.finsetSup {ι : Type} {s : Finset ι} {t : ι → Set X} (ht : ∀ i ∈ s, IsRetrocompact (t i)) : IsRetrocompact (s.sup t) := by induction s using Finset.cons_induction with \| empty => simp \| cons i s ih hi => rw [Finset.sup_cons] exact (ht _ <\| by simp).union <\| hi <\| Finset.forall_of_forall_cons ht set_option linter.docPrime false in lemma IsRetrocompact.finsetSup' {ι : Type} {s : Finset ι} {hs} {t : ι → Set X} (ht : ∀ i ∈ s, IsRetrocompact (t i)) : IsRetrocompact (s.sup' hs t) := by rw [Finset.sup'_eq_sup]; exact .finsetSup ht lemma IsRetrocompact.iUnion [Finite ι] {f : ι → Set X} (hf : ∀ i, IsRetrocompact (f i)) : IsRetrocompact (⋃ i, f i) := supClosed_isRetrocompact.iSup_mem .empty hf lemma IsRetrocompact.sUnion {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsRetrocompact s) : IsRetrocompact (⋃₀ S) := supClosed_isRetrocompact.sSup_mem hS .empty hS' lemma IsRetrocompact.biUnion {ι : Type} {f : ι → Set X} {t : Set ι} (ht : t.Finite) (hf : ∀ i ∈ t, IsRetrocompact (f i)) : IsRetrocompact (⋃ i ∈ t, f i) := supClosed_isRetrocompact.biSup_mem ht .empty hf section T2Space variable [T2Space X] lemma IsRetrocompact.inter (hs : IsRetrocompact s) (ht : IsRetrocompact t) : IsRetrocompact (s ∩ t : Set X) := fun _U hUcomp hUopen ↦ inter_inter_distrib_right .. ▸ (hs hUcomp hUopen).inter (ht hUcomp hUopen) private lemma infClosed_isRetrocompact : InfClosed {s : Set X \| IsRetrocompact s} := fun _s hs _t ht ↦ hs.inter ht lemma IsRetrocompact.finsetInf {ι : Type} {s : Finset ι} {t : ι → Set X} (ht : ∀ i ∈ s, IsRetrocompact (t i)) : IsRetrocompact (s.inf t) := by induction s using Finset.cons_induction with \| empty => simp \| cons i s ih hi => rw [Finset.inf_cons] exact (ht _ <\| by simp).inter <\| hi <\| Finset.forall_of_forall_cons ht set_option linter.docPrime false in lemma IsRetrocompact.finsetInf' {ι : Type} {s : Finset ι} {hs} {t : ι → Set X} (ht : ∀ i ∈ s, IsRetrocompact (t i)) : IsRetrocompact (s.inf' hs t) := by rw [Finset.inf'_eq_inf]; exact .finsetInf ht lemma IsRetrocompact.iInter [Finite ι] {f : ι → Set X} (hf : ∀ i, IsRetrocompact (f i)) : IsRetrocompact (⋂ i, f i) := infClosed_isRetrocompact.iInf_mem .univ hf lemma IsRetrocompact.sInter {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsRetrocompact s) : IsRetrocompact (⋂₀ S) := infClosed_isRetrocompact.sInf_mem hS .univ hS' lemma IsRetrocompact.biInter {ι : Type} {f : ι → Set X} {t : Set ι} (ht : t.Finite) (hf : ∀ i ∈ t, IsRetrocompact (f i)) : IsRetrocompact (⋂ i ∈ t, f i) := infClosed_isRetrocompact.biInf_mem ht .univ hf end T2Space lemma IsRetrocompact.inter_isOpen (hs : IsRetrocompact s) (ht : IsRetrocompact t) (htopen : IsOpen t) : IsRetrocompact (s ∩ t : Set X) := fun _U hUcomp hUopen ↦ inter_assoc .. ▸ hs (ht hUcomp hUopen) (htopen.inter hUopen) lemma IsRetrocompact.isOpen_inter (hs : IsRetrocompact s) (ht : IsRetrocompact t) (hsopen : IsOpen s) : IsRetrocompact (s ∩ t : Set X) := inter_comm .. ▸ ht.inter_isOpen hs hsopen lemma IsRetrocompact_iff_isSpectralMap_subtypeVal : IsRetrocompact s ↔ IsSpectralMap (Subtype.val : s → X) := by refine ⟨fun hs ↦ ⟨continuous_subtype_val, fun t htopen htcomp ↦ ?_⟩, fun hs t htcomp htopen ↦ ?_⟩ · rw [IsEmbedding.subtypeVal.isCompact_iff, image_preimage_eq_inter_range, Subtype.range_coe_subtype, setOf_mem_eq, inter_comm] exact hs htcomp htopen · simpa using (hs.isCompact_preimage_of_isOpen htopen htcomp).image continuous_subtype_val @[stacks 005B] lemma IsRetrocompact.image_of_isEmbedding (hs : IsRetrocompact s) (hfemb : IsEmbedding f) (hfcomp : IsRetrocompact (range f)) : IsRetrocompact (f '' s) := by rintro U hUcomp hUopen rw [← image_inter_preimage, ← hfemb.isCompact_iff] refine hs ?_ <\| hUopen.preimage hfemb.continuous rw [hfemb.isCompact_iff, image_preimage_eq_inter_range, inter_comm] exact hfcomp hUcomp hUopen @[stacks 005J "Extracted from the proof"] lemma IsRetrocompact.preimage_of_isOpenEmbedding {s : Set Y} (hf : IsOpenEmbedding f) (hs : IsRetrocompact s) : IsRetrocompact (f ⁻¹' s) := by rintro U hUcomp hUopen rw [hf.isCompact_iff, image_preimage_inter] exact hs (hUcomp.image hf.continuous) <\| hf.isOpenMap _ hUopen @[stacks 09YE "Extracted from the proof"] lemma IsRetrocompact.preimage_of_isClosedEmbedding {s : Set Y} (hf : IsClosedEmbedding f) (hf' : IsCompact (range f)ᶜ) (hs : IsRetrocompact s) : IsRetrocompact (f ⁻¹' s) := by rintro U hUcomp hUopen have hfUopen : IsOpen (f '' U ∪ (range f)ᶜ) := by simpa [← range_diff_image hf.injective, sdiff_eq, compl_inter, union_comm] using (hf.isClosedMap _ hUopen.isClosed_compl).isOpen_compl have hfUcomp : IsCompact (f '' U ∪ (range f)ᶜ) := (hUcomp.image hf.continuous).union hf' simpa [inter_union_distrib_left, inter_left_comm, inter_eq_right.2 (image_subset_range ..), hf.isCompact_iff, image_preimage_inter] using (hs hfUcomp hfUopen).inter_left hf.isClosed_range /-! ### Constructible sets -/ namespace Topology /-- A constructible set is a set that can be written as the finite union/finite intersection/complement of open retrocompact sets. In other words, constructible sets form the Boolean subalgebra generated by open retrocompact sets. -/ def IsConstructible (s : Set X) : Prop := s ∈ BooleanSubalgebra.closure {U \| IsOpen U ∧ IsRetrocompact U} @[simp] protected lemma IsConstructible.empty : IsConstructible (∅ : Set X) := BooleanSubalgebra.bot_mem @[simp] protected lemma IsConstructible.univ : IsConstructible (univ : Set X) := BooleanSubalgebra.top_mem lemma IsConstructible.union : IsConstructible s → IsConstructible t → IsConstructible (s ∪ t) := BooleanSubalgebra.sup_mem lemma IsConstructible.inter : IsConstructible s → IsConstructible t → IsConstructible (s ∩ t) := BooleanSubalgebra.inf_mem lemma IsConstructible.sdiff : IsConstructible s → IsConstructible t → IsConstructible (s \ t) := BooleanSubalgebra.sdiff_mem lemma IsConstructible.himp : IsConstructible s → IsConstructible t → IsConstructible (s ⇨ t) := BooleanSubalgebra.himp_mem @[simp] lemma isConstructible_compl : IsConstructible sᶜ ↔ IsConstructible s := BooleanSubalgebra.compl_mem_iff alias ⟨IsConstructible.of_compl, IsConstructible.compl⟩ := isConstructible_compl lemma IsConstructible.iUnion [Finite ι] {f : ι → Set X} (hf : ∀ i, IsConstructible (f i)) : IsConstructible (⋃ i, f i) := BooleanSubalgebra.iSup_mem hf lemma IsConstructible.iInter [Finite ι] {f : ι → Set X} (hf : ∀ i, IsConstructible (f i)) : IsConstructible (⋂ i, f i) := BooleanSubalgebra.iInf_mem hf lemma IsConstructible.sUnion {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsConstructible s) : IsConstructible (⋃₀ S) := BooleanSubalgebra.sSup_mem hS hS' lemma IsConstructible.sInter {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsConstructible s) : IsConstructible (⋂₀ S) := BooleanSubalgebra.sInf_mem hS hS' lemma IsConstructible.biUnion {ι : Type} {f : ι → Set X} {t : Set ι} (ht : t.Finite) (hf : ∀ i ∈ t, IsConstructible (f i)) : IsConstructible (⋃ i ∈ t, f i) := BooleanSubalgebra.biSup_mem ht hf lemma IsConstructible.biInter {ι : Type} {f : ι → Set X} {t : Set ι} (ht : t.Finite) (hf : ∀ i ∈ t, IsConstructible (f i)) : IsConstructible (⋂ i ∈ t, f i) := BooleanSubalgebra.biInf_mem ht hf lemma _root_.IsRetrocompact.isConstructible (hUopen : IsOpen U) (hUcomp : IsRetrocompact U) : IsConstructible U := BooleanSubalgebra.subset_closure ⟨hUopen, hUcomp⟩ /-- An induction principle for constructible sets. If `p` holds for all open retrocompact sets, and is preserved under union and complement, then `p` holds for all constructible sets. -/ @[elab_as_elim] lemma IsConstructible.empty_union_induction {p : ∀ s : Set X, IsConstructible s → Prop} (open_retrocompact : ∀ U (hUopen : IsOpen U) (hUcomp : IsRetrocompact U), p U (BooleanSubalgebra.subset_closure ⟨hUopen, hUcomp⟩)) (union : ∀ s hs t ht, p s hs → p t ht → p (s ∪ t) (hs.union ht)) (compl : ∀ s hs, p s hs → p sᶜ hs.compl) {s} (hs : IsConstructible s) : p s hs := by induction hs using BooleanSubalgebra.closure_bot_sup_induction with \| mem U hU => exact open_retrocompact _ hU.1 hU.2 \| bot => exact open_retrocompact _ isOpen_empty .empty \| sup s hs t ht hs' ht' => exact union _ _ _ _ hs' ht' \| compl s hs hs' => exact compl _ _ hs' /-- If `f` is continuous and is such that preimages of open retrocompact sets are retrocompact, then preimages of constructible sets are constructible. -/ @[stacks 005I] lemma IsConstructible.preimage {s : Set Y} (hfcont : Continuous f) (hf : ∀ s, IsOpen s → IsRetrocompact s → IsRetrocompact (f ⁻¹' s)) (hs : IsConstructible s) : IsConstructible (f ⁻¹' s) := by induction hs using IsConstructible.empty_union_induction with \| open_retrocompact U hUopen hUcomp => exact (hf _ hUopen hUcomp).isConstructible <\| hUopen.preimage hfcont \| union s hs t ht hs' ht' => rw [preimage_union]; exact hs'.union ht' \| compl s hs hs' => rw [preimage_compl]; exact hs'.compl @[stacks 005J] lemma IsConstructible.preimage_of_isOpenEmbedding {s : Set Y} (hf : IsOpenEmbedding f) (hs : IsConstructible s) : IsConstructible (f ⁻¹' s) := hs.preimage hf.continuous fun _t _ ht ↦ ht.preimage_of_isOpenEmbedding hf @[stacks 09YE] lemma IsConstructible.preimage_of_isClosedEmbedding {s : Set Y} (hf : IsClosedEmbedding f) (hf' : IsCompact (range f)ᶜ) (hs : IsConstructible s) : IsConstructible (f ⁻¹' s) := hs.preimage hf.continuous fun _t _ ht ↦ ht.preimage_of_isClosedEmbedding hf hf' @[stacks 09YD] lemma IsConstructible.image_of_isOpenEmbedding (hfopen : IsOpenEmbedding f) (hfcomp : IsRetrocompact (range f)) (hs : IsConstructible s) : IsConstructible (f '' s) := by induction hs using IsConstructible.empty_union_induction with \| open_retrocompact U hUopen hUcomp => exact (hUcomp.image_of_isEmbedding hfopen.isEmbedding hfcomp).isConstructible <\| hfopen.isOpenMap _ hUopen \| union s hs t ht hs' ht' => rw [image_union]; exact hs'.union ht' \| compl s hs hs' => rw [← range_diff_image hfopen.injective] exact (hfcomp.isConstructible hfopen.isOpen_range).sdiff hs' @[stacks 09YG] lemma IsConstructible.image_of_isClosedEmbedding (hf : IsClosedEmbedding f) (hfcomp : IsRetrocompact (range f)ᶜ) (hs : IsConstructible s) : IsConstructible (f '' s) := by induction hs using IsConstructible.empty_union_induction with \| open_retrocompact U hUopen hUcomp => have hfU : IsOpen (f '' U ∪ (range f)ᶜ) := by simpa [← range_diff_image hf.injective, sdiff_eq, compl_inter, union_comm] using (hf.isClosedMap _ hUopen.isClosed_compl).isOpen_compl suffices h : IsRetrocompact (f '' U ∪ (range f)ᶜ) by simpa [union_inter_distrib_right, inter_eq_left.2 (image_subset_range ..)] using (h.isConstructible hfU).sdiff (hfcomp.isConstructible hf.isClosed_range.isOpen_compl) rintro V hVcomp hVopen rw [union_inter_distrib_right, ← image_inter_preimage] exact ((hUcomp (hf.isCompact_preimage hVcomp) (hVopen.preimage hf.continuous)).image hf.continuous).union <\| hfcomp hVcomp hVopen \| union s hs t ht hs' ht' => rw [image_union]; exact hs'.union ht' \| compl s hs hs' => rw [← range_diff_image hf.injective] exact (hfcomp.isConstructible hf.isClosed_range.isOpen_compl).of_compl.sdiff hs' lemma isConstructible_preimage_iff_of_isOpenEmbedding {s : Set Y} (hf : IsOpenEmbedding f) (hfcomp : IsRetrocompact (range f)) (hsf : s ⊆ range f) : IsConstructible (f ⁻¹' s) ↔ IsConstructible s where mp hs := by simpa [image_preimage_eq_range_inter, inter_eq_right.2 hsf] using hs.image_of_isOpenEmbedding hf hfcomp mpr := .preimage_of_isOpenEmbedding hf lemma _root_.QuasiSeparatedSpace.of_isOpenCover {ι : Type} {U : ι → Opens X} (hU : IsOpenCover U) (h₁ : ∀ i, IsRetrocompact (X := X) (U i)) (h₂ : ∀ i, IsQuasiSeparated (α := X) (U i)) : QuasiSeparatedSpace X where inter_isCompact V₁ V₂ ho₁ hc₁ ho₂ hc₂ := by obtain ⟨t, ht⟩ := hc₁.elim_finite_subcover _ (fun i ↦ (U i).2) (by simp [hU.iSup_set_eq_univ]) convert t.isCompact_biUnion fun i _ ↦ h₂ i _ _ Set.inter_subset_left ((U i).2.inter ho₁) (h₁ i hc₁ ho₁) Set.inter_subset_left ((U i).2.inter ho₂) (h₁ i hc₂ ho₂) apply subset_antisymm · rintro x ⟨hx₁, hx₂⟩ obtain ⟨i, hi, hxi⟩ := Set.mem_iUnion₂.mp (ht hx₁) exact Set.mem_iUnion₂.mpr ⟨i, hi, by simpa []⟩ · aesop (add simp Set.subset_def) section CompactSpace variable [CompactSpace X] {P : ∀ s : Set X, IsConstructible s → Prop} {B : Set (Set X)} {b : ι → Set X} lemma _root_.IsRetrocompact.isCompact (hs : IsRetrocompact s) : IsCompact s := by simpa using hs CompactSpace.isCompact_univ variable [QuasiSeparatedSpace X] omit [CompactSpace X] in lemma _root_.IsCompact.isRetrocompact (hU' : IsCompact U) (hU : IsOpen U) : IsRetrocompact U := fun _ hV' hV ↦ hU'.inter_of_isOpen hV' hU hV omit [CompactSpace X] in lemma _root_.IsCompact.isConstructible (hU' : IsCompact U) (hU : IsOpen U) : IsConstructible U := (hU'.isRetrocompact hU).isConstructible hU @[stacks 0069 "Iff form of (2). Note that Stacks doesn't define quasi-separated spaces."] lemma _root_.QuasiSeparatedSpace.isRetrocompact_iff_isCompact (hU : IsOpen U) : IsRetrocompact U ↔ IsCompact U := ⟨IsRetrocompact.isCompact, (IsCompact.isRetrocompact · hU)⟩ @[elab_as_elim] lemma IsConstructible.induction_of_isTopologicalBasis {ι : Type} [Nonempty ι] (b : ι → Set X) (basis : IsTopologicalBasis (range b)) (isCompact_basis : ∀ i, IsCompact (b i)) (sdiff : ∀ i s (hs : Set.Finite s), P (b i \ ⋃ j ∈ s, b j) (((isCompact_basis _).isConstructible (basis.isOpen ⟨i, rfl⟩)).sdiff <\| .biUnion hs fun _ _ ↦ ((isCompact_basis _).isConstructible (basis.isOpen ⟨_, rfl⟩)))) (union : ∀ s hs t ht, P s hs → P t ht → P (s ∪ t) (hs.union ht)) (s : Set X) (hs : IsConstructible s) : P s hs := by induction s, hs using BooleanSubalgebra.closure_sdiff_sup_induction with \| isSublattice => exact ⟨fun s hs t ht ↦ ⟨hs.1.union ht.1, hs.2.union ht.2⟩, fun s hs t ht ↦ ⟨hs.1.inter ht.1, hs.2.inter_isOpen ht.2 ht.1⟩⟩ \| bot_mem => exact ⟨isOpen_empty, .empty⟩ \| top_mem => exact ⟨isOpen_univ, .univ⟩ \| sdiff U hU V hV => have := isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis _ basis isCompact_basis obtain ⟨s, hs, rfl⟩ := (this _).1 ⟨hU.2.isCompact, hU.1⟩ obtain ⟨t, ht, rfl⟩ := (this _).1 ⟨hV.2.isCompact, hV.1⟩ simp_rw [iUnion_diff] induction s, hs using Set.Finite.induction_on with \| empty => simpa using sdiff (Classical.arbitrary _) {Classical.arbitrary _} \| @insert i s hi hs ih => simp_rw [biUnion_insert] exact union _ _ _ (.biUnion hs fun i _ ↦ ((isCompact_basis _).isConstructible (basis.isOpen ⟨i, rfl⟩)).sdiff <\| .biUnion ht fun j _ ↦ (isCompact_basis _).isConstructible (basis.isOpen ⟨_, rfl⟩)) (sdiff _ _ ht) (ih ⟨isOpen_biUnion fun _ _ ↦ basis.isOpen ⟨_, rfl⟩, .biUnion hs fun i _ ↦ (isCompact_basis _).isRetrocompact (basis.isOpen ⟨i, rfl⟩)⟩) \| sup s _ t _ hs' ht' => exact union _ _ _ _ hs' ht' end CompactSpace /-! ### Locally constructible sets -/ /-- A set in a topological space is locally constructible, if every point has a neighborhood on which the set is constructible. -/ @[stacks 005G] def IsLocallyConstructible (s : Set X) : Prop := ∀ x, ∃ U ∈ 𝓝 x, IsOpen U ∧ IsConstructible (U ↓∩ s) lemma IsConstructible.isLocallyConstructible (hs : IsConstructible s) : IsLocallyConstructible s := fun _ ↦ ⟨univ, by simp, by simp, (isConstructible_preimage_iff_of_isOpenEmbedding isOpen_univ.isOpenEmbedding_subtypeVal (by simp) (by simp)).2 hs⟩ lemma _root_.IsRetrocompact.isLocallyConstructible (hUopen : IsOpen U) (hUcomp : IsRetrocompact U) : IsLocallyConstructible U := (hUcomp.isConstructible hUopen).isLocallyConstructible @[simp] protected lemma IsLocallyConstructible.empty : IsLocallyConstructible (∅ : Set X) := IsConstructible.empty.isLocallyConstructible @[simp] protected lemma IsLocallyConstructible.univ : IsLocallyConstructible (univ : Set X) := IsConstructible.univ.isLocallyConstructible lemma IsLocallyConstructible.inter (hs : IsLocallyConstructible s) (ht : IsLocallyConstructible t) : IsLocallyConstructible (s ∩ t) := by rintro x obtain ⟨U, hxU, hU, hsU⟩ := hs x obtain ⟨V, hxV, hV, htV⟩ := ht x refine ⟨U ∩ V, Filter.inter_mem hxU hxV, hU.inter hV, ?_⟩ change IsConstructible (inclusion inter_subset_left ⁻¹' (U ↓∩ s) ∩ inclusion inter_subset_right ⁻¹' (V ↓∩ t)) exact .inter (hsU.preimage_of_isOpenEmbedding <\| .inclusion _ <\| .preimage continuous_subtype_val <\| hU.inter hV) (htV.preimage_of_isOpenEmbedding <\| .inclusion _ <\| .preimage continuous_subtype_val <\| hU.inter hV ) lemma IsLocallyConstructible.finsetInf {ι : Type} {s : Finset ι} {t : ι → Set X} (ht : ∀ i ∈ s, IsLocallyConstructible (t i)) : IsLocallyConstructible (s.inf t) := by induction s using Finset.cons_induction with \| empty => simp \| cons i s ih hi => rw [Finset.inf_cons] exact (ht _ <\| by simp).inter <\| hi <\| Finset.forall_of_forall_cons ht set_option linter.docPrime false in lemma IsLocallyConstructible.finsetInf' {ι : Type} {s : Finset ι} {hs} {t : ι → Set X} (ht : ∀ i ∈ s, IsLocallyConstructible (t i)) : IsLocallyConstructible (s.inf' hs t) := by rw [Finset.inf'_eq_inf]; exact .finsetInf ht private lemma infClosed_isLocallyConstructible : InfClosed {s : Set X \| IsLocallyConstructible s} := fun _s hs _t ht ↦ hs.inter ht lemma IsLocallyConstructible.iInter [Finite ι] {f : ι → Set X} (hf : ∀ i, IsLocallyConstructible (f i)) : IsLocallyConstructible (⋂ i, f i) := infClosed_isLocallyConstructible.iInf_mem .univ hf lemma IsLocallyConstructible.sInter {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsLocallyConstructible s) : IsLocallyConstructible (⋂₀ S) := infClosed_isLocallyConstructible.sInf_mem hS .univ hS' lemma IsLocallyConstructible.union (hs : IsLocallyConstructible s) (ht : IsLocallyConstructible t) : IsLocallyConstructible (s ∪ t) := by rintro x obtain ⟨U, hxU, hU, hsU⟩ := hs x obtain ⟨V, hxV, hV, htV⟩ := ht x refine ⟨U ∩ V, Filter.inter_mem hxU hxV, hU.inter hV, ?_⟩ have : (U ∩ V) ↓∩ (s ∪ t) = inclusion inter_subset_left ⁻¹' (U ↓∩ s) ∪ inclusion inter_subset_right ⁻¹' (V ↓∩ t) := by ext; simp rw [this] exact .union (hsU.preimage_of_isOpenEmbedding <\| .inclusion _ <\| .preimage continuous_subtype_val <\| hU.inter hV) (htV.preimage_of_isOpenEmbedding <\| .inclusion _ <\| .preimage continuous_subtype_val <\| hU.inter hV ) lemma IsLocallyConstructible.iUnion [Finite ι] {f : ι → Set X} (hf : ∀ i, IsLocallyConstructible (f i)) : IsLocallyConstructible (⋃ i, f i) := SupClosed.iSup_mem (s := {s \| IsLocallyConstructible s}) (fun _ h₁ _ ↦ h₁.union) .empty hf lemma IsLocallyConstructible.biUnion {ι : Type} {f : ι → Set X} {s : Set ι} (hs : s.Finite) (hf : ∀ i ∈ s, IsLocallyConstructible (f i)) : IsLocallyConstructible (⋃ i ∈ s, f i) := SupClosed.biSup_mem (s := {s \| IsLocallyConstructible s}) (fun _ h₁ _ ↦ h₁.union) hs .empty hf lemma IsLocallyConstructible.sUnion {S : Set (Set X)} (hS : S.Finite) (hS' : ∀ s ∈ S, IsLocallyConstructible s) : IsLocallyConstructible (⋃₀ S) := SupClosed.sSup_mem (s := {s \| IsLocallyConstructible s}) (fun _ h₁ _ ↦ h₁.union) hS .empty hS' lemma IsLocallyConstructible.preimage_of_isOpenEmbedding {s : Set Y} (hs : IsLocallyConstructible s) (hf : IsOpenEmbedding f) : IsLocallyConstructible (f ⁻¹' s) := by intro x obtain ⟨U, hxU, hU, H⟩ := hs (f x) exact ⟨f ⁻¹' U, hf.continuous.continuousAt.preimage_mem_nhds hxU, hU.preimage hf.continuous, (H.preimage_of_isOpenEmbedding (hf.restrictPreimage _) :)⟩ lemma IsLocallyConstructible.isConstructible_of_subset_of_isCompact [PrespectralSpace X] [QuasiSeparatedSpace X] (hs : IsLocallyConstructible s) (hst : s ⊆ t) (ht : IsCompact t) : IsConstructible s := by have (x : _) : ∃ U, IsOpen U ∧ IsCompact U ∧ x ∈ U ∧ IsConstructible (U ∩ s) := have ⟨U, hxU, hU, hUs⟩ := hs x have ⟨V, ⟨hV₁, hV₂⟩, hxV, hVU⟩ := PrespectralSpace.isTopologicalBasis.mem_nhds_iff.mp hxU have : IsConstructible (V ↓∩ s) := (hUs.preimage_of_isOpenEmbedding (IsOpenEmbedding.id.restrict hVU hV₁):) have : IsConstructible (V ∩ s) := by have := this.image_of_isOpenEmbedding hV₁.isOpenEmbedding_subtypeVal (by simpa using hV₂.isRetrocompact hV₁) rwa [Subtype.image_preimage_coe] at this ⟨V, hV₁, hV₂, hxV, this⟩ choose U hU hU' hxU hUs using this obtain ⟨σ, hσ, htσ⟩ := ht.elim_nhds_subcover U (fun x _ ↦ (hU x).mem_nhds (hxU x)) convert IsConstructible.biUnion σ.finite_toSet (fun x _ ↦ hUs x) apply subset_antisymm · rw [← Set.iUnion₂_inter, Set.subset_inter_iff] exact ⟨hst.trans htσ, subset_rfl⟩ · exact Set.iUnion₂_subset fun _ _ ↦ Set.inter_subset_right lemma IsLocallyConstructible.isConstructible [PrespectralSpace X] [QuasiSeparatedSpace X] [CompactSpace X] (hs : IsLocallyConstructible s) : IsConstructible s := hs.isConstructible_of_subset_of_isCompact s.subset_univ isCompact_univ lemma IsLocallyConstructible.inter_of_isOpen_isCompact [PrespectralSpace X] [QuasiSeparatedSpace X] (hs : IsLocallyConstructible s) (ht : IsOpen t) (ht' : IsCompact t) : IsConstructible (s ∩ t) := (hs.inter (ht'.isConstructible ht).isLocallyConstructible).isConstructible_of_subset_of_isCompact Set.inter_subset_right ht' variable {ι : Type*} {U : ι → Opens X} lemma IsLocallyConstructible.of_isOpenCover (hU : IsOpenCover U) (H : ∀ i, IsLocallyConstructible ((U i : Set X) ↓∩ s)) : IsLocallyConstructible s := by intro x have ⟨i, hi⟩ := hU.exists_mem x have ⟨V, hVx, hV, hV'⟩ := H i ⟨x, hi⟩ refine ⟨_, (U i).2.isOpenEmbedding_subtypeVal.image_mem_nhds.mpr hVx, (U i).2.isOpenMap_subtype_val _ hV, ?_⟩ let e : V ≃ₜ Subtype.val '' V := (Equiv.Set.image _ V Subtype.val_injective).toHomeomorphOfIsInducing ((U i).2.isOpenEmbedding_subtypeVal.restrict (by simp [MapsTo]) hV).isInducing convert hV'.preimage_of_isOpenEmbedding e.symm.isOpenEmbedding ext ⟨_, x, hx, rfl⟩ simp [e, Equiv.toHomeomorphOfIsInducing] /-- A variant that requires constructible in the ambient space. This is as strong as the unprimed version only when the open cover consists of retrocompact sets. -/ lemma IsLocallyConstructible.of_isOpenCover' (hU : IsOpenCover U) (H : ∀ i, IsLocallyConstructible (s ∩ U i)) : IsLocallyConstructible s := .of_isOpenCover hU fun i ↦ by rw [← Subtype.preimage_coe_inter_self] exact (H i).preimage_of_isOpenEmbedding (U i).2.isOpenEmbedding_subtypeVal lemma IsLocallyConstructible.iff_of_isOpenCover (hU : IsOpenCover U) : IsLocallyConstructible s ↔ ∀ i, IsLocallyConstructible ((U i : Set X) ↓∩ s) := ⟨fun H i ↦ H.preimage_of_isOpenEmbedding (U i).2.isOpenEmbedding_subtypeVal, fun H ↦ .of_isOpenCover hU H⟩ lemma IsLocallyConstructible.iff_isConstructible_of_isOpenCover [PrespectralSpace X] [QuasiSeparatedSpace X] (hU : IsOpenCover U) (hU' : ∀ i, IsCompact (U i : Set X)) : IsLocallyConstructible s ↔ ∀ i, IsConstructible (s ∩ U i) := ⟨fun H i ↦ H.inter_of_isOpen_isCompact (U i).2 (hU' i), fun H ↦ .of_isOpenCover' hU fun i ↦ (H i).isLocallyConstructible⟩ end Topology
.lake/packages/mathlib/Mathlib/Topology/LocallyFinite.lean	import Mathlib.Order.Filter.SmallSets import Mathlib.Topology.ContinuousOn /-! ### Locally finite families of sets We say that a family of sets in a topological space is locally finite if at every point `x : X`, there is a neighborhood of `x` which meets only finitely many sets in the family. In this file we give the definition and prove basic properties of locally finite families of sets. -/ -- locally finite family [General Topology (Bourbaki, 1995)] open Set Function Filter Topology variable {ι ι' α X Y : Type} [TopologicalSpace X] [TopologicalSpace Y] {f g : ι → Set X} /-- A family of sets in `Set X` is locally finite if at every point `x : X`, there is a neighborhood of `x` which meets only finitely many sets in the family. -/ def LocallyFinite (f : ι → Set X) := ∀ x : X, ∃ t ∈ 𝓝 x, { i \| (f i ∩ t).Nonempty }.Finite theorem locallyFinite_of_finite [Finite ι] (f : ι → Set X) : LocallyFinite f := fun _ => ⟨univ, univ_mem, toFinite _⟩ namespace LocallyFinite theorem point_finite (hf : LocallyFinite f) (x : X) : { b \| x ∈ f b }.Finite := let ⟨_t, hxt, ht⟩ := hf x ht.subset fun _b hb => ⟨x, hb, mem_of_mem_nhds hxt⟩ protected theorem subset (hf : LocallyFinite f) (hg : ∀ i, g i ⊆ f i) : LocallyFinite g := fun a => let ⟨t, ht₁, ht₂⟩ := hf a ⟨t, ht₁, ht₂.subset fun i hi => hi.mono <\| inter_subset_inter (hg i) Subset.rfl⟩ theorem comp_injOn {g : ι' → ι} (hf : LocallyFinite f) (hg : InjOn g { i \| (f (g i)).Nonempty }) : LocallyFinite (f ∘ g) := fun x => by let ⟨t, htx, htf⟩ := hf x refine ⟨t, htx, htf.preimage <\| ?_⟩ exact hg.mono fun i (hi : Set.Nonempty _) => hi.left theorem comp_injective {g : ι' → ι} (hf : LocallyFinite f) (hg : Injective g) : LocallyFinite (f ∘ g) := hf.comp_injOn hg.injOn theorem of_comp_surjective {g : ι' → ι} (hg : Surjective g) (hfg : LocallyFinite (f ∘ g)) : LocallyFinite f := by simpa only [comp_def, surjInv_eq hg] using hfg.comp_injective (injective_surjInv hg) theorem on_range (hf : LocallyFinite f) : LocallyFinite ((↑) : range f → Set X) := of_comp_surjective rangeFactorization_surjective hf theorem _root_.locallyFinite_iff_smallSets : LocallyFinite f ↔ ∀ x, ∀ᶠ s in (𝓝 x).smallSets, { i \| (f i ∩ s).Nonempty }.Finite := forall_congr' fun _ => Iff.symm <\| eventually_smallSets' fun _s _t hst ht => ht.subset fun _i hi => hi.mono <\| inter_subset_inter_right _ hst protected theorem eventually_smallSets (hf : LocallyFinite f) (x : X) : ∀ᶠ s in (𝓝 x).smallSets, { i \| (f i ∩ s).Nonempty }.Finite := locallyFinite_iff_smallSets.mp hf x theorem exists_mem_basis {ι' : Sort} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X} {x : X} (hb : (𝓝 x).HasBasis p s) : ∃ i, p i ∧ { j \| (f j ∩ s i).Nonempty }.Finite := let ⟨i, hpi, hi⟩ := hb.smallSets.eventually_iff.mp (hf.eventually_smallSets x) ⟨i, hpi, hi Subset.rfl⟩ protected theorem nhdsWithin_iUnion (hf : LocallyFinite f) (a : X) : 𝓝[⋃ i, f i] a = ⨆ i, 𝓝[f i] a := by rcases hf a with ⟨U, haU, hfin⟩ refine le_antisymm ?_ (Monotone.le_map_iSup fun _ _ ↦ nhdsWithin_mono _) calc 𝓝[⋃ i, f i] a = 𝓝[⋃ i, f i ∩ U] a := by rw [← iUnion_inter, ← nhdsWithin_inter_of_mem' (nhdsWithin_le_nhds haU)] _ = 𝓝[⋃ i ∈ {j \| (f j ∩ U).Nonempty}, (f i ∩ U)] a := by simp only [mem_setOf_eq, iUnion_nonempty_self] _ = ⨆ i ∈ {j \| (f j ∩ U).Nonempty}, 𝓝[f i ∩ U] a := nhdsWithin_biUnion hfin _ _ _ ≤ ⨆ i, 𝓝[f i ∩ U] a := iSup₂_le_iSup _ _ _ ≤ ⨆ i, 𝓝[f i] a := iSup_mono fun i ↦ nhdsWithin_mono _ inter_subset_left theorem continuousOn_iUnion' {g : X → Y} (hf : LocallyFinite f) (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i) := by rintro x - rw [ContinuousWithinAt, hf.nhdsWithin_iUnion, tendsto_iSup] intro i by_cases hx : x ∈ closure (f i) · exact hc i _ hx · rw [mem_closure_iff_nhdsWithin_neBot, not_neBot] at hx rw [hx] exact tendsto_bot theorem continuousOn_iUnion {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i) := hf.continuousOn_iUnion' fun i x hx ↦ h_cont i x <\| (h_cl i).closure_subset hx protected theorem continuous' {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ) (hc : ∀ i x, x ∈ closure (f i) → ContinuousWithinAt g (f i) x) : Continuous g := continuousOn_univ.1 <\| h_cov ▸ hf.continuousOn_iUnion' hc protected theorem continuous {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = univ) (h_cl : ∀ i, IsClosed (f i)) (h_cont : ∀ i, ContinuousOn g (f i)) : Continuous g := continuousOn_univ.1 <\| h_cov ▸ hf.continuousOn_iUnion h_cl h_cont protected theorem closure (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i) := by intro x rcases hf x with ⟨s, hsx, hsf⟩ refine ⟨interior s, interior_mem_nhds.2 hsx, hsf.subset fun i hi => ?_⟩ exact (hi.mono isOpen_interior.closure_inter).of_closure.mono (inter_subset_inter_right _ interior_subset) theorem closure_iUnion (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i) := by ext x simp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_iUnion, iSup_neBot, mem_iUnion] theorem isClosed_iUnion (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) : IsClosed (⋃ i, f i) := by simp only [← closure_eq_iff_isClosed, hf.closure_iUnion, (hc _).closure_eq] /-- If `f : β → Set α` is a locally finite family of closed sets, then for any `x : α`, the intersection of the complements to `f i`, `x ∉ f i`, is a neighbourhood of `x`. -/ theorem iInter_compl_mem_nhds (hf : LocallyFinite f) (hc : ∀ i, IsClosed (f i)) (x : X) : (⋂ (i) (_ : x ∉ f i), (f i)ᶜ) ∈ 𝓝 x := by refine IsOpen.mem_nhds ?_ (mem_iInter₂.2 fun i => id) suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this exact (hf.comp_injective Subtype.val_injective).isClosed_iUnion fun i => hc _ /-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose that the family of sets `s n = {x \| f (n + 1) x ≠ f n x}` is locally finite. Then there exists a function `F : Π a, β a` such that for any `x`, we have `f n x = F x` on the product of an infinite interval `[N, +∞)` and a neighbourhood of `x`. We formulate the conclusion in terms of the product of filter `Filter.atTop` and `𝓝 x`. -/ theorem exists_forall_eventually_eq_prod {π : X → Sort} {f : ℕ → ∀ x : X, π x} (hf : LocallyFinite fun n => { x \| f (n + 1) x ≠ f n x }) : ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ p : ℕ × X in atTop ×ˢ 𝓝 x, f p.1 p.2 = F p.2 := by choose U hUx hU using hf choose N hN using fun x => (hU x).bddAbove replace hN : ∀ (x), ∀ n > N x, ∀ y ∈ U x, f (n + 1) y = f n y := fun x n hn y hy => by_contra fun hne => hn.lt.not_ge <\| hN x ⟨y, hne, hy⟩ replace hN : ∀ (x), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y := fun x n hn y hy => Nat.le_induction rfl (fun k hle => (hN x _ hle _ hy).trans) n hn refine ⟨fun x => f (N x + 1) x, fun x => ?_⟩ filter_upwards [Filter.prod_mem_prod (eventually_gt_atTop (N x)) (hUx x)] rintro ⟨n, y⟩ ⟨hn : N x < n, hy : y ∈ U x⟩ calc f n y = f (N x + 1) y := hN _ _ hn _ hy _ = f (max (N x + 1) (N y + 1)) y := (hN _ _ (le_max_left _ _) _ hy).symm _ = f (N y + 1) y := hN _ _ (le_max_right _ _) _ (mem_of_mem_nhds <\| hUx y) /-- Let `f : ℕ → Π a, β a` be a sequence of (dependent) functions on a topological space. Suppose that the family of sets `s n = {x \| f (n + 1) x ≠ f n x}` is locally finite. Then there exists a function `F : Π a, β a` such that for any `x`, for sufficiently large values of `n`, we have `f n y = F y` in a neighbourhood of `x`. -/ theorem exists_forall_eventually_atTop_eventually_eq' {π : X → Sort} {f : ℕ → ∀ x : X, π x} (hf : LocallyFinite fun n => { x \| f (n + 1) x ≠ f n x }) : ∃ F : ∀ x : X, π x, ∀ x, ∀ᶠ n : ℕ in atTop, ∀ᶠ y : X in 𝓝 x, f n y = F y := hf.exists_forall_eventually_eq_prod.imp fun _F hF x => (hF x).curry /-- Let `f : ℕ → α → β` be a sequence of functions on a topological space. Suppose that the family of sets `s n = {x \| f (n + 1) x ≠ f n x}` is locally finite. Then there exists a function `F : α → β` such that for any `x`, for sufficiently large values of `n`, we have `f n =ᶠ[𝓝 x] F`. -/ theorem exists_forall_eventually_atTop_eventuallyEq {f : ℕ → X → α} (hf : LocallyFinite fun n => { x \| f (n + 1) x ≠ f n x }) : ∃ F : X → α, ∀ x, ∀ᶠ n : ℕ in atTop, f n =ᶠ[𝓝 x] F := hf.exists_forall_eventually_atTop_eventually_eq' theorem preimage_continuous {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) : LocallyFinite (g ⁻¹' f ·) := fun x => let ⟨s, hsx, hs⟩ := hf (g x) ⟨g ⁻¹' s, hg.continuousAt hsx, hs.subset fun _ ⟨y, hy⟩ => ⟨g y, hy⟩⟩ theorem prod_right (hf : LocallyFinite f) (g : ι → Set Y) : LocallyFinite (fun i ↦ f i ×ˢ g i) := (hf.preimage_continuous continuous_fst).subset fun _ ↦ prod_subset_preimage_fst _ _ theorem prod_left {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set X) : LocallyFinite (fun i ↦ f i ×ˢ g i) := (hg.preimage_continuous continuous_snd).subset fun _ ↦ prod_subset_preimage_snd _ _ end LocallyFinite @[simp] theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f := ⟨fun h => by simpa only [comp_def, e.apply_symm_apply] using h.comp_injective e.symm.injective, fun h => h.comp_injective e.injective⟩ theorem locallyFinite_sum {f : ι ⊕ ι' → Set X} : LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr) := by simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr, preimage_setOf_eq, (· ∘ ·), eventually_and] theorem LocallyFinite.sumElim {g : ι' → Set X} (hf : LocallyFinite f) (hg : LocallyFinite g) : LocallyFinite (Sum.elim f g) := locallyFinite_sum.mpr ⟨hf, hg⟩ theorem locallyFinite_option {f : Option ι → Set X} : LocallyFinite f ↔ LocallyFinite (f ∘ some) := by rw [← (Equiv.optionEquivSumPUnit.{0, _} ι).symm.locallyFinite_comp_iff, locallyFinite_sum] simp only [locallyFinite_of_finite, and_true] rfl theorem LocallyFinite.option_elim' (hf : LocallyFinite f) (s : Set X) : LocallyFinite (Option.elim' s f) := locallyFinite_option.2 hf theorem LocallyFinite.eventually_subset {s : ι → Set X} (hs : LocallyFinite s) (hs' : ∀ i, IsClosed (s i)) (x : X) : ∀ᶠ y in 𝓝 x, {i \| y ∈ s i} ⊆ {i \| x ∈ s i} := by filter_upwards [hs.iInter_compl_mem_nhds hs' x] with y hy i hi simp only [mem_iInter, mem_compl_iff] at hy exact not_imp_not.mp (hy i) hi
.lake/packages/mathlib/Mathlib/Topology/ExtendFrom.lean	import Mathlib.Topology.Separation.Regular /-! # Extending a function from a subset The main definition of this file is `extendFrom A f` where `f : X → Y` and `A : Set X`. This defines a new function `g : X → Y` which maps any `x₀ : X` to the limit of `f` as `x` tends to `x₀`, if such a limit exists. This is analogous to the way `IsDenseInducing.extend` "extends" a function `f : X → Z` to a function `g : Y → Z` along a dense inducing `i : X → Y`. The main theorem we prove about this definition is `continuousOn_extendFrom` which states that, for `extendFrom A f` to be continuous on a set `B ⊆ closure A`, it suffices that `f` converges within `A` at any point of `B`, provided that `f` is a function to a T₃ space. -/ noncomputable section open Topology open Filter Set variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] /-- Extend a function from a set `A`. The resulting function `g` is such that at any `x₀`, if `f` converges to some `y` as `x` tends to `x₀` within `A`, then `g x₀` is defined to be one of these `y`. Else, `g x₀` could be anything. -/ def extendFrom (A : Set X) (f : X → Y) : X → Y := fun x ↦ @limUnder _ _ _ ⟨f x⟩ (𝓝[A] x) f /-- If `f` converges to some `y` as `x` tends to `x₀` within `A`, then `f` tends to `extendFrom A f x` as `x` tends to `x₀`. -/ theorem tendsto_extendFrom {A : Set X} {f : X → Y} {x : X} (h : ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : Tendsto f (𝓝[A] x) (𝓝 <\| extendFrom A f x) := tendsto_nhds_limUnder h theorem extendFrom_eq [T2Space Y] {A : Set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A) (hf : Tendsto f (𝓝[A] x) (𝓝 y)) : extendFrom A f x = y := haveI := mem_closure_iff_nhdsWithin_neBot.mp hx tendsto_nhds_unique (tendsto_nhds_limUnder ⟨y, hf⟩) hf theorem extendFrom_extends [T2Space Y] {f : X → Y} {A : Set X} (hf : ContinuousOn f A) : ∀ x ∈ A, extendFrom A f x = f x := fun x x_in ↦ extendFrom_eq (subset_closure x_in) (hf x x_in) /-- If `f` is a function to a T₃ space `Y` which has a limit within `A` at any point of a set `B ⊆ closure A`, then `extendFrom A f` is continuous on `B`. -/ theorem continuousOn_extendFrom [RegularSpace Y] {f : X → Y} {A B : Set X} (hB : B ⊆ closure A) (hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : ContinuousOn (extendFrom A f) B := by set φ := extendFrom A f intro x x_in suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝[B] x by simpa [ContinuousWithinAt, (closed_nhds_basis (φ x)).tendsto_right_iff] intro V' V'_in V'_closed obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ V ∩ A ⊆ f ⁻¹' V' := by have := tendsto_extendFrom (hf x x_in) rcases (nhdsWithin_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩ exact ⟨V, IsOpen.mem_nhds V_op hxV, V_op, hV⟩ suffices ∀ y ∈ V ∩ B, φ y ∈ V' from mem_of_superset (inter_mem_inf V_in <\| mem_principal_self B) this rintro y ⟨hyV, hyB⟩ haveI := mem_closure_iff_nhdsWithin_neBot.mp (hB hyB) have limy : Tendsto f (𝓝[A] y) (𝓝 <\| φ y) := tendsto_extendFrom (hf y hyB) have hVy : V ∈ 𝓝 y := IsOpen.mem_nhds V_op hyV have : V ∩ A ∈ 𝓝[A] y := by simpa only [inter_comm] using inter_mem_nhdsWithin A hVy exact V'_closed.mem_of_tendsto limy (mem_of_superset this hV) /-- If a function `f` to a T₃ space `Y` has a limit within a dense set `A` for any `x`, then `extendFrom A f` is continuous. -/ theorem continuous_extendFrom [RegularSpace Y] {f : X → Y} {A : Set X} (hA : Dense A) (hf : ∀ x, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : Continuous (extendFrom A f) := by rw [← continuousOn_univ] exact continuousOn_extendFrom (fun x _ ↦ hA x) (by simpa using hf)
.lake/packages/mathlib/Mathlib/Topology/Maps/Basic.lean	import Mathlib.Topology.Order import Mathlib.Topology.NhdsSet /-! # Specific classes of maps between topological spaces This file introduces the following properties of a map `f : X → Y` between topological spaces: * `IsOpenMap f` means the image of an open set under `f` is open. * `IsClosedMap f` means the image of a closed set under `f` is closed. (Open and closed maps need not be continuous.) * `IsInducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`. These behave like embeddings except they need not be injective. Instead, points of `X` which are identified by `f` are also inseparable in the topology on `X`. * `IsEmbedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with a subspace of `Y`. * `IsOpenEmbedding f` means `f` is an embedding with open image, so it identifies `X` with an open subspace of `Y`. Equivalently, `f` is an embedding and an open map. * `IsClosedEmbedding f` similarly means `f` is an embedding with closed image, so it identifies `X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map. * `IsQuotientMap f` is the dual condition to `IsEmbedding f`: `f` is surjective and the topology on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies `Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps. ## References * <https://en.wikipedia.org/wiki/Open_and_closed_maps> * <https://en.wikipedia.org/wiki/Embedding#General_topology> * <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map> ## Tags open map, closed map, embedding, quotient map, identification map -/ open Set Filter Function open TopologicalSpace Topology Filter variable {X : Type} {Y : Type} {Z : Type} {ι : Type} {f : X → Y} {g : Y → Z} namespace Topology section IsInducing variable [TopologicalSpace Y] protected lemma IsInducing.induced (f : X → Y) : @IsInducing X Y (induced f ‹_›) _ f := @IsInducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl variable [TopologicalSpace X] @[fun_prop] protected lemma IsInducing.id : IsInducing (@id X) := ⟨induced_id.symm⟩ variable [TopologicalSpace Z] @[fun_prop] protected lemma IsInducing.comp (hg : IsInducing g) (hf : IsInducing f) : IsInducing (g ∘ f) := ⟨by rw [hf.eq_induced, hg.eq_induced, induced_compose]⟩ lemma IsInducing.of_comp_iff (hg : IsInducing g) : IsInducing (g ∘ f) ↔ IsInducing f := by refine ⟨fun h ↦ ?_, hg.comp⟩ rw [isInducing_iff, hg.eq_induced, induced_compose, h.eq_induced] lemma IsInducing.of_comp (hf : Continuous f) (hg : Continuous g) (hgf : IsInducing (g ∘ f)) : IsInducing f := ⟨le_antisymm hf.le_induced (by grw [hgf.eq_induced, ← induced_compose, ← hg.le_induced])⟩ lemma isInducing_iff_nhds : IsInducing f ↔ ∀ x, 𝓝 x = comap f (𝓝 (f x)) := (isInducing_iff _).trans (induced_iff_nhds_eq f) namespace IsInducing lemma nhds_eq_comap (hf : IsInducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <\| f x) := isInducing_iff_nhds.1 hf lemma basis_nhds {p : ι → Prop} {s : ι → Set Y} (hf : IsInducing f) {x : X} (h_basis : (𝓝 (f x)).HasBasis p s) : (𝓝 x).HasBasis p (preimage f ∘ s) := hf.nhds_eq_comap x ▸ h_basis.comap f lemma nhdsSet_eq_comap (hf : IsInducing f) (s : Set X) : 𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image] lemma map_nhds_eq (hf : IsInducing f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x := hf.eq_induced ▸ map_nhds_induced_eq x lemma map_nhds_of_mem (hf : IsInducing f) (x : X) (h : range f ∈ 𝓝 (f x)) : (𝓝 x).map f = 𝓝 (f x) := hf.eq_induced ▸ map_nhds_induced_of_mem h lemma mapClusterPt_iff (hf : IsInducing f) {x : X} {l : Filter X} : MapClusterPt (f x) l f ↔ ClusterPt x l := by delta MapClusterPt ClusterPt rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff] lemma image_mem_nhdsWithin (hf : IsInducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x) : f '' s ∈ 𝓝[range f] f x := hf.map_nhds_eq x ▸ image_mem_map hs lemma tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsInducing g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := by rw [hg.nhds_eq_comap, tendsto_comap_iff] lemma continuousAt_iff (hg : IsInducing g) {x : X} : ContinuousAt f x ↔ ContinuousAt (g ∘ f) x := hg.tendsto_nhds_iff lemma continuous_iff (hg : IsInducing g) : Continuous f ↔ Continuous (g ∘ f) := by simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff] lemma continuousAt_iff' (hf : IsInducing f) {x : X} (h : range f ∈ 𝓝 (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp] @[fun_prop] protected lemma continuous (hf : IsInducing f) : Continuous f := hf.continuous_iff.mp continuous_id lemma closure_eq_preimage_closure_image (hf : IsInducing f) (s : Set X) : closure s = f ⁻¹' closure (f '' s) := by ext x rw [Set.mem_preimage, ← closure_induced, hf.eq_induced] theorem isClosed_iff (hf : IsInducing f) {s : Set X} : IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.eq_induced, isClosed_induced_iff] theorem isClosed_iff' (hf : IsInducing f) {s : Set X} : IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.eq_induced, isClosed_induced_iff'] theorem isClosed_preimage (h : IsInducing f) (s : Set Y) (hs : IsClosed s) : IsClosed (f ⁻¹' s) := (isClosed_iff h).mpr ⟨s, hs, rfl⟩ theorem isOpen_iff (hf : IsInducing f) {s : Set X} : IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.eq_induced, isOpen_induced_iff] theorem setOf_isOpen (hf : IsInducing f) : {s : Set X \| IsOpen s} = preimage f '' {t \| IsOpen t} := Set.ext fun _ ↦ hf.isOpen_iff theorem dense_iff (hf : IsInducing f) {s : Set X} : Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage] theorem of_subsingleton [Subsingleton X] (f : X → Y) : IsInducing f := ⟨Subsingleton.elim _ _⟩ end IsInducing.IsInducing namespace IsEmbedding lemma induced [t : TopologicalSpace Y] (hf : Injective f) : @IsEmbedding X Y (t.induced f) t f := @IsEmbedding.mk X Y (t.induced f) t _ (.induced f) hf alias _root_.Function.Injective.isEmbedding_induced := IsEmbedding.induced variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] @[fun_prop] lemma isInducing (hf : IsEmbedding f) : IsInducing f := hf.toIsInducing lemma mk' (f : X → Y) (inj : Injective f) (induced : ∀ x, comap f (𝓝 (f x)) = 𝓝 x) : IsEmbedding f := ⟨isInducing_iff_nhds.2 fun x => (induced x).symm, inj⟩ @[fun_prop] protected lemma id : IsEmbedding (@id X) := ⟨.id, fun _ _ h => h⟩ @[fun_prop] protected lemma comp (hg : IsEmbedding g) (hf : IsEmbedding f) : IsEmbedding (g ∘ f) := { hg.isInducing.comp hf.isInducing with injective := fun _ _ h => hf.injective <\| hg.injective h } lemma of_comp_iff (hg : IsEmbedding g) : IsEmbedding (g ∘ f) ↔ IsEmbedding f := by simp_rw [isEmbedding_iff, hg.isInducing.of_comp_iff, hg.injective.of_comp_iff f] protected lemma of_comp (hf : Continuous f) (hg : Continuous g) (hgf : IsEmbedding (g ∘ f)) : IsEmbedding f where toIsInducing := hgf.isInducing.of_comp hf hg injective := hgf.injective.of_comp lemma of_leftInverse {f : X → Y} {g : Y → X} (h : LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsEmbedding g := .of_comp hg hf <\| h.comp_eq_id.symm ▸ .id alias _root_.Function.LeftInverse.isEmbedding := of_leftInverse lemma map_nhds_eq (hf : IsEmbedding f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x := hf.1.map_nhds_eq x lemma map_nhds_of_mem (hf : IsEmbedding f) (x : X) (h : range f ∈ 𝓝 (f x)) : (𝓝 x).map f = 𝓝 (f x) := hf.1.map_nhds_of_mem x h lemma tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsEmbedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := hg.isInducing.tendsto_nhds_iff lemma continuous_iff (hg : IsEmbedding g) : Continuous f ↔ Continuous (g ∘ f) := hg.isInducing.continuous_iff lemma continuous (hf : IsEmbedding f) : Continuous f := hf.isInducing.continuous lemma closure_eq_preimage_closure_image (hf : IsEmbedding f) (s : Set X) : closure s = f ⁻¹' closure (f '' s) := hf.1.closure_eq_preimage_closure_image s /-- The topology induced under an inclusion `f : X → Y` from a discrete topological space `Y` is the discrete topology on `X`. See also `DiscreteTopology.of_continuous_injective`. -/ lemma discreteTopology [DiscreteTopology Y] (hf : IsEmbedding f) : DiscreteTopology X := .of_continuous_injective hf.continuous hf.injective lemma of_subsingleton [Subsingleton X] (f : X → Y) : IsEmbedding f := ⟨.of_subsingleton f, f.injective_of_subsingleton⟩ end IsEmbedding section IsQuotientMap variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] lemma isQuotientMap_iff : IsQuotientMap f ↔ Surjective f ∧ ∀ s, IsOpen s ↔ IsOpen (f ⁻¹' s) := (isQuotientMap_iff' _).trans <\| and_congr Iff.rfl TopologicalSpace.ext_iff theorem isQuotientMap_iff_isClosed : IsQuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsClosed s ↔ IsClosed (f ⁻¹' s) := isQuotientMap_iff.trans <\| Iff.rfl.and <\| compl_surjective.forall.trans <\| by simp only [isOpen_compl_iff, preimage_compl] namespace IsQuotientMap @[fun_prop] protected theorem id : IsQuotientMap (@id X) := ⟨fun x => ⟨x, rfl⟩, coinduced_id.symm⟩ @[fun_prop] protected theorem comp (hg : IsQuotientMap g) (hf : IsQuotientMap f) : IsQuotientMap (g ∘ f) := ⟨hg.surjective.comp hf.surjective, by rw [hg.eq_coinduced, hf.eq_coinduced, coinduced_compose]⟩ protected theorem of_comp (hf : Continuous f) (hg : Continuous g) (hgf : IsQuotientMap (g ∘ f)) : IsQuotientMap g := ⟨hgf.1.of_comp, le_antisymm (by grw [hgf.eq_coinduced, ← coinduced_compose, hf.coinduced_le]) hg.coinduced_le⟩ theorem of_inverse {g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) : IsQuotientMap g := .of_comp hf hg <\| h.comp_eq_id.symm ▸ IsQuotientMap.id protected theorem continuous_iff (hf : IsQuotientMap f) : Continuous g ↔ Continuous (g ∘ f) := by rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.eq_coinduced, coinduced_compose] @[fun_prop] protected theorem continuous (hf : IsQuotientMap f) : Continuous f := hf.continuous_iff.mp continuous_id protected lemma isOpen_preimage (hf : IsQuotientMap f) {s : Set Y} : IsOpen (f ⁻¹' s) ↔ IsOpen s := ((isQuotientMap_iff.1 hf).2 s).symm protected theorem isClosed_preimage (hf : IsQuotientMap f) {s : Set Y} : IsClosed (f ⁻¹' s) ↔ IsClosed s := ((isQuotientMap_iff_isClosed.1 hf).2 s).symm end IsQuotientMap end Topology.IsQuotientMap section OpenMap variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] namespace IsOpenMap protected theorem id : IsOpenMap (@id X) := fun s hs => by rwa [image_id] protected theorem comp (hg : IsOpenMap g) (hf : IsOpenMap f) : IsOpenMap (g ∘ f) := fun s hs => by rw [image_comp]; exact hg _ (hf _ hs) theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by rw [← image_univ] exact hf _ isOpen_univ theorem image_mem_nhds (hf : IsOpenMap f) {x : X} {s : Set X} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) := let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx mem_of_superset (IsOpen.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_mono hts) theorem range_mem_nhds (hf : IsOpenMap f) (x : X) : range f ∈ 𝓝 (f x) := hf.isOpen_range.mem_nhds <\| mem_range_self _ theorem mapsTo_interior (hf : IsOpenMap f) {s : Set X} {t : Set Y} (h : MapsTo f s t) : MapsTo f (interior s) (interior t) := mapsTo_iff_image_subset.2 <\| interior_maximal (h.mono interior_subset Subset.rfl).image_subset (hf _ isOpen_interior) theorem image_interior_subset (hf : IsOpenMap f) (s : Set X) : f '' interior s ⊆ interior (f '' s) := (hf.mapsTo_interior (mapsTo_image f s)).image_subset theorem nhds_le (hf : IsOpenMap f) (x : X) : 𝓝 (f x) ≤ map f (𝓝 x) := le_map fun _ => hf.image_mem_nhds theorem map_nhds_eq (hf : IsOpenMap f) {x : X} (hf' : ContinuousAt f x) : map f (𝓝 x) = 𝓝 (f x) := le_antisymm hf' (hf.nhds_le x) theorem map_nhdsSet_eq (hf : IsOpenMap f) (hf' : Continuous f) (s : Set X) : map f (𝓝ˢ s) = 𝓝ˢ (f '' s) := by rw [← biUnion_of_singleton s] simp_rw [image_iUnion, nhdsSet_iUnion, map_iSup, image_singleton, nhdsSet_singleton, hf.map_nhds_eq hf'.continuousAt] theorem of_nhds_le (hf : ∀ x, 𝓝 (f x) ≤ map f (𝓝 x)) : IsOpenMap f := fun _s hs => isOpen_iff_mem_nhds.2 fun _y ⟨_x, hxs, hxy⟩ => hxy ▸ hf _ (image_mem_map <\| hs.mem_nhds hxs) theorem of_sections (h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f := of_nhds_le fun x => let ⟨g, hgc, hgx, hgf⟩ := h x calc 𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id] _ ≤ map f (𝓝 (g (f x))) := map_mono hgc _ = map f (𝓝 x) := by rw [hgx] theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f') (r_inv : RightInverse f f') : IsOpenMap f := of_sections fun _ => ⟨f', h.continuousAt, r_inv _, l_inv⟩ /-- A continuous surjective open map is a quotient map. -/ theorem isQuotientMap (open_map : IsOpenMap f) (cont : Continuous f) (surj : Surjective f) : IsQuotientMap f := isQuotientMap_iff.2 ⟨surj, fun s => ⟨fun h => h.preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩ theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set Y} : interior (f ⁻¹' s) ⊆ f ⁻¹' interior s := hf.mapsTo_interior (mapsTo_preimage _ _) theorem preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f) (s : Set Y) : f ⁻¹' interior s = interior (f ⁻¹' s) := Subset.antisymm (preimage_interior_subset_interior_preimage hf₂) (interior_preimage_subset_preimage_interior hf₁) theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} : f ⁻¹' closure s ⊆ closure (f ⁻¹' s) := by rw [← compl_subset_compl] simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior] theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) : f ⁻¹' closure s = closure (f ⁻¹' s) := hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s) theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set Y} : f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) := by simpa only [frontier_eq_closure_inter_closure, preimage_inter] using inter_subset_inter hf.preimage_closure_subset_closure_preimage hf.preimage_closure_subset_closure_preimage theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) : f ⁻¹' frontier s = frontier (f ⁻¹' s) := by simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl, hf.preimage_closure_eq_closure_preimage hfc] theorem of_isEmpty [h : IsEmpty X] (f : X → Y) : IsOpenMap f := of_nhds_le h.elim theorem clusterPt_comap (hf : IsOpenMap f) {x : X} {l : Filter Y} (h : ClusterPt (f x) l) : ClusterPt x (comap f l) := by rw [ClusterPt, ← map_neBot_iff, Filter.push_pull] exact h.neBot.mono <\| inf_le_inf_right _ <\| hf.nhds_le _ end IsOpenMap lemma isOpenMap_iff_kernImage : IsOpenMap f ↔ ∀ {u : Set X}, IsClosed u → IsClosed (kernImage f u) := by rw [IsOpenMap, compl_surjective.forall] simp [kernImage_eq_compl] theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ x : X, 𝓝 (f x) ≤ (𝓝 x).map f := ⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩ theorem isOpenMap_iff_clusterPt_comap : IsOpenMap f ↔ ∀ x l, ClusterPt (f x) l → ClusterPt x (comap f l) := by refine ⟨fun hf _ _ ↦ hf.clusterPt_comap, fun h ↦ ?_⟩ simp only [isOpenMap_iff_nhds_le, le_map_iff] intro x s hs contrapose! hs rw [← mem_interior_iff_mem_nhds, mem_interior_iff_not_clusterPt_compl, not_not] at hs ⊢ exact (h _ _ hs).mono <\| by simp [subset_preimage_image] theorem isOpenMap_iff_image_interior : IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) := ⟨IsOpenMap.image_interior_subset, fun hs u hu => subset_interior_iff_isOpen.mp <\| by simpa only [hu.interior_eq] using hs u⟩ @[deprecated (since := "2025-08-30")] alias isOpenMap_iff_interior := isOpenMap_iff_image_interior /-- A map is open if and only if the `Set.kernImage` of every closed set is closed. -/ lemma isOpenMap_iff_closure_kernImage : IsOpenMap f ↔ ∀ {s : Set X}, closure (kernImage f s) ⊆ kernImage f (closure s) := by rw [isOpenMap_iff_image_interior, compl_surjective.forall] simp [kernImage_eq_compl] /-- An inducing map with an open range is an open map. -/ protected lemma Topology.IsInducing.isOpenMap (hi : IsInducing f) (ho : IsOpen (range f)) : IsOpenMap f := IsOpenMap.of_nhds_le fun _ => (hi.map_nhds_of_mem _ <\| IsOpen.mem_nhds ho <\| mem_range_self _).ge /-- Preimage of a dense set under an open map is dense. -/ protected theorem Dense.preimage {s : Set Y} (hs : Dense s) (hf : IsOpenMap f) : Dense (f ⁻¹' s) := fun x ↦ hf.preimage_closure_subset_closure_preimage <\| hs (f x) end OpenMap section IsClosedMap variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] namespace IsClosedMap open Function protected theorem id : IsClosedMap (@id X) := fun s hs => by rwa [image_id] protected theorem comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f) := by intro s hs rw [image_comp] exact hg _ (hf _ hs) protected theorem of_comp_surjective (hf : Surjective f) (hf' : Continuous f) (hfg : IsClosedMap (g ∘ f)) : IsClosedMap g := by intro K hK rw [← image_preimage_eq K hf, ← image_comp] exact hfg _ (hK.preimage hf') theorem closure_image_subset (hf : IsClosedMap f) (s : Set X) : closure (f '' s) ⊆ f '' closure s := closure_minimal (image_mono subset_closure) (hf _ isClosed_closure) theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f') (r_inv : RightInverse f f') : IsClosedMap f := fun s hs => by rw [image_eq_preimage_of_inverse r_inv l_inv] exact hs.preimage h theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) : IsClosedMap f := by intro s hs; rcases eq_empty_or_nonempty s with h2s \| h2s · simp_rw [h2s, image_empty, isClosed_empty] · exact h s hs h2s theorem isClosed_range (hf : IsClosedMap f) : IsClosed (range f) := @image_univ _ _ f ▸ hf _ isClosed_univ theorem isQuotientMap (hcl : IsClosedMap f) (hcont : Continuous f) (hsurj : Surjective f) : IsQuotientMap f := isQuotientMap_iff_isClosed.2 ⟨hsurj, fun s => ⟨fun hs => hs.preimage hcont, fun hs => hsurj.image_preimage s ▸ hcl _ hs⟩⟩ end IsClosedMap /-- A map is closed if and only if the `Set.kernImage` of every open set is open. One way to understand this result is that `f : X → Y` is closed if and only if its fibers vary in an upper hemicontinuous way: for any open subset `U ⊆ X`, the set of all `y ∈ Y` such that `f ⁻¹' {y} ⊆ U` is open in `Y`. -/ lemma isClosedMap_iff_kernImage : IsClosedMap f ↔ ∀ {u : Set X}, IsOpen u → IsOpen (kernImage f u) := by rw [IsClosedMap, compl_surjective.forall] simp [kernImage_eq_compl] lemma Topology.IsInducing.isClosedMap (hf : IsInducing f) (h : IsClosed (range f)) : IsClosedMap f := by intro s hs rcases hf.isClosed_iff.1 hs with ⟨t, ht, rfl⟩ rw [image_preimage_eq_inter_range] exact ht.inter h theorem isClosedMap_iff_closure_image : IsClosedMap f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s := ⟨IsClosedMap.closure_image_subset, fun hs c hc => isClosed_of_closure_subset <\| calc closure (f '' c) ⊆ f '' closure c := hs c _ = f '' c := by rw [hc.closure_eq]⟩ theorem isClosedMap_iff_kernImage_interior : IsClosedMap f ↔ ∀ {s : Set X}, kernImage f (interior s) ⊆ interior (kernImage f s) := by rw [isClosedMap_iff_closure_image, compl_surjective.forall] simp [kernImage_eq_compl] /-- A map `f : X → Y` is closed if and only if for all sets `s`, any cluster point of `f '' s` is the image by `f` of some cluster point of `s`. If you require this for all filters instead of just principal filters, and also that `f` is continuous, you get the notion of proper map. See `isProperMap_iff_clusterPt`. -/ theorem isClosedMap_iff_clusterPt : IsClosedMap f ↔ ∀ s y, MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s) := by simp [MapClusterPt, isClosedMap_iff_closure_image, subset_def, mem_closure_iff_clusterPt, and_comm] theorem isClosedMap_iff_comap_nhdsSet_le : IsClosedMap f ↔ ∀ {s : Set Y}, comap f (𝓝ˢ s) ≤ 𝓝ˢ (f ⁻¹' s) := by simp_rw [Filter.le_def, mem_comap'', ← subset_interior_iff_mem_nhdsSet, ← subset_kernImage_iff, isClosedMap_iff_kernImage_interior] exact ⟨fun H s t hst ↦ hst.trans H, fun H s ↦ H _ subset_rfl⟩ alias ⟨IsClosedMap.comap_nhdsSet_le, _⟩ := isClosedMap_iff_comap_nhdsSet_le theorem isClosedMap_iff_comap_nhds_le : IsClosedMap f ↔ ∀ {y : Y}, comap f (𝓝 y) ≤ 𝓝ˢ (f ⁻¹' {y}) := by rw [isClosedMap_iff_comap_nhdsSet_le] constructor · exact fun H y ↦ nhdsSet_singleton (x := y) ▸ H · intro H s rw [← Set.biUnion_of_singleton s] simp_rw [preimage_iUnion, nhdsSet_iUnion, comap_iSup, nhdsSet_singleton] exact iSup₂_mono fun _ _ ↦ H alias ⟨IsClosedMap.comap_nhds_le, _⟩ := isClosedMap_iff_comap_nhds_le theorem IsClosedMap.comap_nhds_eq (hf : IsClosedMap f) (hf' : Continuous f) (y : Y) : comap f (𝓝 y) = 𝓝ˢ (f ⁻¹' {y}) := le_antisymm (isClosedMap_iff_comap_nhds_le.mp hf) -- Note: below should be an application of `Continuous.tendsto_nhdsSet_nhds`, but this is only -- proven later... (nhdsSet_le.mpr fun x hx ↦ hx ▸ (hf'.tendsto x).le_comap) theorem IsClosedMap.comap_nhdsSet_eq (hf : IsClosedMap f) (hf' : Continuous f) (s : Set Y) : comap f (𝓝ˢ s) = 𝓝ˢ (f ⁻¹' s) := le_antisymm (isClosedMap_iff_comap_nhdsSet_le.mp hf) -- Note: below should be an application of `Continuous.tendsto_nhdsSet_nhdsSet`, but this is only -- proven later... (nhdsSet_le.mpr fun x hx ↦ (hf'.tendsto x).le_comap.trans (comap_mono (nhds_le_nhdsSet hx))) /-- Assume `f` is a closed map. If some property `p` holds around every point in the fiber of `f` at `y₀`, then for any `y` close enough to `y₀` we have that `p` holds on the fiber at `y`. -/ theorem IsClosedMap.eventually_nhds_fiber (hf : IsClosedMap f) {p : X → Prop} (y₀ : Y) (H : ∀ x₀ ∈ f ⁻¹' {y₀}, ∀ᶠ x in 𝓝 x₀, p x) : ∀ᶠ y in 𝓝 y₀, ∀ x ∈ f ⁻¹' {y}, p x := by rw [← eventually_nhdsSet_iff_forall] at H replace H := H.filter_mono hf.comap_nhds_le rwa [eventually_comap] at H /-- Assume `f` is a closed map. If there are points `y` arbitrarily close to `y₀` such that `p` holds for at least some `x ∈ f ⁻¹' {y}`, then one can find `x₀ ∈ f ⁻¹' {y₀}` such that there are points `x` arbitrarily close to `x₀` which satisfy `p`. -/ theorem IsClosedMap.frequently_nhds_fiber (hf : IsClosedMap f) {p : X → Prop} (y₀ : Y) (H : ∃ᶠ y in 𝓝 y₀, ∃ x ∈ f ⁻¹' {y}, p x) : ∃ x₀ ∈ f ⁻¹' {y₀}, ∃ᶠ x in 𝓝 x₀, p x := by /- Note: this result could also be seen as a reformulation of `isClosedMap_iff_clusterPt`. One would then be able to deduce the `eventually` statement, and then go back to `isClosedMap_iff_comap_nhdsSet_le`. Ultimately, this makes no difference. -/ revert H contrapose simpa only [not_frequently, not_exists, not_and] using hf.eventually_nhds_fiber y₀ theorem IsClosedMap.closure_image_eq_of_continuous (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X) : closure (f '' s) = f '' closure s := subset_antisymm (f_closed.closure_image_subset s) (image_closure_subset_closure_image f_cont) theorem IsClosedMap.lift'_closure_map_eq (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) : (map f F).lift' closure = map f (F.lift' closure) := by rw [map_lift'_eq2 (monotone_closure Y), map_lift'_eq (monotone_closure X)] congr 1 ext s : 1 exact f_closed.closure_image_eq_of_continuous f_cont s theorem IsClosedMap.mapClusterPt_iff_lift'_closure {F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y} : MapClusterPt y F f ↔ ((F.lift' closure) ⊓ 𝓟 (f ⁻¹' {y})).NeBot := by rw [MapClusterPt, clusterPt_iff_lift'_closure', f_closed.lift'_closure_map_eq f_cont, ← comap_principal, ← map_neBot_iff f, Filter.push_pull, principal_singleton] end IsClosedMap namespace Topology section IsOpenEmbedding variable [TopologicalSpace X] [TopologicalSpace Y] @[fun_prop] lemma IsOpenEmbedding.isEmbedding (hf : IsOpenEmbedding f) : IsEmbedding f := hf.toIsEmbedding lemma IsOpenEmbedding.isInducing (hf : IsOpenEmbedding f) : IsInducing f := hf.isEmbedding.isInducing lemma IsOpenEmbedding.isOpenMap (hf : IsOpenEmbedding f) : IsOpenMap f := hf.isEmbedding.isInducing.isOpenMap hf.isOpen_range theorem IsOpenEmbedding.map_nhds_eq (hf : IsOpenEmbedding f) (x : X) : map f (𝓝 x) = 𝓝 (f x) := hf.isEmbedding.map_nhds_of_mem _ <\| hf.isOpen_range.mem_nhds <\| mem_range_self _ lemma IsOpenEmbedding.isOpen_iff_image_isOpen (hf : IsOpenEmbedding f) {s : Set X} : IsOpen s ↔ IsOpen (f '' s) where mp := hf.isOpenMap s mpr h := by convert ← h.preimage hf.isEmbedding.continuous; apply preimage_image_eq _ hf.injective theorem IsOpenEmbedding.tendsto_nhds_iff [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsOpenEmbedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := hg.isEmbedding.tendsto_nhds_iff theorem IsOpenEmbedding.tendsto_nhds_iff' (hf : IsOpenEmbedding f) {l : Filter Z} {x : X} : Tendsto (g ∘ f) (𝓝 x) l ↔ Tendsto g (𝓝 (f x)) l := by rw [Tendsto, ← map_map, hf.map_nhds_eq]; rfl theorem IsOpenEmbedding.continuousAt_iff [TopologicalSpace Z] (hf : IsOpenEmbedding f) {x : X} : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := hf.tendsto_nhds_iff' @[fun_prop] theorem IsOpenEmbedding.continuous (hf : IsOpenEmbedding f) : Continuous f := hf.isEmbedding.continuous lemma IsOpenEmbedding.isOpen_iff_preimage_isOpen (hf : IsOpenEmbedding f) {s : Set Y} (hs : s ⊆ range f) : IsOpen s ↔ IsOpen (f ⁻¹' s) := by rw [hf.isOpen_iff_image_isOpen, image_preimage_eq_inter_range, inter_eq_self_of_subset_left hs] @[fun_prop] lemma IsOpenEmbedding.of_isEmbedding_isOpenMap (h₁ : IsEmbedding f) (h₂ : IsOpenMap f) : IsOpenEmbedding f := ⟨h₁, h₂.isOpen_range⟩ /-- A surjective embedding is an `IsOpenEmbedding`. -/ lemma IsEmbedding.isOpenEmbedding_of_surjective (hf : IsEmbedding f) (hsurj : f.Surjective) : IsOpenEmbedding f := ⟨hf, hsurj.range_eq ▸ isOpen_univ⟩ alias IsOpenEmbedding.of_isEmbedding := IsEmbedding.isOpenEmbedding_of_surjective lemma isOpenEmbedding_iff_isEmbedding_isOpenMap : IsOpenEmbedding f ↔ IsEmbedding f ∧ IsOpenMap f := ⟨fun h => ⟨h.1, h.isOpenMap⟩, fun h => .of_isEmbedding_isOpenMap h.1 h.2⟩ theorem IsOpenEmbedding.of_continuous_injective_isOpenMap (h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsOpenMap f) : IsOpenEmbedding f := by simp only [isOpenEmbedding_iff_isEmbedding_isOpenMap, isEmbedding_iff, isInducing_iff_nhds, *, and_true] exact fun x => le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 x) _ h₂ ▸ comap_mono (h₃.nhds_le _)) lemma isOpenEmbedding_iff_continuous_injective_isOpenMap : IsOpenEmbedding f ↔ Continuous f ∧ Injective f ∧ IsOpenMap f := ⟨fun h => ⟨h.continuous, h.injective, h.isOpenMap⟩, fun h => .of_continuous_injective_isOpenMap h.1 h.2.1 h.2.2⟩ namespace IsOpenEmbedding variable [TopologicalSpace Z] @[fun_prop] protected lemma id : IsOpenEmbedding (@id X) := ⟨.id, IsOpenMap.id.isOpen_range⟩ @[fun_prop] protected lemma comp (hg : IsOpenEmbedding g) (hf : IsOpenEmbedding f) : IsOpenEmbedding (g ∘ f) := ⟨hg.1.comp hf.1, (hg.isOpenMap.comp hf.isOpenMap).isOpen_range⟩ theorem isOpenMap_iff (hg : IsOpenEmbedding g) : IsOpenMap f ↔ IsOpenMap (g ∘ f) := by simp_rw [isOpenMap_iff_nhds_le, ← map_map, comp, ← hg.map_nhds_eq, map_le_map_iff hg.injective] theorem of_comp_iff (f : X → Y) (hg : IsOpenEmbedding g) : IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding f := by simp only [isOpenEmbedding_iff_continuous_injective_isOpenMap, ← hg.isOpenMap_iff, ← hg.1.continuous_iff, hg.injective.of_comp_iff] lemma of_comp (f : X → Y) (hg : IsOpenEmbedding g) (h : IsOpenEmbedding (g ∘ f)) : IsOpenEmbedding f := (IsOpenEmbedding.of_comp_iff f hg).1 h theorem of_isEmpty [IsEmpty X] (f : X → Y) : IsOpenEmbedding f := of_isEmbedding_isOpenMap (.of_subsingleton f) (.of_isEmpty f) theorem image_mem_nhds {f : X → Y} (hf : IsOpenEmbedding f) {s : Set X} {x : X} : f '' s ∈ 𝓝 (f x) ↔ s ∈ 𝓝 x := by rw [← hf.map_nhds_eq, mem_map, preimage_image_eq _ hf.injective] end IsOpenEmbedding end IsOpenEmbedding section IsClosedEmbedding variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] namespace IsClosedEmbedding @[fun_prop] lemma isEmbedding (hf : IsClosedEmbedding f) : IsEmbedding f := hf.toIsEmbedding @[fun_prop] lemma isInducing (hf : IsClosedEmbedding f) : IsInducing f := hf.isEmbedding.isInducing @[fun_prop] lemma continuous (hf : IsClosedEmbedding f) : Continuous f := hf.isEmbedding.continuous lemma tendsto_nhds_iff {g : ι → X} {l : Filter ι} {x : X} (hf : IsClosedEmbedding f) : Tendsto g l (𝓝 x) ↔ Tendsto (f ∘ g) l (𝓝 (f x)) := hf.isEmbedding.tendsto_nhds_iff lemma isClosedMap (hf : IsClosedEmbedding f) : IsClosedMap f := hf.isEmbedding.isInducing.isClosedMap hf.isClosed_range lemma isClosed_iff_image_isClosed (hf : IsClosedEmbedding f) {s : Set X} : IsClosed s ↔ IsClosed (f '' s) := ⟨hf.isClosedMap s, fun h => by rw [← preimage_image_eq s hf.injective] exact h.preimage hf.continuous⟩ lemma isClosed_iff_preimage_isClosed (hf : IsClosedEmbedding f) {s : Set Y} (hs : s ⊆ range f) : IsClosed s ↔ IsClosed (f ⁻¹' s) := by rw [hf.isClosed_iff_image_isClosed, image_preimage_eq_of_subset hs] lemma of_isEmbedding_isClosedMap (h₁ : IsEmbedding f) (h₂ : IsClosedMap f) : IsClosedEmbedding f := ⟨h₁, image_univ (f := f) ▸ h₂ univ isClosed_univ⟩ lemma of_continuous_injective_isClosedMap (h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsClosedMap f) : IsClosedEmbedding f := by refine .of_isEmbedding_isClosedMap ⟨⟨?_⟩, h₂⟩ h₃ refine h₁.le_induced.antisymm fun s hs => ?_ refine ⟨(f '' sᶜ)ᶜ, (h₃ _ hs.isClosed_compl).isOpen_compl, ?_⟩ rw [preimage_compl, preimage_image_eq _ h₂, compl_compl] lemma isClosedEmbedding_iff_continuous_injective_isClosedMap {f : X → Y} : IsClosedEmbedding f ↔ Continuous f ∧ Injective f ∧ IsClosedMap f where mp h := ⟨h.continuous, h.injective, h.isClosedMap⟩ mpr h := .of_continuous_injective_isClosedMap h.1 h.2.1 h.2.2 @[fun_prop] protected theorem id : IsClosedEmbedding (@id X) := ⟨.id, IsClosedMap.id.isClosed_range⟩ @[fun_prop] theorem comp (hg : IsClosedEmbedding g) (hf : IsClosedEmbedding f) : IsClosedEmbedding (g ∘ f) := ⟨hg.isEmbedding.comp hf.isEmbedding, (hg.isClosedMap.comp hf.isClosedMap).isClosed_range⟩ lemma of_comp_iff (hg : IsClosedEmbedding g) : IsClosedEmbedding (g ∘ f) ↔ IsClosedEmbedding f := by simp_rw [isClosedEmbedding_iff, hg.isEmbedding.of_comp_iff, Set.range_comp, ← hg.isClosed_iff_image_isClosed] theorem closure_image_eq (hf : IsClosedEmbedding f) (s : Set X) : closure (f '' s) = f '' closure s := hf.isClosedMap.closure_image_eq_of_continuous hf.continuous s end Topology.IsClosedEmbedding.IsClosedEmbedding
.lake/packages/mathlib/Mathlib/Topology/Maps/OpenQuotient.lean	import Mathlib.Topology.Maps.Basic /-! # Open quotient maps An open quotient map is an open map `f : X → Y` which is both an open map and a quotient map. Equivalently, it is a surjective continuous open map. We use the latter characterization as a definition. Many important quotient maps are open quotient maps, including - the quotient map from a topological space to its quotient by the action of a group; - the quotient map from a topological group to its quotient by a normal subgroup; - the quotient map from a topological space to its separation quotient. Contrary to general quotient maps, the category of open quotient maps is closed under `Prod.map`. -/ open Filter Function Set Topology variable {X Y Z : Type} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y} namespace IsOpenQuotientMap protected theorem id : IsOpenQuotientMap (id : X → X) := ⟨surjective_id, continuous_id, .id⟩ /-- An open quotient map is a quotient map. -/ theorem isQuotientMap (h : IsOpenQuotientMap f) : IsQuotientMap f := h.isOpenMap.isQuotientMap h.continuous h.surjective theorem iff_isOpenMap_isQuotientMap : IsOpenQuotientMap f ↔ IsOpenMap f ∧ IsQuotientMap f := ⟨fun h ↦ ⟨h.isOpenMap, h.isQuotientMap⟩, fun ⟨ho, hq⟩ ↦ ⟨hq.surjective, hq.continuous, ho⟩⟩ theorem of_isOpenMap_isQuotientMap (ho : IsOpenMap f) (hq : IsQuotientMap f) : IsOpenQuotientMap f := iff_isOpenMap_isQuotientMap.2 ⟨ho, hq⟩ theorem comp {g : Y → Z} (hg : IsOpenQuotientMap g) (hf : IsOpenQuotientMap f) : IsOpenQuotientMap (g ∘ f) := ⟨.comp hg.1 hf.1, .comp hg.2 hf.2, .comp hg.3 hf.3⟩ theorem map_nhds_eq (h : IsOpenQuotientMap f) (x : X) : map f (𝓝 x) = 𝓝 (f x) := le_antisymm h.continuous.continuousAt <\| h.isOpenMap.nhds_le _ theorem continuous_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} : Continuous (g ∘ f) ↔ Continuous g := h.isQuotientMap.continuous_iff.symm theorem continuousAt_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} {x : X} : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by simp only [ContinuousAt, ← h.map_nhds_eq, tendsto_map'_iff, comp_def] theorem dense_preimage_iff (h : IsOpenQuotientMap f) {s : Set Y} : Dense (f ⁻¹' s) ↔ Dense s := ⟨fun hs ↦ h.surjective.denseRange.dense_of_mapsTo h.continuous hs (mapsTo_preimage _ _), fun hs ↦ hs.preimage h.isOpenMap⟩ end IsOpenQuotientMap theorem Topology.IsInducing.isOpenQuotientMap_of_surjective (ind : IsInducing f) (surj : Function.Surjective f) : IsOpenQuotientMap f where surjective := surj continuous := ind.continuous isOpenMap U U_open := by obtain ⟨V, hV, rfl⟩ := ind.isOpen_iff.mp U_open rwa [V.image_preimage_eq surj] theorem Topology.IsInducing.isQuotientMap_of_surjective (ind : IsInducing f) (surj : Function.Surjective f) : IsQuotientMap f := (ind.isOpenQuotientMap_of_surjective surj).isQuotientMap section Subquotient variable {A B C D : Type} variable [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D] variable (f : A → B) (g : C → D) (p : A → C) (q : B → D) omit [TopologicalSpace C] in /-- Given the following diagram with `f` inducing, `p` surjective, `q` an open quotient map, and `g` injective. Suppose the image of `A` in `B` is stable under the equivalence mod `q`, then the coinduced topology on `C` (from `A`) coincides with the induced topology (from `D`). ``` A -f→ B ∣ ∣ p q ↓ ↓ C -g→ D ``` A typical application is when `K ≤ H` are subgroups of `G`, then the quotient topology on `H/K` is also the subspace topology from `G/K`. -/ lemma coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing (h : g ∘ p = q ∘ f) (hf : IsInducing f) (hp : Function.Surjective p) (hq : IsOpenQuotientMap q) (hg : Function.Injective g) (H : q ⁻¹' (q '' (Set.range f)) ⊆ Set.range f) : ‹TopologicalSpace A›.coinduced p = ‹TopologicalSpace D›.induced g := by ext U change IsOpen (p ⁻¹' U) ↔ ∃ V, _ simp_rw [hf.isOpen_iff, (Set.image_surjective.mpr hq.surjective).exists, ← hq.isQuotientMap.isOpen_preimage] constructor · rintro ⟨V, hV, e⟩ refine ⟨V, hq.continuous.1 _ (hq.isOpenMap _ hV), ?_⟩ ext x obtain ⟨x, rfl⟩ := hp x constructor · rintro ⟨y, hy, e'⟩ obtain ⟨y, rfl⟩ := H ⟨_, ⟨x, rfl⟩, (e'.trans (congr_fun h x)).symm⟩ rw [← hg ((congr_fun h y).trans e')] exact e.le hy · intro H exact ⟨f x, e.ge H, congr_fun h.symm x⟩ · rintro ⟨V, hV, rfl⟩ refine ⟨_, hV, ?_⟩ simp_rw [← Set.preimage_comp, h] lemma isEmbedding_of_isOpenQuotientMap_of_isInducing (h : g ∘ p = q ∘ f) (hf : IsInducing f) (hp : IsQuotientMap p) (hq : IsOpenQuotientMap q) (hg : Function.Injective g) (H : q ⁻¹' (q '' (Set.range f)) ⊆ Set.range f) : IsEmbedding g := ⟨⟨hp.eq_coinduced.trans (coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing f g p q h hf hp.surjective hq hg H)⟩, hg⟩ lemma isQuotientMap_of_isOpenQuotientMap_of_isInducing (h : g ∘ p = q ∘ f) (hf : IsInducing f) (hp : Surjective p) (hq : IsOpenQuotientMap q) (hg : IsEmbedding g) (H : q ⁻¹' (q '' (Set.range f)) ⊆ Set.range f) : IsQuotientMap p := ⟨hp, hg.eq_induced.trans ((coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing f g p q h hf hp hq hg.injective H)).symm⟩ end Subquotient
.lake/packages/mathlib/Mathlib/Topology/Maps/Proper/CompactlyGenerated.lean	import Mathlib.Topology.Compactness.CompactlyGeneratedSpace import Mathlib.Topology.Maps.Proper.Basic /-! # A map is proper iff preimage of compact sets are compact This file proves that if `Y` is a Hausdorff and compactly generated space, a continuous map `f : X → Y` is proper if and only if preimage of compact sets are compact. -/ open Set Filter variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] variable [T2Space Y] [CompactlyGeneratedSpace Y] variable {f : X → Y} /-- If `Y` is Hausdorff and compactly generated, then proper maps `X → Y` are exactly continuous maps such that the preimage of any compact set is compact. This is in particular true if `Y` is Hausdorff and sequential or locally compact. There was an older version of this theorem which was changed to this one to make use of the `CompactlyGeneratedSpace` typeclass. (since 2024-11-10) -/ theorem isProperMap_iff_isCompact_preimage : IsProperMap f ↔ Continuous f ∧ ∀ ⦃K⦄, IsCompact K → IsCompact (f ⁻¹' K) where mp hf := ⟨hf.continuous, fun _ ↦ hf.isCompact_preimage⟩ mpr := fun ⟨hf, h⟩ ↦ isProperMap_iff_isClosedMap_and_compact_fibers.2 ⟨hf, fun _ hs ↦ CompactlyGeneratedSpace.isClosed fun _ hK ↦ image_inter_preimage .. ▸ (((h hK).inter_left hs).image hf).isClosed, fun _ ↦ h isCompact_singleton⟩ /-- Version of `isProperMap_iff_isCompact_preimage` in terms of `cocompact`. There was an older version of this theorem which was changed to this one to make use of the `CompactlyGeneratedSpace` typeclass. (since 2024-11-10) -/ lemma isProperMap_iff_tendsto_cocompact : IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y) := by simp_rw [isProperMap_iff_isCompact_preimage, hasBasis_cocompact.tendsto_right_iff, ← mem_preimage, eventually_mem_set, preimage_compl] refine and_congr_right fun f_cont ↦ ⟨fun H K hK ↦ (H hK).compl_mem_cocompact, fun H K hK ↦ ?_⟩ rcases mem_cocompact.mp (H K hK) with ⟨K', hK', hK'y⟩ exact hK'.of_isClosed_subset (hK.isClosed.preimage f_cont) (compl_le_compl_iff_le.mp hK'y)
.lake/packages/mathlib/Mathlib/Topology/Maps/Proper/Basic.lean	import Mathlib.Topology.Homeomorph.Lemmas /-! # Proper maps between topological spaces This file develops the basic theory of proper maps between topological spaces. A map `f : X → Y` between two topological spaces is said to be proper if it is continuous and satisfies the following equivalent conditions: 1. `f` is closed and has compact fibers. 2. `f` is universally closed, in the sense that for any topological space `Z`, the map `Prod.map f id : X × Z → Y × Z` is closed. 3. For any `ℱ : Filter X`, all cluster points of `map f ℱ` are images by `f` of some cluster point of `ℱ`. We take 3 as the definition in `IsProperMap`, and we show the equivalence with 1, 2, and some other variations. ## Main statements * `isProperMap_iff_ultrafilter`: characterization of proper maps in terms of limits of ultrafilters instead of cluster points of filters. * `IsProperMap.pi_map`: any product of proper maps is proper. * `isProperMap_iff_isClosedMap_and_compact_fibers`: a map is proper if and only if it is continuous, closed, and has compact fibers ## Implementation notes In algebraic geometry, it is common to also ask that proper maps are separated, in the sense of [Stacks: definition OCY1](https://stacks.math.columbia.edu/tag/0CY1). We don't follow this convention because it is unclear whether it would give the right notion in all cases, and in particular for the theory of proper group actions. That means that our terminology does NOT align with that of [Stacks: Characterizing proper maps](https://stacks.math.columbia.edu/tag/005M), instead our definition of `IsProperMap` coincides with what they call "Bourbaki-proper". Regarding the proofs, we don't really follow Bourbaki and go for more filter-heavy proofs, as usual. In particular, their arguments rely heavily on restriction of closed maps (see `IsClosedMap.restrictPreimage`), which makes them somehow annoying to formalize in type theory. In contrast, the filter-based proofs work really well thanks to the existing API. In fact, the filter proofs work so well that I thought this would be a great pedagogical resource about how we use filters. For that reason, all interesting proofs in this file are commented, so don't hesitate to have a look! ## TODO * prove the equivalence with condition 3 of [Stacks: Theorem 005R](https://stacks.math.columbia.edu/tag/005R). Note that they mean something different by "universally closed". ## References * [N. Bourbaki, General Topology][bourbaki1966] * [Stacks: Characterizing proper maps](https://stacks.math.columbia.edu/tag/005M) -/ assert_not_exists StoneCech open Filter Topology Function Set open Prod (fst snd) variable {X Y Z W ι : Type} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace W] {f : X → Y} {g : Y → Z} /-- A map `f : X → Y` between two topological spaces is said to be proper* if it is continuous and, for all `ℱ : Filter X`, any cluster point of `map f ℱ` is the image by `f` of a cluster point of `ℱ`. -/ @[mk_iff isProperMap_iff_clusterPt, fun_prop] structure IsProperMap (f : X → Y) : Prop extends Continuous f where /-- By definition, if `f` is a proper map and `ℱ` is any filter on `X`, then any cluster point of `map f ℱ` is the image by `f` of some cluster point of `ℱ`. -/ clusterPt_of_mapClusterPt : ∀ ⦃ℱ : Filter X⦄, ∀ ⦃y : Y⦄, MapClusterPt y ℱ f → ∃ x, f x = y ∧ ClusterPt x ℱ /-- Definition of proper maps. See also `isClosedMap_iff_clusterPt` for a related criterion for closed maps. -/ add_decl_doc isProperMap_iff_clusterPt /-- By definition, a proper map is continuous. -/ @[fun_prop] lemma IsProperMap.continuous (h : IsProperMap f) : Continuous f := h.toContinuous /-- A proper map is closed. -/ lemma IsProperMap.isClosedMap (h : IsProperMap f) : IsClosedMap f := by rw [isClosedMap_iff_clusterPt] exact fun s y ↦ h.clusterPt_of_mapClusterPt (ℱ := 𝓟 s) (y := y) /-- Characterization of proper maps by ultrafilters. -/ lemma isProperMap_iff_ultrafilter : IsProperMap f ↔ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x := by -- This is morally trivial since ultrafilters give all the information about cluster points. rw [isProperMap_iff_clusterPt] refine and_congr_right (fun _ ↦ ?_) constructor <;> intro H · intro 𝒰 y (hY : (Ultrafilter.map f 𝒰 : Filter Y) ≤ _) simp_rw [← Ultrafilter.clusterPt_iff] at hY ⊢ exact H hY · simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, ← exists_ultrafilter_iff, forall_exists_index] intro ℱ y 𝒰 hy rcases H (tendsto_iff_comap.mpr <\| hy.trans inf_le_left) with ⟨x, hxy, hx⟩ exact ⟨x, hxy, 𝒰, le_inf hx (hy.trans inf_le_right)⟩ lemma isProperMap_iff_ultrafilter_of_t2 [T2Space Y] : IsProperMap f ↔ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄, ∀ ⦃y : Y⦄, Tendsto f 𝒰 (𝓝 y) → ∃ x, 𝒰.1 ≤ 𝓝 x := isProperMap_iff_ultrafilter.trans <\| and_congr_right fun hc ↦ forall₃_congr fun _𝒰 _y hy ↦ exists_congr fun x ↦ and_iff_right_of_imp fun h ↦ tendsto_nhds_unique ((hc.tendsto x).mono_left h) hy /-- If `f` is proper and converges to `y` along some ultrafilter `𝒰`, then `𝒰` converges to some `x` such that `f x = y`. -/ lemma IsProperMap.ultrafilter_le_nhds_of_tendsto (h : IsProperMap f) ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄ (hy : Tendsto f 𝒰 (𝓝 y)) : ∃ x, f x = y ∧ 𝒰 ≤ 𝓝 x := (isProperMap_iff_ultrafilter.mp h).2 hy /-- The composition of two proper maps is proper. -/ lemma IsProperMap.comp (hg : IsProperMap g) (hf : IsProperMap f) : IsProperMap (g ∘ f) := by refine ⟨by fun_prop, fun ℱ z h ↦ ?_⟩ rw [mapClusterPt_comp] at h rcases hg.clusterPt_of_mapClusterPt h with ⟨y, rfl, hy⟩ rcases hf.clusterPt_of_mapClusterPt hy with ⟨x, rfl, hx⟩ use x, rfl /-- If the composition of two continuous functions `g ∘ f` is proper and `f` is surjective, then `g` is proper. -/ lemma isProperMap_of_comp_of_surj (hf : Continuous f) (hg : Continuous g) (hgf : IsProperMap (g ∘ f)) (f_surj : f.Surjective) : IsProperMap g := by refine ⟨hg, fun ℱ z h ↦ ?_⟩ rw [← ℱ.map_comap_of_surjective f_surj, ← mapClusterPt_comp] at h rcases hgf.clusterPt_of_mapClusterPt h with ⟨x, rfl, hx⟩ rw [← ℱ.map_comap_of_surjective f_surj] exact ⟨f x, rfl, hx.map hf.continuousAt tendsto_map⟩ /-- If the composition of two continuous functions `g ∘ f` is proper and `g` is injective, then `f` is proper. -/ lemma isProperMap_of_comp_of_inj {f : X → Y} {g : Y → Z} (hf : Continuous f) (hg : Continuous g) (hgf : IsProperMap (g ∘ f)) (g_inj : g.Injective) : IsProperMap f := by refine ⟨hf, fun ℱ y h ↦ ?_⟩ rcases hgf.clusterPt_of_mapClusterPt (h.map hg.continuousAt tendsto_map) with ⟨x, hx1, hx2⟩ exact ⟨x, g_inj hx1, hx2⟩ /-- If the composition of two continuous functions `f : X → Y` and `g : Y → Z` is proper and `Y` is T2, then `f` is proper. -/ lemma isProperMap_of_comp_of_t2 [T2Space Y] (hf : Continuous f) (hg : Continuous g) (hgf : IsProperMap (g ∘ f)) : IsProperMap f := by rw [isProperMap_iff_ultrafilter_of_t2] refine ⟨hf, fun 𝒰 y h ↦ ?_⟩ rw [isProperMap_iff_ultrafilter] at hgf rcases hgf.2 ((hg.tendsto y).comp h) with ⟨x, -, hx⟩ exact ⟨x, hx⟩ /-- A binary product of proper maps is proper. -/ lemma IsProperMap.prodMap {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) : IsProperMap (Prod.map f g) := by simp_rw [isProperMap_iff_ultrafilter] at hf hg ⊢ constructor -- Continuity is clear. · exact hf.1.prodMap hg.1 -- Let `𝒰 : Ultrafilter (X × Z)`, and assume that `f × g` tends to some `(y, w) : Y × W` -- along `𝒰`. · intro 𝒰 ⟨y, w⟩ hyw -- That means that `f` tends to `y` along `map fst 𝒰` and `g` tends to `w` along `map snd 𝒰`. simp_rw [nhds_prod_eq, tendsto_prod_iff'] at hyw -- Thus, by properness of `f` and `g`, we get some `x : X` and `z : Z` such that `f x = y`, -- `g z = w`, `map fst 𝒰` tends to `x`, and `map snd 𝒰` tends to `y`. rcases hf.2 (show Tendsto f (Ultrafilter.map fst 𝒰) (𝓝 y) by simpa using hyw.1) with ⟨x, hxy, hx⟩ rcases hg.2 (show Tendsto g (Ultrafilter.map snd 𝒰) (𝓝 w) by simpa using hyw.2) with ⟨z, hzw, hz⟩ -- By the properties of the product topology, that means that `𝒰` tends to `(x, z)`, -- which completes the proof since `(f × g)(x, z) = (y, w)`. refine ⟨⟨x, z⟩, Prod.ext hxy hzw, ?_⟩ rw [nhds_prod_eq, le_prod] exact ⟨hx, hz⟩ /-- Any product of proper maps is proper. -/ lemma IsProperMap.pi_map {X Y : ι → Type} [∀ i, TopologicalSpace (X i)] [∀ i, TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (h : ∀ i, IsProperMap (f i)) : IsProperMap (fun (x : ∀ i, X i) i ↦ f i (x i)) := by simp_rw [isProperMap_iff_ultrafilter] at h ⊢ constructor -- Continuity is clear. · exact continuous_pi fun i ↦ (h i).1.comp (continuous_apply i) -- Let `𝒰 : Ultrafilter (Π i, X i)`, and assume that `Π i, f i` tends to some `y : Π i, Y i` -- along `𝒰`. · intro 𝒰 y hy -- That means that each `f i` tends to `y i` along `map (eval i) 𝒰`. have : ∀ i, Tendsto (f i) (Ultrafilter.map (eval i) 𝒰) (𝓝 (y i)) := by simpa [tendsto_pi_nhds] using hy -- Thus, by properness of all the `f i`s, we can choose some `x : Π i, X i` such that, for all -- `i`, `f i (x i) = y i` and `map (eval i) 𝒰` tends to `x i`. choose x hxy hx using fun i ↦ (h i).2 (this i) -- By the properties of the product topology, that means that `𝒰` tends to `x`, -- which completes the proof since `(Π i, f i) x = y`. refine ⟨x, funext hxy, ?_⟩ rwa [nhds_pi, le_pi] /-- The preimage of a compact set by a proper map is again compact. See also `isProperMap_iff_isCompact_preimage` which proves that this property completely characterizes proper map when the codomain is compactly generated and Hausdorff. -/ lemma IsProperMap.isCompact_preimage (h : IsProperMap f) {K : Set Y} (hK : IsCompact K) : IsCompact (f ⁻¹' K) := by rw [isCompact_iff_ultrafilter_le_nhds] -- Let `𝒰 ≤ 𝓟 (f ⁻¹' K)` an ultrafilter. intro 𝒰 h𝒰 -- In other words, we have `map f 𝒰 ≤ 𝓟 K` rw [← comap_principal, ← map_le_iff_le_comap, ← Ultrafilter.coe_map] at h𝒰 -- Thus, by compactness of `K`, the ultrafilter `map f 𝒰` tends to some `y ∈ K`. rcases hK.ultrafilter_le_nhds _ h𝒰 with ⟨y, hyK, hy⟩ -- Then, by properness of `f`, that means that `𝒰` tends to some `x ∈ f ⁻¹' {y} ⊆ f ⁻¹' K`, -- which completes the proof. rcases h.ultrafilter_le_nhds_of_tendsto hy with ⟨x, rfl, hx⟩ exact ⟨x, hyK, hx⟩ /-- A map is proper if and only if it is closed and its fibers are compact. -/ theorem isProperMap_iff_isClosedMap_and_compact_fibers : IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ ∀ y, IsCompact (f ⁻¹' {y}) := by constructor <;> intro H -- Note: In Bourbaki, the direct implication is proved by going through universally closed maps. -- We could do the same (using a `TFAE` cycle) but proving it directly from -- `IsProperMap.isCompact_preimage` is nice enough already so we don't bother with that. · exact ⟨H.continuous, H.isClosedMap, fun y ↦ H.isCompact_preimage isCompact_singleton⟩ · rw [isProperMap_iff_clusterPt] -- Let `ℱ : Filter X` and `y` some cluster point of `map f ℱ`. refine ⟨H.1, fun ℱ y hy ↦ ?_⟩ -- That means that the singleton `pure y` meets the "closure" of `map f ℱ`, by which we mean -- `Filter.lift' (map f ℱ) closure`. But `f` is closed, so -- `closure (map f ℱ) = map f (closure ℱ)` (see `IsClosedMap.lift'_closure_map_eq`). -- Thus `map f (closure ℱ ⊓ 𝓟 (f ⁻¹' {y})) = map f (closure ℱ) ⊓ 𝓟 {y} ≠ ⊥`, hence -- `closure ℱ ⊓ 𝓟 (f ⁻¹' {y}) ≠ ⊥`. rw [H.2.1.mapClusterPt_iff_lift'_closure H.1] at hy -- Now, applying the compactness of `f ⁻¹' {y}` to the nontrivial filter -- `closure ℱ ⊓ 𝓟 (f ⁻¹' {y})`, we obtain that it has a cluster point `x ∈ f ⁻¹' {y}`. rcases H.2.2 y (f := Filter.lift' ℱ closure ⊓ 𝓟 (f ⁻¹' {y})) inf_le_right with ⟨x, hxy, hx⟩ refine ⟨x, hxy, ?_⟩ -- In particular `x` is a cluster point of `closure ℱ`. Since cluster points of `closure ℱ` -- are exactly cluster points of `ℱ` (see `clusterPt_lift'_closure_iff`), this completes -- the proof. rw [← clusterPt_lift'_closure_iff] exact hx.mono inf_le_left /-- An injective and continuous function is proper if and only if it is closed. -/ lemma isProperMap_iff_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) : IsProperMap f ↔ IsClosedMap f := by refine ⟨fun h ↦ h.isClosedMap, fun h ↦ ?_⟩ rw [isProperMap_iff_isClosedMap_and_compact_fibers] exact ⟨f_cont, h, fun y ↦ (subsingleton_singleton.preimage f_inj).isCompact⟩ /-- A injective continuous and closed map is proper. -/ lemma isProperMap_of_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) (f_closed : IsClosedMap f) : IsProperMap f := (isProperMap_iff_isClosedMap_of_inj f_cont f_inj).2 f_closed /-- A homeomorphism is proper. -/ @[simp] lemma Homeomorph.isProperMap (e : X ≃ₜ Y) : IsProperMap e := isProperMap_of_isClosedMap_of_inj e.continuous e.injective e.isClosedMap protected lemma IsHomeomorph.isProperMap (hf : IsHomeomorph f) : IsProperMap f := isProperMap_of_isClosedMap_of_inj hf.continuous hf.injective hf.isClosedMap /-- The identity is proper. -/ @[simp] lemma isProperMap_id : IsProperMap (id : X → X) := IsHomeomorph.id.isProperMap /-- A closed embedding is proper. -/ lemma Topology.IsClosedEmbedding.isProperMap (hf : IsClosedEmbedding f) : IsProperMap f := isProperMap_of_isClosedMap_of_inj hf.continuous hf.injective hf.isClosedMap /-- The coercion from a closed subset is proper. -/ lemma IsClosed.isProperMap_subtypeVal {C : Set X} (hC : IsClosed C) : IsProperMap ((↑) : C → X) := hC.isClosedEmbedding_subtypeVal.isProperMap /-- The restriction of a proper map to a closed subset is proper. -/ lemma IsProperMap.restrict {C : Set X} (hf : IsProperMap f) (hC : IsClosed C) : IsProperMap fun x : C ↦ f x := hf.comp hC.isProperMap_subtypeVal /-- The range of a proper map is closed. -/ lemma IsProperMap.isClosed_range (hf : IsProperMap f) : IsClosed (range f) := hf.isClosedMap.isClosed_range /-- Version of `isProperMap_iff_isClosedMap_and_compact_fibers` in terms of `cofinite` and `cocompact`. Only works when the codomain is `T1`. -/ lemma isProperMap_iff_isClosedMap_and_tendsto_cofinite [T1Space Y] : IsProperMap f ↔ Continuous f ∧ IsClosedMap f ∧ Tendsto f (cocompact X) cofinite := by simp_rw [isProperMap_iff_isClosedMap_and_compact_fibers, Tendsto, le_cofinite_iff_compl_singleton_mem, mem_map, preimage_compl] refine and_congr_right fun f_cont ↦ and_congr_right fun _ ↦ ⟨fun H y ↦ (H y).compl_mem_cocompact, fun H y ↦ ?_⟩ rcases mem_cocompact.mp (H y) with ⟨K, hK, hKy⟩ exact hK.of_isClosed_subset (isClosed_singleton.preimage f_cont) (compl_le_compl_iff_le.mp hKy) /-- A continuous map from a compact space to a T₂ space is a proper map. -/ theorem Continuous.isProperMap [CompactSpace X] [T2Space Y] (hf : Continuous f) : IsProperMap f := isProperMap_iff_isClosedMap_and_tendsto_cofinite.2 ⟨hf, hf.isClosedMap, by simp⟩ /-- A constant map to a T₁ space is proper if and only if its domain is compact. -/ theorem isProperMap_const_iff [T1Space Y] (y : Y) : IsProperMap (fun _ : X ↦ y) ↔ CompactSpace X := by classical rw [isProperMap_iff_isClosedMap_and_compact_fibers] constructor · rintro ⟨-, -, h⟩ exact ⟨by simpa using h y⟩ · intro H refine ⟨continuous_const, isClosedMap_const, fun y' ↦ ?_⟩ simp [preimage_const, mem_singleton_iff, apply_ite, isCompact_univ] theorem isProperMap_const [h : CompactSpace X] [T1Space Y] (y : Y) : IsProperMap (fun _ : X ↦ y) := isProperMap_const_iff y \|>.mpr h /-- If `Y` is a compact topological space, then `Prod.fst : X × Y → X` is a proper map. -/ theorem isProperMap_fst_of_compactSpace [CompactSpace Y] : IsProperMap (Prod.fst : X × Y → X) := Homeomorph.prodPUnit X \|>.isProperMap.comp (isProperMap_id.prodMap (isProperMap_const ())) /-- If `X` is a compact topological space, then `Prod.snd : X × Y → Y` is a proper map. -/ theorem isProperMap_snd_of_compactSpace [CompactSpace X] : IsProperMap (Prod.snd : X × Y → Y) := Homeomorph.punitProd Y \|>.isProperMap.comp ((isProperMap_const ()).prodMap isProperMap_id) /-- If `Y` is a compact topological space, then `Prod.fst : X × Y → X` is a closed map. -/ theorem isClosedMap_fst_of_compactSpace [CompactSpace Y] : IsClosedMap (Prod.fst : X × Y → X) := isProperMap_fst_of_compactSpace.isClosedMap /-- If `X` is a compact topological space, then `Prod.snd : X × Y → Y` is a closed map. -/ theorem isClosedMap_snd_of_compactSpace [CompactSpace X] : IsClosedMap (Prod.snd : X × Y → Y) := isProperMap_snd_of_compactSpace.isClosedMap /-- A proper map `f : X → Y` is universally closed*: for any topological space `Z`, the map `Prod.map f id : X × Z → Y × Z` is closed. We will prove in `isProperMap_iff_universally_closed` that proper maps are exactly continuous maps which have this property, but this result should be easier to use because it allows `Z` to live in any universe. -/ theorem IsProperMap.universally_closed (Z) [TopologicalSpace Z] (h : IsProperMap f) : IsClosedMap (Prod.map f id : X × Z → Y × Z) := -- `f × id` is proper as a product of proper maps, hence closed. (h.prodMap isProperMap_id).isClosedMap
.lake/packages/mathlib/Mathlib/Topology/Maps/Proper/UniversallyClosed.lean	import Mathlib.Topology.Compactification.StoneCech import Mathlib.Topology.Filter import Mathlib.Topology.Maps.Proper.Basic /-! # A map is proper iff it is continuous and universally closed -/ open Filter universe u v /-- A map `f : X → Y` is proper if and only if it is continuous and the map `(Prod.map f id : X × Filter X → Y × Filter X)` is closed. This is stronger than `isProperMap_iff_universally_closed` since it shows that there's only one space to check to get properness, but in most cases it doesn't matter. -/ theorem isProperMap_iff_isClosedMap_filter {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : IsProperMap f ↔ Continuous f ∧ IsClosedMap (Prod.map f id : X × Filter X → Y × Filter X) := by constructor <;> intro H -- The direct implication is clear. · exact ⟨H.continuous, H.universally_closed _⟩ · rw [isProperMap_iff_ultrafilter] -- Let `𝒰 : Ultrafilter X`, and assume that `f` tends to some `y` along `𝒰`. refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩ -- In `X × Filter X`, consider the closed set `F := closure {(x, ℱ) \| ℱ = pure x}` let F : Set (X × Filter X) := closure {xℱ \| xℱ.2 = pure xℱ.1} -- Since `f × id` is closed, the set `(f × id) '' F` is also closed. have := H.2 F isClosed_closure -- Let us show that `(y, 𝒰) ∈ (f × id) '' F`. have : (y, ↑𝒰) ∈ Prod.map f id '' F := -- Note that, by the properties of the topology on `Filter X`, the function `pure : X → Filter X` -- tends to the point `𝒰` of `Filter X` along the filter `𝒰` on `X`. Since `f` tends to `y` along -- `𝒰`, we get that the function `(f, pure) : X → (Y, Filter X)` tends to `(y, 𝒰)` along -- `𝒰`. Furthermore, each `(f, pure)(x) = (f × id)(x, pure x)` is clearly an element of -- the closed set `(f × id) '' F`, thus the limit `(y, 𝒰)` also belongs to that set. this.mem_of_tendsto (hy.prodMk_nhds (Filter.tendsto_pure_self (𝒰 : Filter X))) (Eventually.of_forall fun x ↦ ⟨⟨x, pure x⟩, subset_closure rfl, rfl⟩) -- The above shows that `(y, 𝒰) = (f x, 𝒰)`, for some `x : X` such that `(x, 𝒰) ∈ F`. rcases this with ⟨⟨x, _⟩, hx, ⟨_, _⟩⟩ -- We already know that `f x = y`, so to finish the proof we just have to check that `𝒰` tends -- to `x`. So, for `U ∈ 𝓝 x` arbitrary, let's show that `U ∈ 𝒰`. Since `𝒰` is a ultrafilter, -- it is enough to show that `Uᶜ` is not in `𝒰`. refine ⟨x, rfl, fun U hU ↦ Ultrafilter.compl_notMem_iff.mp fun hUc ↦ ?_⟩ rw [mem_closure_iff_nhds] at hx -- Indeed, if that was the case, the set `V := {𝒢 : Filter X \| Uᶜ ∈ 𝒢}` would be a neighborhood -- of `𝒰` in `Filter X`, hence `U ×ˢ V` would be a neighborhood of `(x, 𝒰) : X × Filter X`. -- But recall that `(x, 𝒰) ∈ F = closure {(x, ℱ) \| ℱ = pure x}`, so the neighborhood `U ×ˢ V` -- must contain some element of the form `(z, pure z)`. In other words, we have `z ∈ U` and -- `Uᶜ ∈ pure z`, which means `z ∈ Uᶜ` by the definition of pure. -- This is a contradiction, which completes the proof. rcases hx (U ×ˢ {𝒢 \| Uᶜ ∈ 𝒢}) (prod_mem_nhds hU (isOpen_setOf_mem.mem_nhds hUc)) with ⟨⟨z, 𝒢⟩, ⟨⟨hz : z ∈ U, hz' : Uᶜ ∈ 𝒢⟩, rfl : 𝒢 = pure z⟩⟩ exact hz' hz /-- A map `f : X → Y` is proper if and only if it is continuous and the map `(Prod.map f id : X × Ultrafilter X → Y × Ultrafilter X)` is closed. This is stronger than `isProperMap_iff_universally_closed` since it shows that there's only one space to check to get properness, but in most cases it doesn't matter. -/ theorem isProperMap_iff_isClosedMap_ultrafilter {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : IsProperMap f ↔ Continuous f ∧ IsClosedMap (Prod.map f id : X × Ultrafilter X → Y × Ultrafilter X) := by -- The proof is basically the same as above. constructor <;> intro H · exact ⟨H.continuous, H.universally_closed _⟩ · rw [isProperMap_iff_ultrafilter] refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩ let F : Set (X × Ultrafilter X) := closure {xℱ \| xℱ.2 = pure xℱ.1} have := H.2 F isClosed_closure have : (y, 𝒰) ∈ Prod.map f id '' F := this.mem_of_tendsto (hy.prodMk_nhds (Ultrafilter.tendsto_pure_self 𝒰)) (Eventually.of_forall fun x ↦ ⟨⟨x, pure x⟩, subset_closure rfl, rfl⟩) rcases this with ⟨⟨x, _⟩, hx, ⟨_, _⟩⟩ refine ⟨x, rfl, fun U hU ↦ Ultrafilter.compl_notMem_iff.mp fun hUc ↦ ?_⟩ rw [mem_closure_iff_nhds] at hx rcases hx (U ×ˢ {𝒢 \| Uᶜ ∈ 𝒢}) (prod_mem_nhds hU ((ultrafilter_isOpen_basic _).mem_nhds hUc)) with ⟨⟨y, 𝒢⟩, ⟨⟨hy : y ∈ U, hy' : Uᶜ ∈ 𝒢⟩, rfl : 𝒢 = pure y⟩⟩ exact hy' hy /-- A map `f : X → Y` is proper if and only if it is continuous and universally closed, in the sense that for any topological space `Z`, the map `Prod.map f id : X × Z → Y × Z` is closed. Note that `Z` lives in the same universe as `X` here, but `IsProperMap.universally_closed` does not have this restriction. This is taken as the definition of properness in [N. Bourbaki, General Topology][bourbaki1966]. -/ theorem isProperMap_iff_universally_closed {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : IsProperMap f ↔ Continuous f ∧ ∀ (Z : Type u) [TopologicalSpace Z], IsClosedMap (Prod.map f id : X × Z → Y × Z) := by constructor <;> intro H · exact ⟨H.continuous, fun Z ↦ H.universally_closed _⟩ · rw [isProperMap_iff_isClosedMap_ultrafilter] exact ⟨H.1, H.2 _⟩
.lake/packages/mathlib/Mathlib/Topology/Order/IsLocallyClosed.lean	import Mathlib.Topology.Order.OrderClosed import Mathlib.Topology.LocallyClosed /-! # Intervals are locally closed We prove that the intervals on a topological ordered space are locally closed. -/ variable {X : Type*} [TopologicalSpace X] {a b : X} theorem isLocallyClosed_Icc [Preorder X] [OrderClosedTopology X] : IsLocallyClosed (Set.Icc a b) := isClosed_Icc.isLocallyClosed theorem isLocallyClosed_Ioo [LinearOrder X] [OrderClosedTopology X] : IsLocallyClosed (Set.Ioo a b) := isOpen_Ioo.isLocallyClosed theorem isLocallyClosed_Ici [Preorder X] [ClosedIciTopology X] : IsLocallyClosed (Set.Ici a) := isClosed_Ici.isLocallyClosed theorem isLocallyClosed_Iic [Preorder X] [ClosedIicTopology X] : IsLocallyClosed (Set.Iic a) := isClosed_Iic.isLocallyClosed theorem isLocallyClosed_Ioi [LinearOrder X] [ClosedIicTopology X] : IsLocallyClosed (Set.Ioi a) := isOpen_Ioi.isLocallyClosed theorem isLocallyClosed_Iio [LinearOrder X] [ClosedIciTopology X] : IsLocallyClosed (Set.Iio a) := isOpen_Iio.isLocallyClosed theorem isLocallyClosed_Ioc [LinearOrder X] [ClosedIicTopology X] : IsLocallyClosed (Set.Ioc a b) := by rw [← Set.Iic_inter_Ioi] exact isLocallyClosed_Iic.inter isLocallyClosed_Ioi theorem isLocallyClosed_Ico [LinearOrder X] [ClosedIciTopology X] : IsLocallyClosed (Set.Ico a b) := by rw [← Set.Iio_inter_Ici] exact isLocallyClosed_Iio.inter isLocallyClosed_Ici
.lake/packages/mathlib/Mathlib/Topology/Order/MonotoneContinuity.lean	import Mathlib.Topology.Homeomorph.Defs import Mathlib.Topology.Order.LeftRightNhds /-! # Continuity of monotone functions In this file we prove the following fact: if `f` is a monotone function on a neighborhood of `a` and the image of this neighborhood is a neighborhood of `f a`, then `f` is continuous at `a`, see `continuousWithinAt_of_monotoneOn_of_image_mem_nhds`, as well as several similar facts. We also prove that an `OrderIso` is continuous. ## Tags continuous, monotone -/ open Set Filter open Topology section LinearOrder variable {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] variable [LinearOrder β] [TopologicalSpace β] [OrderTopology β] /-- If `f` is a function strictly monotone on a right neighborhood of `a` and the image of this neighborhood under `f` meets every interval `(f a, b]`, `b > f a`, then `f` is continuous at `a` from the right. The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` is required because otherwise the function `f : ℝ → ℝ` given by `f x = if x ≤ 0 then x else x + 1` would be a counter-example at `a = 0`. -/ theorem StrictMonoOn.continuousWithinAt_right_of_exists_between {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) : ContinuousWithinAt f (Ici a) a := by have ha : a ∈ Ici a := left_mem_Ici have has : a ∈ s := mem_of_mem_nhdsWithin ha hs refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩ · filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le ((h_mono.le_iff_le has hxs).2 hxa) · rcases hfs b hb with ⟨c, hcs, hac, hcb⟩ rw [h_mono.lt_iff_lt has hcs] at hac filter_upwards [hs, Ico_mem_nhdsGE hac] rintro x hx ⟨_, hxc⟩ exact ((h_mono.lt_iff_lt hx hcs).2 hxc).trans_le hcb /-- If `f` is a monotone function on a right neighborhood of `a` and the image of this neighborhood under `f` meets every interval `(f a, b)`, `b > f a`, then `f` is continuous at `a` from the right. The assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b` cannot be replaced by the weaker assumption `hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b` we use for strictly monotone functions because otherwise the function `ceil : ℝ → ℤ` would be a counter-example at `a = 0`. -/ theorem continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) : ContinuousWithinAt f (Ici a) a := by have ha : a ∈ Ici a := left_mem_Ici have has : a ∈ s := mem_of_mem_nhdsWithin ha hs refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩ · filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le (h_mono has hxs hxa) · rcases hfs b hb with ⟨c, hcs, hac, hcb⟩ have : a < c := not_le.1 fun h => hac.not_ge <\| h_mono hcs has h filter_upwards [hs, Ico_mem_nhdsGE this] rintro x hx ⟨_, hxc⟩ exact (h_mono hx hcs hxc.le).trans_lt hcb /-- If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at `a` from the right. -/ theorem continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a := by refine continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono hs fun b hb => ?_ rcases (mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩ rcases exists_between hab' with ⟨c', hc'⟩ rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') isOpen_Ioo hc' with ⟨_, hc, ⟨c, hcs, rfl⟩⟩ exact ⟨c, hcs, hc.1, hc.2.trans_le hbb'⟩ /-- If a function `f` with a densely ordered codomain is monotone on a right neighborhood of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at `a` from the right. -/ theorem continuousWithinAt_right_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a := continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono hs <\| mem_of_superset hfs subset_closure /-- If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood of `a` and the closure of the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at `a` from the right. -/ theorem StrictMonoOn.continuousWithinAt_right_of_closure_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : closure (f '' s) ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a := continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin (fun _ hx _ hy => (h_mono.le_iff_le hx hy).2) hs hfs /-- If a function `f` with a densely ordered codomain is strictly monotone on a right neighborhood of `a` and the image of this neighborhood under `f` is a right neighborhood of `f a`, then `f` is continuous at `a` from the right. -/ theorem StrictMonoOn.continuousWithinAt_right_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : f '' s ∈ 𝓝[≥] f a) : ContinuousWithinAt f (Ici a) a := h_mono.continuousWithinAt_right_of_closure_image_mem_nhdsWithin hs (mem_of_superset hfs subset_closure) /-- If a function `f` is strictly monotone on a right neighborhood of `a` and the image of this neighborhood under `f` includes `Ioi (f a)`, then `f` is continuous at `a` from the right. -/ theorem StrictMonoOn.continuousWithinAt_right_of_surjOn {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : SurjOn f s (Ioi (f a))) : ContinuousWithinAt f (Ici a) a := h_mono.continuousWithinAt_right_of_exists_between hs fun _ hb => let ⟨c, hcs, hcb⟩ := hfs hb ⟨c, hcs, hcb.symm ▸ hb, hcb.le⟩ /-- If `f` is a strictly monotone function on a left neighborhood of `a` and the image of this neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, then `f` is continuous at `a` from the left. The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` is required because otherwise the function `f : ℝ → ℝ` given by `f x = if x < 0 then x else x + 1` would be a counter-example at `a = 0`. -/ theorem StrictMonoOn.continuousWithinAt_left_of_exists_between {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) : ContinuousWithinAt f (Iic a) a := h_mono.dual.continuousWithinAt_right_of_exists_between hs fun b hb => let ⟨c, hcs, hcb, hca⟩ := hfs b hb ⟨c, hcs, hca, hcb⟩ /-- If `f` is a monotone function on a left neighborhood of `a` and the image of this neighborhood under `f` meets every interval `(b, f a)`, `b < f a`, then `f` is continuous at `a` from the left. The assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)` cannot be replaced by the weaker assumption `hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)` we use for strictly monotone functions because otherwise the function `floor : ℝ → ℤ` would be a counter-example at `a = 0`. -/ theorem continuousWithinAt_left_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) : ContinuousWithinAt f (Iic a) a := @continuousWithinAt_right_of_monotoneOn_of_exists_between αᵒᵈ βᵒᵈ _ _ _ _ _ _ f s a hf.dual hs fun b hb => let ⟨c, hcs, hcb, hca⟩ := hfs b hb ⟨c, hcs, hca, hcb⟩ /-- If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at `a` from the left -/ theorem continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (hf : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a := @continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ f s a hf.dual hs hfs /-- If a function `f` with a densely ordered codomain is monotone on a left neighborhood of `a` and the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at `a` from the left. -/ theorem continuousWithinAt_left_of_monotoneOn_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a := continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono hs (mem_of_superset hfs subset_closure) /-- If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of `a` and the closure of the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at `a` from the left. -/ theorem StrictMonoOn.continuousWithinAt_left_of_closure_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : closure (f '' s) ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a := h_mono.dual.continuousWithinAt_right_of_closure_image_mem_nhdsWithin hs hfs /-- If a function `f` with a densely ordered codomain is strictly monotone on a left neighborhood of `a` and the image of this neighborhood under `f` is a left neighborhood of `f a`, then `f` is continuous at `a` from the left. -/ theorem StrictMonoOn.continuousWithinAt_left_of_image_mem_nhdsWithin [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : f '' s ∈ 𝓝[≤] f a) : ContinuousWithinAt f (Iic a) a := h_mono.dual.continuousWithinAt_right_of_image_mem_nhdsWithin hs hfs /-- If a function `f` is strictly monotone on a left neighborhood of `a` and the image of this neighborhood under `f` includes `Iio (f a)`, then `f` is continuous at `a` from the left. -/ theorem StrictMonoOn.continuousWithinAt_left_of_surjOn {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝[≤] a) (hfs : SurjOn f s (Iio (f a))) : ContinuousWithinAt f (Iic a) a := h_mono.dual.continuousWithinAt_right_of_surjOn hs hfs /-- If a function `f` is strictly monotone on a neighborhood of `a` and the image of this neighborhood under `f` meets every interval `[b, f a)`, `b < f a`, and every interval `(f a, b]`, `b > f a`, then `f` is continuous at `a`. -/ theorem StrictMonoOn.continuousAt_of_exists_between {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ico b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b) : ContinuousAt f a := continuousAt_iff_continuous_left_right.2 ⟨h_mono.continuousWithinAt_left_of_exists_between (mem_nhdsWithin_of_mem_nhds hs) hfs_l, h_mono.continuousWithinAt_right_of_exists_between (mem_nhdsWithin_of_mem_nhds hs) hfs_r⟩ /-- If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a` and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. -/ theorem StrictMonoOn.continuousAt_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a := continuousAt_iff_continuous_left_right.2 ⟨h_mono.continuousWithinAt_left_of_closure_image_mem_nhdsWithin (mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs), h_mono.continuousWithinAt_right_of_closure_image_mem_nhdsWithin (mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs)⟩ /-- If a function `f` with a densely ordered codomain is strictly monotone on a neighborhood of `a` and the image of this set under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. -/ theorem StrictMonoOn.continuousAt_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : StrictMonoOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) : ContinuousAt f a := h_mono.continuousAt_of_closure_image_mem_nhds hs (mem_of_superset hfs subset_closure) /-- If `f` is a monotone function on a neighborhood of `a` and the image of this neighborhood under `f` meets every interval `(b, f a)`, `b < f a`, and every interval `(f a, b)`, `b > f a`, then `f` is continuous at `a`. -/ theorem continuousAt_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs_l : ∀ b < f a, ∃ c ∈ s, f c ∈ Ioo b (f a)) (hfs_r : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) : ContinuousAt f a := continuousAt_iff_continuous_left_right.2 ⟨continuousWithinAt_left_of_monotoneOn_of_exists_between h_mono (mem_nhdsWithin_of_mem_nhds hs) hfs_l, continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono (mem_nhdsWithin_of_mem_nhds hs) hfs_r⟩ /-- If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the closure of the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. -/ theorem continuousAt_of_monotoneOn_of_closure_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : closure (f '' s) ∈ 𝓝 (f a)) : ContinuousAt f a := continuousAt_iff_continuous_left_right.2 ⟨continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono (mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs), continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin h_mono (mem_nhdsWithin_of_mem_nhds hs) (mem_nhdsWithin_of_mem_nhds hfs)⟩ /-- If a function `f` with a densely ordered codomain is monotone on a neighborhood of `a` and the image of this neighborhood under `f` is a neighborhood of `f a`, then `f` is continuous at `a`. -/ theorem continuousAt_of_monotoneOn_of_image_mem_nhds [DenselyOrdered β] {f : α → β} {s : Set α} {a : α} (h_mono : MonotoneOn f s) (hs : s ∈ 𝓝 a) (hfs : f '' s ∈ 𝓝 (f a)) : ContinuousAt f a := continuousAt_of_monotoneOn_of_closure_image_mem_nhds h_mono hs (mem_of_superset hfs subset_closure) /-- A monotone function with densely ordered codomain and a dense range is continuous. -/ theorem Monotone.continuous_of_denseRange [DenselyOrdered β] {f : α → β} (h_mono : Monotone f) (h_dense : DenseRange f) : Continuous f := continuous_iff_continuousAt.mpr fun a => continuousAt_of_monotoneOn_of_closure_image_mem_nhds (fun _ _ _ _ hxy => h_mono hxy) univ_mem <\| by simp only [image_univ, h_dense.closure_eq, univ_mem] /-- A monotone surjective function with a densely ordered codomain is continuous. -/ theorem Monotone.continuous_of_surjective [DenselyOrdered β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) : Continuous f := h_mono.continuous_of_denseRange h_surj.denseRange end LinearOrder /-! ### Continuity of order isomorphisms In this section we prove that an `OrderIso` is continuous, hence it is a `Homeomorph`. We prove this for an `OrderIso` between to partial orders with order topology. -/ namespace OrderIso variable {α β : Type} [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [OrderTopology α] [OrderTopology β] protected theorem continuous (e : α ≃o β) : Continuous e := by rw [‹OrderTopology β›.topology_eq_generate_intervals, continuous_generateFrom_iff] rintro s ⟨a, rfl \| rfl⟩ · rw [e.preimage_Ioi] apply isOpen_lt' · rw [e.preimage_Iio] apply isOpen_gt' instance : HomeomorphClass (α ≃o β) α β where map_continuous := OrderIso.continuous inv_continuous e := e.symm.continuous /-- An order isomorphism between two linear order `OrderTopology` spaces is a homeomorphism. -/ abbrev toHomeomorph (e : α ≃o β) : α ≃ₜ β := HomeomorphClass.toHomeomorph e theorem coe_toHomeomorph (e : α ≃o β) : ⇑e.toHomeomorph = e := rfl --Simp can prove this too @[simp] theorem coe_toHomeomorph_symm (e : α ≃o β) : ⇑e.toHomeomorph.symm = e.symm := rfl end OrderIso
.lake/packages/mathlib/Mathlib/Topology/Order/LowerUpperTopology.lean	import Mathlib.Order.Hom.CompleteLattice import Mathlib.Topology.Homeomorph.Defs import Mathlib.Topology.Order.Lattice /-! # Lower and Upper topology This file introduces the lower topology on a preorder as the topology generated by the complements of the left-closed right-infinite intervals. For completeness we also introduce the dual upper topology, generated by the complements of the right-closed left-infinite intervals. ## Main statements - `IsLower.t0Space` - the lower topology on a partial order is T₀ - `IsLower.isTopologicalBasis` - the complements of the upper closures of finite subsets form a basis for the lower topology - `IsLower.continuousInf` - the inf map is continuous with respect to the lower topology ## Implementation notes A type synonym `WithLower` is introduced and for a preorder `α`, `WithLower α` is made an instance of `TopologicalSpace` by the topology generated by the complements of the closed intervals to infinity. We define a mixin class `IsLower` for the class of types which are both a preorder and a topology and where the topology is generated by the complements of the closed intervals to infinity. It is shown that `WithLower α` is an instance of `IsLower`. Similarly for the upper topology. ## Motivation The lower topology is used with the `Scott` topology to define the Lawson topology. The restriction of the lower topology to the spectrum of a complete lattice coincides with the hull-kernel topology. ## References * [Gierz et al, A Compendium of Continuous Lattices][GierzEtAl1980] ## Tags lower topology, upper topology, preorder -/ open Set TopologicalSpace Topology namespace Topology /-- The lower topology is the topology generated by the complements of the left-closed right-infinite intervals. -/ def lower (α : Type) [Preorder α] : TopologicalSpace α := generateFrom {s \| ∃ a, (Ici a)ᶜ = s} /-- The upper topology is the topology generated by the complements of the right-closed left-infinite intervals. -/ def upper (α : Type) [Preorder α] : TopologicalSpace α := generateFrom {s \| ∃ a, (Iic a)ᶜ = s} /-- Type synonym for a preorder equipped with the lower set topology. -/ def WithLower (α : Type) := α variable {α β : Type} namespace WithLower /-- `toLower` is the identity function to the `WithLower` of a type. -/ @[match_pattern] def toLower : α ≃ WithLower α := Equiv.refl _ /-- `ofLower` is the identity function from the `WithLower` of a type. -/ @[match_pattern] def ofLower : WithLower α ≃ α := Equiv.refl _ @[simp] lemma toLower_symm : (@toLower α).symm = ofLower := rfl @[simp] lemma ofLower_symm : (@ofLower α).symm = toLower := rfl @[simp] lemma toLower_ofLower (a : WithLower α) : toLower (ofLower a) = a := rfl @[simp] lemma ofLower_toLower (a : α) : ofLower (toLower a) = a := rfl lemma toLower_inj {a b : α} : toLower a = toLower b ↔ a = b := Iff.rfl theorem ofLower_inj {a b : WithLower α} : ofLower a = ofLower b ↔ a = b := Iff.rfl /-- A recursor for `WithLower`. Use as `induction x`. -/ @[elab_as_elim, cases_eliminator, induction_eliminator] protected def rec {β : WithLower α → Sort} (h : ∀ a, β (toLower a)) : ∀ a, β a := fun a => h (ofLower a) instance [Nonempty α] : Nonempty (WithLower α) := ‹Nonempty α› instance [Inhabited α] : Inhabited (WithLower α) := ‹Inhabited α› section Preorder variable [Preorder α] {s : Set α} instance : Preorder (WithLower α) := ‹Preorder α› instance : TopologicalSpace (WithLower α) := lower (WithLower α) @[simp] lemma toLower_le_toLower {x y : α} : toLower x ≤ toLower y ↔ x ≤ y := .rfl @[simp] lemma toLower_lt_toLower {x y : α} : toLower x < toLower y ↔ x < y := .rfl @[simp] lemma ofLower_le_ofLower {x y : WithLower α} : ofLower x ≤ ofLower y ↔ x ≤ y := .rfl @[simp] lemma ofLower_lt_ofLower {x y : WithLower α} : ofLower x < ofLower y ↔ x < y := .rfl lemma isOpen_preimage_ofLower : IsOpen (ofLower ⁻¹' s) ↔ IsOpen[lower α] s := Iff.rfl lemma isOpen_def (T : Set (WithLower α)) : IsOpen T ↔ IsOpen[lower α] (WithLower.toLower ⁻¹' T) := Iff.rfl theorem continuous_toLower [TopologicalSpace α] [ClosedIciTopology α] : Continuous (toLower : α → WithLower α) := continuous_generateFrom_iff.mpr <\| by rintro _ ⟨a, rfl⟩; exact isClosed_Ici.isOpen_compl end Preorder instance [PartialOrder α] : PartialOrder (WithLower α) := ‹PartialOrder α› instance [LinearOrder α] : LinearOrder (WithLower α) := ‹LinearOrder α› end WithLower /-- Type synonym for a preorder equipped with the upper topology. -/ def WithUpper (α : Type) := α namespace WithUpper /-- `toUpper` is the identity function to the `WithUpper` of a type. -/ @[match_pattern] def toUpper : α ≃ WithUpper α := Equiv.refl _ /-- `ofUpper` is the identity function from the `WithUpper` of a type. -/ @[match_pattern] def ofUpper : WithUpper α ≃ α := Equiv.refl _ @[simp] lemma toUpper_symm {α} : (@toUpper α).symm = ofUpper := rfl @[simp] lemma ofUpper_symm : (@ofUpper α).symm = toUpper := rfl @[simp] lemma toUpper_ofUpper (a : WithUpper α) : toUpper (ofUpper a) = a := rfl @[simp] lemma ofUpper_toUpper (a : α) : ofUpper (toUpper a) = a := rfl lemma toUpper_inj {a b : α} : toUpper a = toUpper b ↔ a = b := Iff.rfl lemma ofUpper_inj {a b : WithUpper α} : ofUpper a = ofUpper b ↔ a = b := Iff.rfl /-- A recursor for `WithUpper`. Use as `induction x`. -/ @[elab_as_elim, cases_eliminator, induction_eliminator] protected def rec {β : WithUpper α → Sort} (h : ∀ a, β (toUpper a)) : ∀ a, β a := fun a => h (ofUpper a) instance [Nonempty α] : Nonempty (WithUpper α) := ‹Nonempty α› instance [Inhabited α] : Inhabited (WithUpper α) := ‹Inhabited α› section Preorder variable [Preorder α] {s : Set α} instance : Preorder (WithUpper α) := ‹Preorder α› instance : TopologicalSpace (WithUpper α) := upper (WithUpper α) @[simp] lemma toUpper_le_toUpper {x y : α} : toUpper x ≤ toUpper y ↔ x ≤ y := .rfl @[simp] lemma toUpper_lt_toUpper {x y : α} : toUpper x < toUpper y ↔ x < y := .rfl @[simp] lemma ofUpper_le_ofUpper {x y : WithUpper α} : ofUpper x ≤ ofUpper y ↔ x ≤ y := .rfl @[simp] lemma ofUpper_lt_ofUpper {x y : WithUpper α} : ofUpper x < ofUpper y ↔ x < y := .rfl lemma isOpen_preimage_ofUpper : IsOpen (ofUpper ⁻¹' s) ↔ (upper α).IsOpen s := Iff.rfl lemma isOpen_def {s : Set (WithUpper α)} : IsOpen s ↔ (upper α).IsOpen (toUpper ⁻¹' s) := Iff.rfl theorem continuous_toUpper [TopologicalSpace α] [ClosedIicTopology α] : Continuous (toUpper : α → WithUpper α) := continuous_generateFrom_iff.mpr <\| by rintro _ ⟨a, rfl⟩; exact isClosed_Iic.isOpen_compl end Preorder instance [PartialOrder α] : PartialOrder (WithUpper α) := ‹PartialOrder α› instance [LinearOrder α] : LinearOrder (WithUpper α) := ‹LinearOrder α› end WithUpper /-- The lower topology is the topology generated by the complements of the left-closed right-infinite intervals. -/ class IsLower (α : Type) [t : TopologicalSpace α] [Preorder α] : Prop where topology_eq_lowerTopology : t = lower α attribute [nolint docBlame] IsLower.topology_eq_lowerTopology /-- The upper topology is the topology generated by the complements of the right-closed left-infinite intervals. -/ class IsUpper (α : Type) [t : TopologicalSpace α] [Preorder α] : Prop where topology_eq_upperTopology : t = upper α attribute [nolint docBlame] IsUpper.topology_eq_upperTopology instance [Preorder α] : IsLower (WithLower α) := ⟨rfl⟩ instance [Preorder α] : IsUpper (WithUpper α) := ⟨rfl⟩ /-- The lower topology is homeomorphic to the upper topology on the dual order -/ def WithLower.toDualHomeomorph [Preorder α] : WithLower α ≃ₜ WithUpper αᵒᵈ where toFun := OrderDual.toDual invFun := OrderDual.ofDual left_inv := OrderDual.toDual_ofDual right_inv := OrderDual.ofDual_toDual continuous_toFun := continuous_coinduced_rng continuous_invFun := continuous_coinduced_rng namespace IsLower /-- The complements of the upper closures of finite sets are a collection of lower sets which form a basis for the lower topology. -/ def lowerBasis (α : Type) [Preorder α] := { s : Set α \| ∃ t : Set α, t.Finite ∧ (upperClosure t : Set α)ᶜ = s } section Preorder variable (α) variable [Preorder α] [TopologicalSpace α] [IsLower α] {s : Set α} lemma topology_eq : ‹_› = lower α := topology_eq_lowerTopology variable {α} /-- If `α` is equipped with the lower topology, then it is homeomorphic to `WithLower α`. -/ def withLowerHomeomorph : WithLower α ≃ₜ α := WithLower.ofLower.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ theorem isOpen_iff_generate_Ici_compl : IsOpen s ↔ GenerateOpen { t \| ∃ a, (Ici a)ᶜ = t } s := by rw [topology_eq α]; rfl instance _root_.OrderDual.instIsUpper [Preorder α] [TopologicalSpace α] [IsLower α] : IsUpper αᵒᵈ where topology_eq_upperTopology := topology_eq_lowerTopology (α := α) /-- Left-closed right-infinite intervals [a, ∞) are closed in the lower topology. -/ instance : ClosedIciTopology α := ⟨fun a ↦ isOpen_compl_iff.1 <\| isOpen_iff_generate_Ici_compl.2 <\| GenerateOpen.basic _ ⟨a, rfl⟩⟩ /-- The upper closure of a finite set is closed in the lower topology. -/ theorem isClosed_upperClosure (h : s.Finite) : IsClosed (upperClosure s : Set α) := by simp only [← UpperSet.iInf_Ici, UpperSet.coe_iInf] exact h.isClosed_biUnion fun _ _ => isClosed_Ici /-- Every set open in the lower topology is a lower set. -/ theorem isLowerSet_of_isOpen (h : IsOpen s) : IsLowerSet s := by replace h := isOpen_iff_generate_Ici_compl.1 h induction h with \| basic u h' => obtain ⟨a, rfl⟩ := h'; exact (isUpperSet_Ici a).compl \| univ => exact isLowerSet_univ \| inter u v _ _ hu2 hv2 => exact hu2.inter hv2 \| sUnion _ _ ih => exact isLowerSet_sUnion ih theorem isUpperSet_of_isClosed (h : IsClosed s) : IsUpperSet s := isLowerSet_compl.1 <\| isLowerSet_of_isOpen h.isOpen_compl theorem tendsto_nhds_iff_not_le {β : Type} {f : β → α} {l : Filter β} {x : α} : Filter.Tendsto f l (𝓝 x) ↔ ∀ y, ¬y ≤ x → ∀ᶠ z in l, ¬y ≤ f z := by simp [topology_eq_lowerTopology, tendsto_nhds_generateFrom_iff, Filter.Eventually, Ici, compl_setOf] /-- The closure of a singleton `{a}` in the lower topology is the left-closed right-infinite interval [a, ∞). -/ @[simp] theorem closure_singleton (a : α) : closure {a} = Ici a := Subset.antisymm ((closure_minimal fun _ h => h.ge) <\| isClosed_Ici) <\| (isUpperSet_of_isClosed isClosed_closure).Ici_subset <\| subset_closure rfl protected theorem isTopologicalBasis : IsTopologicalBasis (lowerBasis α) := by convert isTopologicalBasis_of_subbasis (topology_eq α) simp_rw [lowerBasis, coe_upperClosure, compl_iUnion] ext s constructor · rintro ⟨F, hF, rfl⟩ refine ⟨(fun a => (Ici a)ᶜ) '' F, ⟨hF.image _, image_subset_iff.2 fun _ _ => ⟨_, rfl⟩⟩, ?_⟩ simp only [sInter_image] · rintro ⟨F, ⟨hF, hs⟩, rfl⟩ haveI := hF.to_subtype rw [subset_def, Subtype.forall'] at hs choose f hf using hs exact ⟨_, finite_range f, by simp_rw [biInter_range, hf, sInter_eq_iInter]⟩ /-- A function `f : β → α` with lower topology in the codomain is continuous if and only if the preimage of every interval `Set.Ici a` is a closed set. -/ lemma continuous_iff_Ici [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Ici a)) := by obtain rfl := IsLower.topology_eq α simp [continuous_generateFrom_iff] end Preorder section PartialOrder variable [PartialOrder α] [TopologicalSpace α] [IsLower α] -- see Note [lower instance priority] /-- The lower topology on a partial order is T₀. -/ instance (priority := 90) t0Space : T0Space α := (t0Space_iff_inseparable α).2 fun x y h => Ici_injective <\| by simpa only [inseparable_iff_closure_eq, closure_singleton] using h end PartialOrder section LinearOrder variable [LinearOrder α] [TopologicalSpace α] [IsLower α] lemma isTopologicalBasis_insert_univ_subbasis : IsTopologicalBasis (insert univ {s : Set α \| ∃ a, (Ici a)ᶜ = s}) := isTopologicalBasis_of_subbasis_of_inter (by rw [topology_eq α, lower]) (by rintro _ ⟨b, rfl⟩ _ ⟨c, rfl⟩ use b ⊓ c rw [compl_Ici, compl_Ici, compl_Ici, Iio_inter_Iio]) theorem tendsto_nhds_iff_lt {β : Type} {f : β → α} {l : Filter β} {x : α} : Filter.Tendsto f l (𝓝 x) ↔ ∀ y, x < y → ∀ᶠ z in l, f z < y := by simp only [tendsto_nhds_iff_not_le, not_le] end LinearOrder section CompleteLinearOrder variable [CompleteLinearOrder α] [t : TopologicalSpace α] [IsLower α] lemma isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Ici a)ᶜ = U := by by_cases hU : U = univ · simp only [hU, isOpen_univ, compl_Ici, true_or] refine ⟨?_, isTopologicalBasis_insert_univ_subbasis.isOpen⟩ intro hO apply Or.inr convert IsTopologicalBasis.open_eq_sUnion isTopologicalBasis_insert_univ_subbasis hO constructor · intro ⟨a, ha⟩ use {U} constructor · apply subset_trans (singleton_subset_iff.mpr _) (subset_insert _ _) use a · rw [sUnion_singleton] · intro ⟨S, hS1, hS2⟩ have hUS : univ ∉ S := by by_contra hUS' apply hU rw [hS2] exact sUnion_eq_univ_iff.mpr (fun a => ⟨univ, hUS', trivial⟩) use sSup {a \| (Ici a)ᶜ ∈ S} rw [hS2, sUnion_eq_compl_sInter_compl, compl_inj_iff] apply le_antisymm · intro b hb simp only [sInter_image, mem_iInter, mem_compl_iff] intro s hs obtain ⟨a, ha⟩ := (subset_insert_iff_of_notMem hUS).mp hS1 hs subst hS2 ha simp_all only [compl_Ici, mem_Ici, sSup_le_iff, mem_setOf_eq, mem_Iio, not_lt] · intro b hb rw [mem_Ici, sSup_le_iff] intro c hc simp only [sInter_image, mem_iInter] at hb rw [← not_lt, ← mem_Iio, ← compl_Ici] exact hb _ hc end CompleteLinearOrder end IsLower namespace IsUpper /-- The complements of the lower closures of finite sets are a collection of upper sets which form a basis for the upper topology. -/ def upperBasis (α : Type) [Preorder α] := { s : Set α \| ∃ t : Set α, t.Finite ∧ (lowerClosure t : Set α)ᶜ = s } section Preorder variable (α) variable [Preorder α] [TopologicalSpace α] [IsUpper α] {s : Set α} lemma topology_eq : ‹_› = upper α := topology_eq_upperTopology variable {α} /-- If `α` is equipped with the upper topology, then it is homeomorphic to `WithUpper α`. -/ def withUpperHomeomorph : WithUpper α ≃ₜ α := WithUpper.ofUpper.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ theorem isOpen_iff_generate_Iic_compl : IsOpen s ↔ GenerateOpen { t \| ∃ a, (Iic a)ᶜ = t } s := by rw [topology_eq α]; rfl instance _root_.OrderDual.instIsLower [Preorder α] [TopologicalSpace α] [IsUpper α] : IsLower αᵒᵈ where topology_eq_lowerTopology := topology_eq_upperTopology (α := α) /-- Left-infinite right-closed intervals (-∞,a] are closed in the upper topology. -/ instance : ClosedIicTopology α := ⟨fun a ↦ isOpen_compl_iff.1 <\| isOpen_iff_generate_Iic_compl.2 <\| GenerateOpen.basic _ ⟨a, rfl⟩⟩ /-- The lower closure of a finite set is closed in the upper topology. -/ theorem isClosed_lowerClosure (h : s.Finite) : IsClosed (lowerClosure s : Set α) := IsLower.isClosed_upperClosure (α := αᵒᵈ) h /-- Every set open in the upper topology is a upper set. -/ theorem isUpperSet_of_isOpen (h : IsOpen s) : IsUpperSet s := IsLower.isLowerSet_of_isOpen (α := αᵒᵈ) h theorem isLowerSet_of_isClosed (h : IsClosed s) : IsLowerSet s := isUpperSet_compl.1 <\| isUpperSet_of_isOpen h.isOpen_compl theorem tendsto_nhds_iff_not_le {β : Type} {f : β → α} {l : Filter β} {x : α} : Filter.Tendsto f l (𝓝 x) ↔ ∀ y, ¬x ≤ y → ∀ᶠ z in l, ¬f z ≤ y := IsLower.tendsto_nhds_iff_not_le (α := αᵒᵈ) /-- The closure of a singleton `{a}` in the upper topology is the left-infinite right-closed interval (-∞,a]. -/ @[simp] theorem closure_singleton (a : α) : closure {a} = Iic a := IsLower.closure_singleton (α := αᵒᵈ) _ protected theorem isTopologicalBasis : IsTopologicalBasis (upperBasis α) := IsLower.isTopologicalBasis (α := αᵒᵈ) /-- A function `f : β → α` with upper topology in the codomain is continuous if and only if the preimage of every interval `Set.Iic a` is a closed set. -/ lemma continuous_iff_Iic [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Iic a)) := IsLower.continuous_iff_Ici (α := αᵒᵈ) end Preorder section PartialOrder variable [PartialOrder α] [TopologicalSpace α] [IsUpper α] -- see Note [lower instance priority] /-- The upper topology on a partial order is T₀. -/ instance (priority := 90) t0Space : T0Space α := IsLower.t0Space (α := αᵒᵈ) end PartialOrder section LinearOrder variable [LinearOrder α] [TopologicalSpace α] [IsUpper α] lemma isTopologicalBasis_insert_univ_subbasis : IsTopologicalBasis (insert univ {s : Set α \| ∃ a, (Iic a)ᶜ = s}) := IsLower.isTopologicalBasis_insert_univ_subbasis (α := αᵒᵈ) theorem tendsto_nhds_iff_lt {β : Type*} {f : β → α} {l : Filter β} {x : α} : Filter.Tendsto f l (𝓝 x) ↔ ∀ y < x, ∀ᶠ z in l, y < f z := IsLower.tendsto_nhds_iff_lt (α := αᵒᵈ) end LinearOrder section CompleteLinearOrder variable [CompleteLinearOrder α] [t : TopologicalSpace α] [IsUpper α] lemma isTopologicalSpace_basis (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U := IsLower.isTopologicalSpace_basis (α := αᵒᵈ) U end CompleteLinearOrder end IsUpper instance instIsLowerProd [Preorder α] [TopologicalSpace α] [IsLower α] [OrderBot α] [Preorder β] [TopologicalSpace β] [IsLower β] [OrderBot β] : IsLower (α × β) where topology_eq_lowerTopology := by refine le_antisymm (le_generateFrom ?_) ?_ · rintro _ ⟨x, rfl⟩ exact (isClosed_Ici.prod isClosed_Ici).isOpen_compl rw [(IsLower.isTopologicalBasis.prod IsLower.isTopologicalBasis).eq_generateFrom, le_generateFrom_iff_subset_isOpen, image2_subset_iff] rintro _ ⟨s, hs, rfl⟩ _ ⟨t, ht, rfl⟩ dsimp simp_rw [coe_upperClosure, compl_iUnion, prod_eq, preimage_iInter, preimage_compl] -- without `let`, `refine` tries to use the product topology and fails let _ : TopologicalSpace (α × β) := lower (α × β) refine (hs.isOpen_biInter fun a _ => ?_).inter (ht.isOpen_biInter fun b _ => ?_) · exact GenerateOpen.basic _ ⟨(a, ⊥), by simp [Ici_prod_eq, prod_univ]⟩ · exact GenerateOpen.basic _ ⟨(⊥, b), by simp [Ici_prod_eq, univ_prod]⟩ instance instIsUpperProd [Preorder α] [TopologicalSpace α] [IsUpper α] [OrderTop α] [Preorder β] [TopologicalSpace β] [IsUpper β] [OrderTop β] : IsUpper (α × β) where topology_eq_upperTopology := by suffices IsLower (α × β)ᵒᵈ from IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ) exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ) section CompleteLattice_IsLower variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [IsLower α] [TopologicalSpace β] [IsLower β] protected lemma _root_.sInfHom.continuous (f : sInfHom α β) : Continuous f := by refine IsLower.continuous_iff_Ici.2 fun b => ?_ convert isClosed_Ici (a := sInf <\| f ⁻¹' Ici b) refine Subset.antisymm (fun a => sInf_le) fun a ha => le_trans ?_ <\| OrderHomClass.mono (f : α →o β) ha refine LE.le.trans ?_ (map_sInf f _).ge simp -- see Note [lower instance priority] instance (priority := 90) IsLower.toContinuousInf : ContinuousInf α := ⟨(infsInfHom : sInfHom (α × α) α).continuous⟩ end CompleteLattice_IsLower section CompleteLattice_IsUpper variable [CompleteLattice α] [CompleteLattice β] [TopologicalSpace α] [IsUpper α] [TopologicalSpace β] [IsUpper β] protected lemma _root_.sSupHom.continuous (f : sSupHom α β) : Continuous f := sInfHom.continuous (α := αᵒᵈ) (β := βᵒᵈ) (sSupHom.dual.toFun f) -- see Note [lower instance priority] instance (priority := 90) IsUpper.toContinuousInf : ContinuousSup α := ⟨(supsSupHom : sSupHom (α × α) α).continuous⟩ end CompleteLattice_IsUpper lemma isUpper_orderDual [Preorder α] [TopologicalSpace α] : IsUpper αᵒᵈ ↔ IsLower α := by constructor · apply OrderDual.instIsLower · apply OrderDual.instIsUpper lemma isLower_orderDual [Preorder α] [TopologicalSpace α] : IsLower αᵒᵈ ↔ IsUpper α := isUpper_orderDual.symm end Topology /-- The Sierpiński topology on `Prop` is the upper topology -/ instance : IsUpper Prop where topology_eq_upperTopology := by rw [Topology.upper, sierpinskiSpace, ← generateFrom_insert_empty] congr exact le_antisymm (fun h hs => by simp only [compl_Iic, mem_setOf_eq] rw [← Ioi_True, ← Ioi_False] at hs rcases hs with (rfl \| rfl) · use True · use False) (by rintro _ ⟨a, rfl⟩; by_cases a <;> aesop (add simp [Ioi, lt_iff_le_not_ge]))
.lake/packages/mathlib/Mathlib/Topology/Order/WithTop.lean	import Mathlib.Topology.Order.Basic import Mathlib.Data.Fintype.WithTopBot /-! # Order topology on `WithTop ι` When `ι` is a topological space with the order topology, we also endow `WithTop ι` with the order topology. If `ι` is second countable, we prove that `WithTop ι` also is. -/ open Set Filter open scoped Topology namespace TopologicalSpace variable {ι : Type} [Preorder ι] @[nolint unusedArguments] instance [TopologicalSpace ι] [OrderTopology ι] : TopologicalSpace (WithTop ι) := Preorder.topology _ instance [TopologicalSpace ι] [OrderTopology ι] : OrderTopology (WithTop ι) := ⟨rfl⟩ instance [ts : TopologicalSpace ι] [ht : OrderTopology ι] [SecondCountableTopology ι] : SecondCountableTopology (WithTop ι) := by classical rcases isEmpty_or_nonempty ι with hι \| ⟨⟨x₀⟩⟩ · infer_instance /- Let `c` be a countable set in `ι` such that the topology is generated by the sets `Iio a` and `Ioi a` for `a ∈ c`, by second-countability. Let `c'` be a dense set in `ι`, again by second-countability. Let `d` in `WithTop ι` be obtained from `c ∪ c'`, by adding `⊤` and a point `x₁ : ι` with `Ioi x₁ = ∅` if there is one. We will show that the topology on `WithTop ι` is generated by the intervals `Iio a` and `Ioi a` for `a ∈ d`, which will conclude the proof by countability of `d`. -/ obtain ⟨c, c_count, hc⟩ : ∃ (c : Set ι), c.Countable ∧ ts = generateFrom { s \| ∃ a ∈ c, s = Ioi a ∨ s = Iio a } := exists_countable_generateFrom_Ioi_Iio ι obtain ⟨c', c'_count, hc'⟩ : ∃ c' : Set ι, c'.Countable ∧ Dense c' := SeparableSpace.exists_countable_dense let x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀ let d : Set (WithTop ι) := (↑)'' c ∪ (↑)'' c' ∪ {⊤} ∪ {(x₁ : WithTop ι)} suffices H : instWithTopOfOrderTopology = generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a} by refine ⟨{s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}, ?_, by rw [← H]⟩ have d_count : d.Countable := (((c_count.image _).union (c'_count.image _)).union (by simp)).union (by simp) have : {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a} = Ioi '' d ∪ Iio '' d := by ext; simp; grind rw [this] exact (d_count.image _).union (d_count.image _) -- We should check the easy direction that all the elements in our generating set are open. -- This is trivial as these are all intervals. apply le_antisymm · apply le_generateFrom_iff_subset_isOpen.2 simp only [setOf_subset_setOf, forall_exists_index, and_imp] grind [isOpen_Iio', isOpen_Ioi'] /- Now, let us check that our sets indeed generates the topology. As the topology is generated by the open half-infinite intervals by definition, we should check that open half-infinite intervals are covered by finite intersections of our sets. -/ let basis : Set (Set ι) := {s \| ∃ (f g : Set ι), f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ (⋂ a ∈ g, Iio a)} have h_basis : IsTopologicalBasis basis := isTopologicalBasis_biInter_Ioi_Iio_of_generateFrom c hc rw [OrderTopology.topology_eq_generate_intervals (α := WithTop ι)] apply le_generateFrom_iff_subset_isOpen.2 simp only [setOf_subset_setOf, forall_exists_index] rintro u a (rfl \| rfl) -- Consider an interval of the form `Ioi a`. We should cover it by finite intersections of -- our sets. · induction a with \| top => -- for `a = ⊤`, this is trivial as `Ioi ⊤` is in our set by design apply isOpen_generateFrom_of_mem exact ⟨⊤, by simp [d]⟩ \| coe a => -- for `a : ι`, we consider an element `b ∈ Ioi a`, and seek a finite intersection of our -- sets containing it and contained in `Ioi a`. rw [@isOpen_iff_forall_mem_open] intro b hb induction b with \| top => -- if `b = ⊤`, then either `Ioi a` is empty in `ι` and then we use `Ioi (↑x₁)` which is -- in our set and reduced to `⊤` rcases eq_empty_or_nonempty (Ioi a) with ha \| ha · refine ⟨Ioi x₁, ?_, ?_, by simp⟩ · have : Ioi x₁ = ∅ := by have h : ∃ x, Ioi x = ∅ := ⟨a, ha⟩ simp only [x₁, h, ↓reduceDIte] exact h.choose_spec simp [WithTop.Ioi_coe, this] · apply isOpen_generateFrom_of_mem simp [d] -- If `Ioi a` is not empty in `ι`, it contains a point `b` in the dense set `c'`, and then -- we use `Ioi (↑b)` which is in our set, contained in `Ioi (↑a)` and contains `⊤`. · obtain ⟨b, bc', hb⟩ : ∃ b ∈ c', b ∈ Ioi a := hc'.exists_mem_open (isOpen_Ioi' a) ha refine ⟨Ioi b, ?_, ?_, by simp⟩ · apply Ioi_subset_Ioi simpa using hb.le · apply isOpen_generateFrom_of_mem exact ⟨b, by simp [d, bc'], Or.inl rfl⟩ \| coe b => -- if `b` comes from `ι`, then in `ι` we can find a finite intersection of our sets -- containing `b` and contained in `Ioi a`. We lift it to `WithTop ι`. simp only [mem_Ioi, WithTop.coe_lt_coe] at hb obtain ⟨t, ⟨f, g, hfc, hgc, f_fin, g_fin, rfl⟩, hb, hfg⟩ : ∃ t ∈ basis, b ∈ t ∧ t ⊆ Ioi a := h_basis.isOpen_iff.1 (isOpen_Ioi' a) b hb refine ⟨(⋂ z ∈ f, Ioi z) ∩ ⋂ z ∈ g, Iio z, ?_, ?_, by simpa using hb⟩ · intro w hw induction w with \| top => simp \| coe w => simp only [mem_Ioi, WithTop.coe_lt_coe] apply hfg simpa using hw · apply @IsOpen.inter _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}) · apply @Finite.isOpen_biInter _ _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}) · exact f_fin · intro i hi apply isOpen_generateFrom_of_mem grind · apply @Finite.isOpen_biInter _ _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}) · exact g_fin · intro i hi apply isOpen_generateFrom_of_mem grind -- Consider an interval of the form `Iio a`. We should cover it by finite intersections of -- our sets. · induction a with \| top => -- for `a = ⊤`, this is trivial as `Iio ⊤` is in our set by design apply isOpen_generateFrom_of_mem exact ⟨⊤, by simp [d]⟩ \| coe a => -- for `a : ι`, we consider an element `b ∈ Iio a`, and seek a finite intersection of our -- sets containing it and contained in `Iio a`. rw [@isOpen_iff_forall_mem_open] intro b hb induction b with \| top => simp at hb \| coe b => -- `b` can not be equal to `⊤`, so it comes from `ι`. In `ι` we can find a finite -- intersection of our sets containing `b` and contained in `Iio a`. We lift it -- to `WithTop ι`, and intersect with `Iio ⊤` (which is also in our set) to exclude `⊤`. simp only [mem_Iio, WithTop.coe_lt_coe] at hb obtain ⟨t, ⟨f, g, hfc, hgc, f_fin, g_fin, rfl⟩, hb, hfg⟩ : ∃ t ∈ basis, b ∈ t ∧ t ⊆ Iio a := h_basis.isOpen_iff.1 (isOpen_Iio' a) b hb refine ⟨(⋂ z ∈ f, Ioi z) ∩ (⋂ z ∈ g, Iio z) ∩ Iio ⊤, ?_, ?_, by simpa using hb⟩ · intro w hw induction w with \| top => simp at hw \| coe w => simp only [mem_Iio, WithTop.coe_lt_coe] apply hfg simpa using hw · apply @IsOpen.inter _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}); swap · apply isOpen_generateFrom_of_mem simp [d] apply @IsOpen.inter _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}) · apply @Finite.isOpen_biInter _ _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}) · exact f_fin · intro i hi apply isOpen_generateFrom_of_mem grind · apply @Finite.isOpen_biInter _ _ (generateFrom {s \| ∃ a ∈ d, s = Ioi a ∨ s = Iio a}) · exact g_fin · intro i hi apply isOpen_generateFrom_of_mem grind end TopologicalSpace namespace WithTop variable {ι : Type} [LinearOrder ι] [TopologicalSpace ι] [OrderTopology ι] section Coe lemma isEmbedding_coe : Topology.IsEmbedding ((↑) : ι → WithTop ι) := by refine WithTop.coe_strictMono.isEmbedding_of_ordConnected (α := ι) ?_ rw [WithTop.range_coe] exact Set.ordConnected_Iio lemma isOpenEmbedding_coe : Topology.IsOpenEmbedding ((↑) : ι → WithTop ι) := ⟨isEmbedding_coe, by rw [WithTop.range_coe]; exact isOpen_Iio⟩ lemma nhds_coe {r : ι} : 𝓝 (r : WithTop ι) = (𝓝 r).map (↑) := (isOpenEmbedding_coe.map_nhds_eq r).symm @[fun_prop, continuity] lemma continuous_coe : Continuous ((↑) : ι → WithTop ι) := isEmbedding_coe.continuous end Coe section ContinuousUnTop lemma tendsto_untopD (d : ι) {a : WithTop ι} (ha : a ≠ ⊤) : Tendsto (untopD d) (𝓝 a) (𝓝 (untopD d a)) := by lift a to ι using ha rw [nhds_coe, tendsto_map'_iff] exact tendsto_id lemma continuousOn_untopD (d : ι) : ContinuousOn (untopD d) { a : WithTop ι \| a ≠ ⊤ } := fun _a ha ↦ ContinuousAt.continuousWithinAt (tendsto_untopD d ha) lemma tendsto_untopA [Nonempty ι] {a : WithTop ι} (ha : a ≠ ⊤) : Tendsto untopA (𝓝 a) (𝓝 a.untopA) := tendsto_untopD _ ha lemma continuousOn_untopA [Nonempty ι] : ContinuousOn untopA { a : WithTop ι \| a ≠ ⊤ } := continuousOn_untopD _ lemma tendsto_untop (a : {a : WithTop ι \| a ≠ ⊤}) : Tendsto (fun x ↦ untop x.1 x.2) (𝓝 a) (𝓝 (untop a.1 a.2)) := by have : Nonempty ι := ⟨untop a.1 a.2⟩ simp only [← untopA_eq_untop, ne_eq, coe_setOf, mem_setOf_eq] exact (tendsto_untopA a.2).comp <\| tendsto_subtype_rng.mp tendsto_id lemma continuous_untop : Continuous (fun x : {a : WithTop ι \| a ≠ ⊤} ↦ untop x.1 x.2) := continuous_iff_continuousAt.mpr tendsto_untop end ContinuousUnTop variable (ι) in /-- Homeomorphism between the non-top elements of `WithTop ι` and `ι`. -/ noncomputable def neTopHomeomorph : { a : WithTop ι \| a ≠ ⊤ } ≃ₜ ι where toEquiv := Equiv.withTopSubtypeNe continuous_toFun := continuous_untop continuous_invFun := continuous_coe.subtype_mk _ variable (ι) in /-- If `ι` has a top element, then `WithTop ι` is homeomorphic to `ι ⊕ Unit`. -/ noncomputable def sumHomeomorph [OrderTop ι] : WithTop ι ≃ₜ ι ⊕ Unit where toFun x := if h : x = ⊤ then Sum.inr () else Sum.inl x.untopA invFun x := match x with \| Sum.inl i => (i : WithTop ι) \| Sum.inr () => ⊤ left_inv x := by cases x <;> simp right_inv x := by cases x <;> simp continuous_toFun := by have h_fr : frontier ({⊤} : Set (WithTop ι)) = ∅ := by simp only [frontier, Set.finite_singleton, Set.Finite.isClosed, IsClosed.closure_eq] suffices interior ({⊤} : Set (WithTop ι)) = {⊤} by simp [this] rw [interior_eq_iff_isOpen] have : {⊤} = Set.Ioi ((⊤ : ι) : WithTop ι) := by ext; simp rw [this] exact isOpen_Ioi refine continuous_if' (by simp [h_fr]) (by simp [h_fr]) (by simp) ?_ exact Continuous.comp_continuousOn (by fun_prop) continuousOn_untopA continuous_invFun := continuous_sum_dom.mpr ⟨by fun_prop, by fun_prop⟩ end WithTop
.lake/packages/mathlib/Mathlib/Topology/Order/DenselyOrdered.lean	import Mathlib.Topology.Order.IsLUB /-! # Order topology on a densely ordered set -/ open Set Filter TopologicalSpace Topology Function open OrderDual (toDual ofDual) variable {α β : Type*} section DenselyOrdered variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} {s : Set α} /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`, unless `a` is a top element. -/ theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by apply Subset.antisymm · exact closure_minimal Ioi_subset_Ici_self isClosed_Ici · rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff] exact isGLB_Ioi.mem_closure h /-- The closure of the interval `(a, +∞)` is the closed interval `[a, +∞)`. -/ @[simp] theorem closure_Ioi (a : α) [NoMaxOrder α] : closure (Ioi a) = Ici a := closure_Ioi' nonempty_Ioi /-- The closure of the interval `(-∞, a)` is the closed interval `(-∞, a]`, unless `a` is a bottom element. -/ theorem closure_Iio' (h : (Iio a).Nonempty) : closure (Iio a) = Iic a := closure_Ioi' (α := αᵒᵈ) h /-- The closure of the interval `(-∞, a)` is the interval `(-∞, a]`. -/ @[simp] theorem closure_Iio (a : α) [NoMinOrder α] : closure (Iio a) = Iic a := closure_Iio' nonempty_Iio /-- The closure of the open interval `(a, b)` is the closed interval `[a, b]`. -/ @[simp] theorem closure_Ioo {a b : α} (hab : a ≠ b) : closure (Ioo a b) = Icc a b := by apply Subset.antisymm · exact closure_minimal Ioo_subset_Icc_self isClosed_Icc · rcases hab.lt_or_gt with hab \| hab · rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le] have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab simp only [insert_subset_iff, singleton_subset_iff] exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩ · rw [Icc_eq_empty_of_lt hab] exact empty_subset _ /-- The closure of the interval `(a, b]` is the closed interval `[a, b]`. -/ @[simp] theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by apply Subset.antisymm · exact closure_minimal Ioc_subset_Icc_self isClosed_Icc · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self) rw [closure_Ioo hab] /-- The closure of the interval `[a, b)` is the closed interval `[a, b]`. -/ @[simp] theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by apply Subset.antisymm · exact closure_minimal Ico_subset_Icc_self isClosed_Icc · apply Subset.trans _ (closure_mono Ioo_subset_Ico_self) rw [closure_Ioo hab] @[simp] theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic] theorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a := interior_Ici' nonempty_Iio @[simp] theorem interior_Iic' {a : α} (ha : (Ioi a).Nonempty) : interior (Iic a) = Iio a := interior_Ici' (α := αᵒᵈ) ha theorem interior_Iic [NoMaxOrder α] {a : α} : interior (Iic a) = Iio a := interior_Iic' nonempty_Ioi @[simp] theorem interior_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} : interior (Icc a b) = Ioo a b := by rw [← Ici_inter_Iic, interior_inter, interior_Ici, interior_Iic, Ioi_inter_Iio] @[simp] theorem Icc_mem_nhds_iff [NoMinOrder α] [NoMaxOrder α] {a b x : α} : Icc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by rw [← interior_Icc, mem_interior_iff_mem_nhds] @[simp] theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio] @[simp] theorem Ico_mem_nhds_iff [NoMinOrder α] {a b x : α} : Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by rw [← interior_Ico, mem_interior_iff_mem_nhds] @[simp] theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio] @[simp] theorem Ioc_mem_nhds_iff [NoMaxOrder α] {a b x : α} : Ioc a b ∈ 𝓝 x ↔ x ∈ Ioo a b := by rw [← interior_Ioc, mem_interior_iff_mem_nhds] theorem closure_interior_Icc {a b : α} (h : a ≠ b) : closure (interior (Icc a b)) = Icc a b := (closure_minimal interior_subset isClosed_Icc).antisymm <\| calc Icc a b = closure (Ioo a b) := (closure_Ioo h).symm _ ⊆ closure (interior (Icc a b)) := closure_mono (interior_maximal Ioo_subset_Icc_self isOpen_Ioo) theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by rcases eq_or_ne a b with (rfl \| h) · simp · calc Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self _ = closure (Ioo a b) := (closure_Ioo h).symm _ ⊆ closure (interior (Ioc a b)) := closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo) theorem Ico_subset_closure_interior (a b : α) : Ico a b ⊆ closure (interior (Ico a b)) := by simpa only [Ioc_toDual] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a) @[simp] theorem frontier_Ici' {a : α} (ha : (Iio a).Nonempty) : frontier (Ici a) = {a} := by simp [frontier, ha] theorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} := frontier_Ici' nonempty_Iio @[simp] theorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by simp [frontier, ha] theorem frontier_Iic [NoMaxOrder α] {a : α} : frontier (Iic a) = {a} := frontier_Iic' nonempty_Ioi @[simp] theorem frontier_Ioi' {a : α} (ha : (Ioi a).Nonempty) : frontier (Ioi a) = {a} := by simp [frontier, closure_Ioi' ha] theorem frontier_Ioi [NoMaxOrder α] {a : α} : frontier (Ioi a) = {a} := frontier_Ioi' nonempty_Ioi @[simp] theorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by simp [frontier, closure_Iio' ha] theorem frontier_Iio [NoMinOrder α] {a : α} : frontier (Iio a) = {a} := frontier_Iio' nonempty_Iio @[simp] theorem frontier_Icc [NoMinOrder α] [NoMaxOrder α] {a b : α} (h : a ≤ b) : frontier (Icc a b) = {a, b} := by simp [frontier, h, Icc_diff_Ioo_same] @[simp] theorem frontier_Ioo {a b : α} (h : a < b) : frontier (Ioo a b) = {a, b} := by rw [frontier, closure_Ioo h.ne, interior_Ioo, Icc_diff_Ioo_same h.le] @[simp] theorem frontier_Ico [NoMinOrder α] {a b : α} (h : a < b) : frontier (Ico a b) = {a, b} := by rw [frontier, closure_Ico h.ne, interior_Ico, Icc_diff_Ioo_same h.le] @[simp] theorem frontier_Ioc [NoMaxOrder α] {a b : α} (h : a < b) : frontier (Ioc a b) = {a, b} := by rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le] theorem nhdsWithin_Ioi_neBot' {a b : α} (H₁ : (Ioi a).Nonempty) (H₂ : a ≤ b) : NeBot (𝓝[Ioi a] b) := mem_closure_iff_nhdsWithin_neBot.1 <\| by rwa [closure_Ioi' H₁] theorem nhdsWithin_Ioi_neBot [NoMaxOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Ioi a] b) := nhdsWithin_Ioi_neBot' nonempty_Ioi H theorem nhdsGT_neBot_of_exists_gt {a : α} (H : ∃ b, a < b) : NeBot (𝓝[>] a) := nhdsWithin_Ioi_neBot' H (le_refl a) instance nhdsGT_neBot [NoMaxOrder α] (a : α) : NeBot (𝓝[>] a) := nhdsWithin_Ioi_neBot le_rfl theorem nhdsWithin_Iio_neBot' {b c : α} (H₁ : (Iio c).Nonempty) (H₂ : b ≤ c) : NeBot (𝓝[Iio c] b) := mem_closure_iff_nhdsWithin_neBot.1 <\| by rwa [closure_Iio' H₁] theorem nhdsWithin_Iio_neBot [NoMinOrder α] {a b : α} (H : a ≤ b) : NeBot (𝓝[Iio b] a) := nhdsWithin_Iio_neBot' nonempty_Iio H theorem nhdsWithin_Iio_self_neBot' {b : α} (H : (Iio b).Nonempty) : NeBot (𝓝[<] b) := nhdsWithin_Iio_neBot' H (le_refl b) instance nhdsLT_neBot [NoMinOrder α] (a : α) : NeBot (𝓝[<] a) := nhdsWithin_Iio_neBot (le_refl a) theorem right_nhdsWithin_Ico_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ico a b] b) := (isLUB_Ico H).nhdsWithin_neBot (nonempty_Ico.2 H) theorem left_nhdsWithin_Ioc_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioc a b] a) := (isGLB_Ioc H).nhdsWithin_neBot (nonempty_Ioc.2 H) theorem left_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] a) := (isGLB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H) theorem right_nhdsWithin_Ioo_neBot {a b : α} (H : a < b) : NeBot (𝓝[Ioo a b] b) := (isLUB_Ioo H).nhdsWithin_neBot (nonempty_Ioo.2 H) theorem comap_coe_nhdsLT_of_Ioo_subset (hb : s ⊆ Iio b) (hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by nontriviality haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_› rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩ ext u; constructor · rintro ⟨t, ht, hts⟩ obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ := (mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset h).mp ht obtain ⟨y, hxy, hyb⟩ := exists_between hxb refine mem_of_superset (mem_atTop ⟨y, hs ⟨hxa.trans_lt hxy, hyb⟩⟩) ?_ rintro ⟨z, hzs⟩ (hyz : y ≤ z) exact hts (hxt ⟨hxy.trans_le hyz, hb hzs⟩) · intro hu obtain ⟨x : s, hx : ∀ z, x ≤ z → z ∈ u⟩ := mem_atTop_sets.1 hu exact ⟨Ioo x b, Ioo_mem_nhdsLT (hb x.2), fun z hz => hx _ hz.1.le⟩ theorem comap_coe_nhdsGT_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : s.Nonempty → ∃ b > a, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[>] a) = atBot := by apply comap_coe_nhdsLT_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) simp only [OrderDual.exists, Ioo_toDual] exact hs theorem map_coe_atTop_of_Ioo_subset (hb : s ⊆ Iio b) (hs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s) : map ((↑) : s → α) atTop = 𝓝[<] b := by rcases eq_empty_or_nonempty (Iio b) with (hb' \| ⟨a, ha⟩) · have : IsEmpty s := ⟨fun x => hb'.subset (hb x.2)⟩ rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty] · rw [← comap_coe_nhdsLT_of_Ioo_subset hb fun _ => hs a ha, map_comap_of_mem] rw [Subtype.range_val] exact (mem_nhdsLT_iff_exists_Ioo_subset' ha).2 (hs a ha) theorem map_coe_atBot_of_Ioo_subset (ha : s ⊆ Ioi a) (hs : ∀ b' > a, ∃ b > a, Ioo a b ⊆ s) : map ((↑) : s → α) atBot = 𝓝[>] a := by -- the elaborator gets stuck without `(... :)` refine (map_coe_atTop_of_Ioo_subset (show ofDual ⁻¹' s ⊆ Iio (toDual a) from ha) fun b' hb' => ?_ :) simpa using hs b' hb' /-- The `atTop` filter for an open interval `Ioo a b` comes from the left-neighbourhoods filter at the right endpoint in the ambient order. -/ theorem comap_coe_Ioo_nhdsLT (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[<] b) = atTop := comap_coe_nhdsLT_of_Ioo_subset Ioo_subset_Iio_self fun h => ⟨a, nonempty_Ioo.1 h, Subset.refl _⟩ /-- The `atBot` filter for an open interval `Ioo a b` comes from the right-neighbourhoods filter at the left endpoint in the ambient order. -/ theorem comap_coe_Ioo_nhdsGT (a b : α) : comap ((↑) : Ioo a b → α) (𝓝[>] a) = atBot := comap_coe_nhdsGT_of_Ioo_subset Ioo_subset_Ioi_self fun h => ⟨b, nonempty_Ioo.1 h, Subset.refl _⟩ theorem comap_coe_Ioi_nhdsGT (a : α) : comap ((↑) : Ioi a → α) (𝓝[>] a) = atBot := comap_coe_nhdsGT_of_Ioo_subset (Subset.refl _) fun ⟨x, hx⟩ => ⟨x, hx, Ioo_subset_Ioi_self⟩ theorem comap_coe_Iio_nhdsLT (a : α) : comap ((↑) : Iio a → α) (𝓝[<] a) = atTop := comap_coe_Ioi_nhdsGT (α := αᵒᵈ) a @[simp] theorem map_coe_Ioo_atTop {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atTop = 𝓝[<] b := map_coe_atTop_of_Ioo_subset Ioo_subset_Iio_self fun _ _ => ⟨_, h, Subset.refl _⟩ @[simp] theorem map_coe_Ioo_atBot {a b : α} (h : a < b) : map ((↑) : Ioo a b → α) atBot = 𝓝[>] a := map_coe_atBot_of_Ioo_subset Ioo_subset_Ioi_self fun _ _ => ⟨_, h, Subset.refl _⟩ @[simp] theorem map_coe_Ioi_atBot (a : α) : map ((↑) : Ioi a → α) atBot = 𝓝[>] a := map_coe_atBot_of_Ioo_subset (Subset.refl _) fun b hb => ⟨b, hb, Ioo_subset_Ioi_self⟩ @[simp] theorem map_coe_Iio_atTop (a : α) : map ((↑) : Iio a → α) atTop = 𝓝[<] a := map_coe_Ioi_atBot (α := αᵒᵈ) _ variable {l : Filter β} {f : α → β} @[simp] theorem tendsto_comp_coe_Ioo_atTop (h : a < b) : Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl @[simp] theorem tendsto_comp_coe_Ioo_atBot (h : a < b) : Tendsto (fun x : Ioo a b => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by rw [← map_coe_Ioo_atBot h, tendsto_map'_iff]; rfl @[simp] theorem tendsto_comp_coe_Ioi_atBot : Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl @[simp] theorem tendsto_comp_coe_Iio_atTop : Tendsto (fun x : Iio a => f x) atTop l ↔ Tendsto f (𝓝[<] a) l := by rw [← map_coe_Iio_atTop, tendsto_map'_iff]; rfl @[simp] theorem tendsto_Ioo_atTop {f : β → Ioo a b} : Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by rw [← comap_coe_Ioo_nhdsLT, tendsto_comap_iff]; rfl @[simp] theorem tendsto_Ioo_atBot {f : β → Ioo a b} : Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by rw [← comap_coe_Ioo_nhdsGT, tendsto_comap_iff]; rfl @[simp] theorem tendsto_Ioi_atBot {f : β → Ioi a} : Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l (𝓝[>] a) := by rw [← comap_coe_Ioi_nhdsGT, tendsto_comap_iff]; rfl @[simp] theorem tendsto_Iio_atTop {f : β → Iio a} : Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by rw [← comap_coe_Iio_nhdsLT, tendsto_comap_iff]; rfl instance (x : α) [Nontrivial α] : NeBot (𝓝[≠] x) := by refine forall_mem_nonempty_iff_neBot.1 fun s hs => ?_ obtain ⟨u, u_open, xu, us⟩ : ∃ u : Set α, IsOpen u ∧ x ∈ u ∧ u ∩ {x}ᶜ ⊆ s := mem_nhdsWithin.1 hs obtain ⟨a, b, a_lt_b, hab⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ u := u_open.exists_Ioo_subset ⟨x, xu⟩ obtain ⟨y, hy⟩ : ∃ y, a < y ∧ y < b := exists_between a_lt_b rcases ne_or_eq x y with (xy \| rfl) · exact ⟨y, us ⟨hab hy, xy.symm⟩⟩ obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1 exact ⟨z, us ⟨hab ⟨hz.1, hz.2.trans hy.2⟩, hz.2.ne⟩⟩ /-- If the order topology for a dense linear ordering is discrete, the space has at most one point. We would prefer for this to be an instance but even at `(priority := 100)` this was problematic so we have deferred this issue. TODO Promote this to an `instance`! -/ lemma DenselyOrdered.subsingleton_of_discreteTopology [DiscreteTopology α] : Subsingleton α := by suffices ∀ a b : α, b ≤ a from ⟨fun a b ↦ le_antisymm (this b a) (this a b)⟩ intro a b by_contra! contra suffices b ∈ Ioo a b by grind rw [← (isClosed_discrete (Ioo a b)).closure_eq, closure_Ioo contra.ne] grind /-- Let `s` be a dense set in a nontrivial dense linear order `α`. If `s` is a separable space (e.g., if `α` has a second countable topology), then there exists a countable dense subset `t ⊆ s` such that `t` does not contain bottom/top elements of `α`. -/ theorem Dense.exists_countable_dense_subset_no_bot_top [Nontrivial α] {s : Set α} [SeparableSpace s] (hs : Dense s) : ∃ t, t ⊆ s ∧ t.Countable ∧ Dense t ∧ (∀ x, IsBot x → x ∉ t) ∧ ∀ x, IsTop x → x ∉ t := by rcases hs.exists_countable_dense_subset with ⟨t, hts, htc, htd⟩ refine ⟨t \ ({ x \| IsBot x } ∪ { x \| IsTop x }), ?_, ?_, ?_, fun x hx => ?_, fun x hx => ?_⟩ · exact diff_subset.trans hts · exact htc.mono diff_subset · exact htd.diff_finite ((subsingleton_isBot α).finite.union (subsingleton_isTop α).finite) · simp [hx] · simp [hx] variable (α) in /-- If `α` is a nontrivial separable dense linear order, then there exists a countable dense set `s : Set α` that contains neither top nor bottom elements of `α`. For a dense set containing both bot and top elements, see `exists_countable_dense_bot_top`. -/ theorem exists_countable_dense_no_bot_top [SeparableSpace α] [Nontrivial α] : ∃ s : Set α, s.Countable ∧ Dense s ∧ (∀ x, IsBot x → x ∉ s) ∧ ∀ x, IsTop x → x ∉ s := by simpa using dense_univ.exists_countable_dense_subset_no_bot_top /-- `Set.Ico a b` is only closed if it is empty. -/ @[simp] theorem isClosed_Ico_iff {a b : α} : IsClosed (Set.Ico a b) ↔ b ≤ a := by refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩ have := h.closure_eq rw [closure_Ico hab.ne, Icc_eq_Ico_same_iff] at this exact this hab.le /-- `Set.Ioc a b` is only closed if it is empty. -/ @[simp] theorem isClosed_Ioc_iff {a b : α} : IsClosed (Set.Ioc a b) ↔ b ≤ a := by refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩ have := h.closure_eq rw [closure_Ioc hab.ne, Icc_eq_Ioc_same_iff] at this exact this hab.le /-- `Set.Ioo a b` is only closed if it is empty. -/ @[simp] theorem isClosed_Ioo_iff {a b : α} : IsClosed (Set.Ioo a b) ↔ b ≤ a := by refine ⟨fun h => le_of_not_gt fun hab => ?_, by simp_all⟩ have := h.closure_eq rw [closure_Ioo hab.ne, Icc_eq_Ioo_same_iff] at this exact this hab.le end DenselyOrdered
.lake/packages/mathlib/Mathlib/Topology/Order/OrderClosedExtr.lean	import Mathlib.Topology.Order.OrderClosed import Mathlib.Topology.Order.LocalExtr /-! # Local maxima from monotonicity and antitonicity In this file we prove a lemma that is useful for the First Derivative Test in calculus, and its dual. ## Main statements * `isLocalMax_of_mono_anti` : if a function `f` is monotone to the left of `x` and antitone to the right of `x` then `f` has a local maximum at `x`. * `isLocalMin_of_anti_mono` : the dual statement for minima. * `isLocalMax_of_mono_anti'` : a version of `isLocalMax_of_mono_anti` for filters. * `isLocalMin_of_anti_mono'` : a version of `isLocalMax_of_mono_anti'` for minima. -/ open Set Topology Filter /-- If `f` is monotone on `(a,b]` and antitone on `[b,c)` then `f` has a local maximum at `b`. -/ lemma isLocalMax_of_mono_anti {α : Type} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {β : Type} [Preorder β] {a b c : α} (g₀ : a < b) (g₁ : b < c) {f : α → β} (h₀ : MonotoneOn f (Ioc a b)) (h₁ : AntitoneOn f (Ico b c)) : IsLocalMax f b := isLocalMax_of_mono_anti' (Ioc_mem_nhdsLE g₀) (Ico_mem_nhdsGE g₁) h₀ h₁ /-- If `f` is antitone on `(a,b]` and monotone on `[b,c)` then `f` has a local minimum at `b`. -/ lemma isLocalMin_of_anti_mono {α : Type} [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {β : Type} [Preorder β] {a b c : α} (g₀ : a < b) (g₁ : b < c) {f : α → β} (h₀ : AntitoneOn f (Ioc a b)) (h₁ : MonotoneOn f (Ico b c)) : IsLocalMin f b := mem_of_superset (Ioo_mem_nhds g₀ g₁) (fun x hx => by rcases le_total x b <;> aesop)
.lake/packages/mathlib/Mathlib/Topology/Order/Bornology.lean	import Mathlib.Topology.Bornology.Constructions /-! # Bornology of order-bounded sets This file relates the notion of bornology-boundedness (sets that lie in a bornology) to the notion of order-boundedness (sets that are bounded above and below). ## Main declarations * `orderBornology`: The bornology of order-bounded sets of a nonempty lattice. * `IsOrderBornology`: Typeclass predicate for a preorder to be equipped with its order-bornology. -/ open Bornology Set variable {α : Type} {s t : Set α} section Lattice variable [Lattice α] [Nonempty α] /-- Order-bornology on a nonempty lattice. The bounded sets are the sets that are bounded both above and below. -/ def orderBornology : Bornology α := .ofBounded {s \| BddBelow s ∧ BddAbove s} (by simp) (fun _ hs _ hst ↦ ⟨hs.1.mono hst, hs.2.mono hst⟩) (fun _ hs _ ht ↦ ⟨hs.1.union ht.1, hs.2.union ht.2⟩) (by simp) @[simp] lemma orderBornology_isBounded : orderBornology.IsBounded s ↔ BddBelow s ∧ BddAbove s := by simp [IsBounded, IsCobounded, -isCobounded_compl_iff] end Lattice variable [Bornology α] variable (α) [Preorder α] in /-- Predicate for a preorder to be equipped with its order-bornology, namely for its bounded sets to be the ones that are bounded both above and below. -/ class IsOrderBornology : Prop where protected isBounded_iff_bddBelow_bddAbove (s : Set α) : IsBounded s ↔ BddBelow s ∧ BddAbove s lemma isOrderBornology_iff_eq_orderBornology [Lattice α] [Nonempty α] : IsOrderBornology α ↔ ‹Bornology α› = orderBornology := by refine ⟨fun h ↦ ?_, fun h ↦ ⟨fun s ↦ by rw [h, orderBornology_isBounded]⟩⟩ ext s exact isBounded_compl_iff.symm.trans (h.1 _) section Preorder variable [Preorder α] [IsOrderBornology α] lemma isBounded_iff_bddBelow_bddAbove : IsBounded s ↔ BddBelow s ∧ BddAbove s := IsOrderBornology.isBounded_iff_bddBelow_bddAbove _ protected lemma Bornology.IsBounded.bddBelow (hs : IsBounded s) : BddBelow s := (isBounded_iff_bddBelow_bddAbove.1 hs).1 protected lemma Bornology.IsBounded.bddAbove (hs : IsBounded s) : BddAbove s := (isBounded_iff_bddBelow_bddAbove.1 hs).2 protected lemma BddBelow.isBounded (hs₀ : BddBelow s) (hs₁ : BddAbove s) : IsBounded s := isBounded_iff_bddBelow_bddAbove.2 ⟨hs₀, hs₁⟩ protected lemma BddAbove.isBounded (hs₀ : BddAbove s) (hs₁ : BddBelow s) : IsBounded s := isBounded_iff_bddBelow_bddAbove.2 ⟨hs₁, hs₀⟩ lemma BddBelow.isBounded_inter (hs : BddBelow s) (ht : BddAbove t) : IsBounded (s ∩ t) := (hs.mono inter_subset_left).isBounded <\| ht.mono inter_subset_right lemma BddAbove.isBounded_inter (hs : BddAbove s) (ht : BddBelow t) : IsBounded (s ∩ t) := (hs.mono inter_subset_left).isBounded <\| ht.mono inter_subset_right instance OrderDual.instIsOrderBornology : IsOrderBornology αᵒᵈ where isBounded_iff_bddBelow_bddAbove s := by rw [← isBounded_preimage_toDual, ← bddBelow_preimage_toDual, ← bddAbove_preimage_toDual, isBounded_iff_bddBelow_bddAbove, and_comm] instance Prod.instIsOrderBornology {β : Type} [Preorder β] [Bornology β] [IsOrderBornology β] : IsOrderBornology (α × β) where isBounded_iff_bddBelow_bddAbove s := by rw [← isBounded_image_fst_and_snd, bddBelow_prod, bddAbove_prod, and_and_and_comm, isBounded_iff_bddBelow_bddAbove, isBounded_iff_bddBelow_bddAbove] instance Pi.instIsOrderBornology {ι : Type} {α : ι → Type} [∀ i, Preorder (α i)] [∀ i, Bornology (α i)] [∀ i, IsOrderBornology (α i)] : IsOrderBornology (∀ i, α i) where isBounded_iff_bddBelow_bddAbove s := by simp_rw [← forall_isBounded_image_eval_iff, bddBelow_pi, bddAbove_pi, ← forall_and, isBounded_iff_bddBelow_bddAbove] end Preorder section ConditionallyCompleteLattice variable [ConditionallyCompleteLattice α] [IsOrderBornology α] {s : Set α} protected lemma Bornology.IsBounded.subset_Icc_sInf_sSup (hs : IsBounded s) : s ⊆ Icc (sInf s) (sSup s) := subset_Icc_csInf_csSup hs.bddBelow hs.bddAbove end ConditionallyCompleteLattice
.lake/packages/mathlib/Mathlib/Topology/Order/Compact.lean	import Mathlib.Topology.Algebra.Support import Mathlib.Topology.Order.IntermediateValue import Mathlib.Topology.Order.IsLUB import Mathlib.Topology.Order.LocalExtr /-! # Compactness of a closed interval In this file we prove that a closed interval in a conditionally complete linear ordered type with order topology (or a product of such types) is compact. We prove the extreme value theorem (`IsCompact.exists_isMinOn`, `IsCompact.exists_isMaxOn`): a continuous function on a compact set takes its minimum and maximum values. We provide many variations of this theorem. We also prove that the image of a closed interval under a continuous map is a closed interval, see `ContinuousOn.image_Icc`. ## Tags compact, extreme value theorem -/ open Filter OrderDual TopologicalSpace Function Set open scoped Filter Topology /-! ### Compactness of a closed interval In this section we define a typeclass `CompactIccSpace α` saying that all closed intervals in `α` are compact. Then we provide an instance for a `ConditionallyCompleteLinearOrder` and prove that the product (both `α × β` and an indexed product) of spaces with this property inherits the property. We also prove some simple lemmas about spaces with this property. -/ /-- This typeclass says that all closed intervals in `α` are compact. This is true for all conditionally complete linear orders with order topology and products (finite or infinite) of such spaces. -/ class CompactIccSpace (α : Type) [TopologicalSpace α] [Preorder α] : Prop where /-- A closed interval `Set.Icc a b` is a compact set for all `a` and `b`. -/ isCompact_Icc : ∀ {a b : α}, IsCompact (Icc a b) export CompactIccSpace (isCompact_Icc) variable {α : Type} -- TODO: make it the definition lemma CompactIccSpace.mk' [TopologicalSpace α] [Preorder α] (h : ∀ {a b : α}, a ≤ b → IsCompact (Icc a b)) : CompactIccSpace α where isCompact_Icc {a b} := by_cases h fun hab => by rw [Icc_eq_empty hab]; exact isCompact_empty -- TODO: drop one `'` lemma CompactIccSpace.mk'' [TopologicalSpace α] [PartialOrder α] (h : ∀ {a b : α}, a < b → IsCompact (Icc a b)) : CompactIccSpace α := .mk' fun hab => hab.eq_or_lt.elim (by rintro rfl; simp) h instance [TopologicalSpace α] [Preorder α] [CompactIccSpace α] : CompactIccSpace (αᵒᵈ) where isCompact_Icc := by intro a b convert isCompact_Icc (α := α) (a := b) (b := a) using 1 exact Icc_toDual (α := α) /-- A closed interval in a conditionally complete linear order is compact. -/ instance (priority := 100) ConditionallyCompleteLinearOrder.toCompactIccSpace (α : Type) [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] : CompactIccSpace α := by refine .mk'' fun {a b} hlt => ?_ rcases le_or_gt a b with hab \| hab swap · simp [hab] refine isCompact_iff_ultrafilter_le_nhds.2 fun f hf => ?_ contrapose! hf rw [le_principal_iff] have hpt : ∀ x ∈ Icc a b, {x} ∉ f := fun x hx hxf => hf x hx ((le_pure_iff.2 hxf).trans (pure_le_nhds x)) set s := { x ∈ Icc a b \| Icc a x ∉ f } have hsb : b ∈ upperBounds s := fun x hx => hx.1.2 have sbd : BddAbove s := ⟨b, hsb⟩ have ha : a ∈ s := by simp [s, hpt, hab] rcases hab.eq_or_lt with (rfl \| _hlt) · exact ha.2 let c := sSup s have hsc : IsLUB s c := isLUB_csSup ⟨a, ha⟩ ⟨b, hsb⟩ have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩ specialize hf c hc have hcs : c ∈ s := by -- rcases ... with (rfl \| ... ) fails here, rewrite manually. rcases hc.1.eq_or_lt with (h \| hlt) · rwa [h] at ha refine ⟨hc, fun hcf => hf fun U hU => ?_⟩ rcases (mem_nhdsLE_iff_exists_Ioc_subset' hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with ⟨x, hxc, hxU⟩ rcases ((hsc.frequently_mem ⟨a, ha⟩).and_eventually (Ioc_mem_nhdsLE hxc)).exists with ⟨y, ⟨_hyab, hyf⟩, hy⟩ refine mem_of_superset (f.diff_mem_iff.2 ⟨hcf, hyf⟩) (Subset.trans ?_ hxU) rw [diff_subset_iff] exact Subset.trans Icc_subset_Icc_union_Ioc <\| union_subset_union Subset.rfl <\| Ioc_subset_Ioc_left hy.1.le -- rcases ... with (rfl \| ... ) fails here, rewrite manually. rcases hc.2.eq_or_lt with (h \| hlt) · simpa [h] using hcs.2 exfalso refine hf fun U hU => ?_ rcases (mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset hlt).1 (mem_nhdsWithin_of_mem_nhds hU) with ⟨y, hxy, hyU⟩ refine mem_of_superset ?_ hyU; clear! U have hy : y ∈ Icc a b := ⟨hc.1.trans hxy.1.le, hxy.2⟩ by_cases hay : Icc a y ∈ f · refine mem_of_superset (f.diff_mem_iff.2 ⟨f.diff_mem_iff.2 ⟨hay, hcs.2⟩, hpt y hy⟩) ?_ rw [diff_subset_iff, union_comm, Ico_union_right hxy.1.le, diff_subset_iff] exact Icc_subset_Icc_union_Icc · exact ((hsc.1 ⟨hy, hay⟩).not_gt hxy.1).elim instance {ι : Type} {α : ι → Type} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)] [∀ i, CompactIccSpace (α i)] : CompactIccSpace (∀ i, α i) := ⟨fun {a b} => (pi_univ_Icc a b ▸ isCompact_univ_pi) fun _ => isCompact_Icc⟩ instance Pi.compact_Icc_space' {α β : Type} [Preorder β] [TopologicalSpace β] [CompactIccSpace β] : CompactIccSpace (α → β) := inferInstance instance {α β : Type} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] [Preorder β] [TopologicalSpace β] [CompactIccSpace β] : CompactIccSpace (α × β) := ⟨fun {a b} => (Icc_prod_eq a b).symm ▸ isCompact_Icc.prod isCompact_Icc⟩ /-- An unordered closed interval is compact. -/ theorem isCompact_uIcc {α : Type} [LinearOrder α] [TopologicalSpace α] [CompactIccSpace α] {a b : α} : IsCompact (uIcc a b) := isCompact_Icc -- See note [lower instance priority] /-- A complete linear order is a compact space. We do not register an instance for a `[CompactIccSpace α]` because this would only add instances for products (indexed or not) of complete linear orders, and we have instances with higher priority that cover these cases. -/ instance (priority := 100) compactSpace_of_completeLinearOrder {α : Type} [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] : CompactSpace α := ⟨by simp only [← Icc_bot_top, isCompact_Icc]⟩ section variable {α : Type} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] instance compactSpace_Icc (a b : α) : CompactSpace (Icc a b) := isCompact_iff_compactSpace.mp isCompact_Icc end section openIntervals variable {α : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] /-- `Set.Ico a b` is only compact if it is empty. -/ @[simp] theorem isCompact_Ico_iff {a b : α} : IsCompact (Set.Ico a b) ↔ b ≤ a := ⟨fun h => isClosed_Ico_iff.mp h.isClosed, by simp_all⟩ /-- `Set.Ioc a b` is only compact if it is empty. -/ @[simp] theorem isCompact_Ioc_iff {a b : α} : IsCompact (Set.Ioc a b) ↔ b ≤ a := ⟨fun h => isClosed_Ioc_iff.mp h.isClosed, by simp_all⟩ /-- `Set.Ioo a b` is only compact if it is empty. -/ @[simp] theorem isCompact_Ioo_iff {a b : α} : IsCompact (Set.Ioo a b) ↔ b ≤ a := ⟨fun h => isClosed_Ioo_iff.mp h.isClosed, by simp_all⟩ end openIntervals /-! ### Extreme value theorem -/ section LinearOrder variable {α β γ : Type} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] theorem IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : ∃ x, IsLeast s x := by haveI : Nonempty s := ne_s.to_subtype suffices (s ∩ ⋂ x ∈ s, Iic x).Nonempty from ⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩ rw [biInter_eq_iInter] by_contra H rw [not_nonempty_iff_eq_empty] at H rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Iic) H (Monotone.directed_ge fun _ _ h => Iic_subset_Iic.mpr h) with ⟨x, hx⟩ exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩ theorem IsCompact.exists_isGreatest [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : ∃ x, IsGreatest s x := IsCompact.exists_isLeast (α := αᵒᵈ) hs ne_s theorem IsCompact.exists_isGLB [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : ∃ x ∈ s, IsGLB s x := (hs.exists_isLeast ne_s).imp (fun x (hx : IsLeast s x) => ⟨hx.1, hx.isGLB⟩) theorem IsCompact.exists_isLUB [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : ∃ x ∈ s, IsLUB s x := IsCompact.exists_isGLB (α := αᵒᵈ) hs ne_s theorem cocompact_le_atBot_atTop [CompactIccSpace α] : cocompact α ≤ atBot ⊔ atTop := by refine fun s hs ↦ mem_cocompact.mpr <\| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro · exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩ · obtain ⟨t, ht⟩ := mem_atBot_sets.mp hs.1 obtain ⟨u, hu⟩ := mem_atTop_sets.mp hs.2 refine ⟨Icc t u, isCompact_Icc, fun x hx ↦ ?_⟩ exact (not_and_or.mp hx).casesOn (fun h ↦ ht x (le_of_not_ge h)) fun h ↦ hu x (le_of_not_ge h) theorem cocompact_le_atBot [OrderTop α] [CompactIccSpace α] : cocompact α ≤ atBot := by refine fun _ hs ↦ mem_cocompact.mpr <\| (isEmpty_or_nonempty α).casesOn ?_ ?_ <;> intro · exact ⟨∅, isCompact_empty, fun x _ ↦ (IsEmpty.false x).elim⟩ · obtain ⟨t, ht⟩ := mem_atBot_sets.mp hs refine ⟨Icc t ⊤, isCompact_Icc, fun _ hx ↦ ?_⟩ exact (not_and_or.mp hx).casesOn (fun h ↦ ht _ (le_of_not_ge h)) (fun h ↦ (h le_top).elim) theorem cocompact_le_atTop [OrderBot α] [CompactIccSpace α] : cocompact α ≤ atTop := cocompact_le_atBot (α := αᵒᵈ) theorem atBot_le_cocompact [NoMinOrder α] [ClosedIicTopology α] : atBot ≤ cocompact α := by refine fun s hs ↦ ?_ obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs refine (Set.eq_empty_or_nonempty t).casesOn (fun h_empty ↦ ?_) (fun h_nonempty ↦ ?_) · rewrite [compl_univ_iff.mpr h_empty, univ_subset_iff] at hts convert univ_mem · haveI := h_nonempty.nonempty obtain ⟨a, ha⟩ := ht.exists_isLeast h_nonempty obtain ⟨b, hb⟩ := exists_lt a exact Filter.mem_atBot_sets.mpr ⟨b, fun b' hb' ↦ hts <\| Classical.byContradiction fun hc ↦ LT.lt.false <\| hb'.trans_lt <\| hb.trans_le <\| ha.2 (not_notMem.mp hc)⟩ theorem atTop_le_cocompact [NoMaxOrder α] [ClosedIciTopology α] : atTop ≤ cocompact α := atBot_le_cocompact (α := αᵒᵈ) theorem atBot_atTop_le_cocompact [NoMinOrder α] [NoMaxOrder α] [OrderClosedTopology α] : atBot ⊔ atTop ≤ cocompact α := sup_le atBot_le_cocompact atTop_le_cocompact @[simp 900] theorem cocompact_eq_atBot_atTop [NoMaxOrder α] [NoMinOrder α] [OrderClosedTopology α] [CompactIccSpace α] : cocompact α = atBot ⊔ atTop := cocompact_le_atBot_atTop.antisymm atBot_atTop_le_cocompact @[simp] theorem cocompact_eq_atBot [NoMinOrder α] [OrderTop α] [ClosedIicTopology α] [CompactIccSpace α] : cocompact α = atBot := cocompact_le_atBot.antisymm atBot_le_cocompact @[simp] theorem cocompact_eq_atTop [NoMaxOrder α] [OrderBot α] [ClosedIciTopology α] [CompactIccSpace α] : cocompact α = atTop := cocompact_le_atTop.antisymm atTop_le_cocompact /-- The extreme value theorem: a continuous function realizes its minimum on a compact set. -/ theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩ refine ⟨x, hxs, forall_mem_image.1 (fun _ hb => hx <\| mem_image_of_mem f ?_)⟩ rwa [(image_id' s).symm] /-- If a continuous function lies strictly above `a` on a compact set, it has a lower bound strictly above `a`. -/ theorem IsCompact.exists_forall_le' [ClosedIicTopology α] [NoMaxOrder α] {f : β → α} {s : Set β} (hs : IsCompact s) (hf : ContinuousOn f s) {a : α} (hf' : ∀ b ∈ s, a < f b) : ∃ a', a < a' ∧ ∀ b ∈ s, a' ≤ f b := by rcases s.eq_empty_or_nonempty with (rfl \| hs') · obtain ⟨a', ha'⟩ := exists_gt a exact ⟨a', ha', fun _ a ↦ a.elim⟩ · obtain ⟨x, hx, hx'⟩ := hs.exists_isMinOn hs' hf exact ⟨f x, hf' x hx, hx'⟩ /-- The extreme value theorem: a continuous function realizes its maximum on a compact set. -/ theorem IsCompact.exists_isMaxOn [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMaxOn f s x := IsCompact.exists_isMinOn (α := αᵒᵈ) hs ne_s hf /-- The extreme value theorem: if a function `f` is continuous on a closed set `s` and it is larger than a value in its image away from compact sets, then it has a minimum on this set. -/ theorem ContinuousOn.exists_isMinOn' [ClosedIicTopology α] {s : Set β} {f : β → α} (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x₀ ≤ f x) : ∃ x ∈ s, IsMinOn f s x := by rcases (hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with ⟨K, hK, hKf⟩ have hsub : insert x₀ (K ∩ s) ⊆ s := insert_subset_iff.2 ⟨h₀, inter_subset_right⟩ obtain ⟨x, hx, hxf⟩ : ∃ x ∈ insert x₀ (K ∩ s), ∀ y ∈ insert x₀ (K ∩ s), f x ≤ f y := ((hK.inter_right hsc).insert x₀).exists_isMinOn (insert_nonempty _ _) (hf.mono hsub) refine ⟨x, hsub hx, fun y hy => ?_⟩ by_cases hyK : y ∈ K exacts [hxf _ (Or.inr ⟨hyK, hy⟩), (hxf _ (Or.inl rfl)).trans (hKf ⟨hyK, hy⟩)] /-- The extreme value theorem: if a function `f` is continuous on a closed set `s` and it is smaller than a value in its image away from compact sets, then it has a maximum on this set. -/ theorem ContinuousOn.exists_isMaxOn' [ClosedIciTopology α] {s : Set β} {f : β → α} (hf : ContinuousOn f s) (hsc : IsClosed s) {x₀ : β} (h₀ : x₀ ∈ s) (hc : ∀ᶠ x in cocompact β ⊓ 𝓟 s, f x ≤ f x₀) : ∃ x ∈ s, IsMaxOn f s x := ContinuousOn.exists_isMinOn' (α := αᵒᵈ) hf hsc h₀ hc /-- The extreme value theorem: if a continuous function `f` is larger than a value in its range away from compact sets, then it has a global minimum. -/ theorem Continuous.exists_forall_le' [ClosedIicTopology α] {f : β → α} (hf : Continuous f) (x₀ : β) (h : ∀ᶠ x in cocompact β, f x₀ ≤ f x) : ∃ x : β, ∀ y : β, f x ≤ f y := let ⟨x, _, hx⟩ := hf.continuousOn.exists_isMinOn' isClosed_univ (mem_univ x₀) (by rwa [principal_univ, inf_top_eq]) ⟨x, fun y => hx (mem_univ y)⟩ /-- The extreme value theorem: if a continuous function `f` is smaller than a value in its range away from compact sets, then it has a global maximum. -/ theorem Continuous.exists_forall_ge' [ClosedIciTopology α] {f : β → α} (hf : Continuous f) (x₀ : β) (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x := Continuous.exists_forall_le' (α := αᵒᵈ) hf x₀ h /-- The extreme value theorem: if a continuous function `f` tends to infinity away from compact sets, then it has a global minimum. -/ theorem Continuous.exists_forall_le [ClosedIicTopology α] [Nonempty β] {f : β → α} (hf : Continuous f) (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y := by inhabit β exact hf.exists_forall_le' default (hlim.eventually <\| eventually_ge_atTop _) /-- The extreme value theorem: if a continuous function `f` tends to negative infinity away from compact sets, then it has a global maximum. -/ theorem Continuous.exists_forall_ge [ClosedIciTopology α] [Nonempty β] {f : β → α} (hf : Continuous f) (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x := Continuous.exists_forall_le (α := αᵒᵈ) hf hlim /-- A continuous function with compact support has a global minimum. -/ @[to_additive /-- A continuous function with compact support has a global minimum. -/] theorem Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology α] [Nonempty β] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f x ≤ f y := by obtain ⟨_, ⟨x, rfl⟩, hx⟩ := (h.isCompact_range hf).exists_isLeast (range_nonempty _) rw [mem_lowerBounds, forall_mem_range] at hx exact ⟨x, hx⟩ /-- A continuous function with compact support has a global maximum. -/ @[to_additive /-- A continuous function with compact support has a global maximum. -/] theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [ClosedIciTopology α] [Nonempty β] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : ∃ x : β, ∀ y : β, f y ≤ f x := Continuous.exists_forall_le_of_hasCompactMulSupport (α := αᵒᵈ) hf h /-- A compact set is bounded below -/ theorem IsCompact.bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) : BddBelow s := by rcases s.eq_empty_or_nonempty with rfl \| hne · exact bddBelow_empty · obtain ⟨a, -, has⟩ := hs.exists_isLeast hne exact ⟨a, has⟩ /-- A compact set is bounded above -/ theorem IsCompact.bddAbove [ClosedIciTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) : BddAbove s := IsCompact.bddBelow (α := αᵒᵈ) hs /-- A continuous function is bounded below on a compact set. -/ theorem IsCompact.bddBelow_image [ClosedIicTopology α] [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K) (hf : ContinuousOn f K) : BddBelow (f '' K) := (hK.image_of_continuousOn hf).bddBelow /-- A continuous function is bounded above on a compact set. -/ theorem IsCompact.bddAbove_image [ClosedIciTopology α] [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K) (hf : ContinuousOn f K) : BddAbove (f '' K) := IsCompact.bddBelow_image (α := αᵒᵈ) hK hf /-- A continuous function with compact support is bounded below. -/ @[to_additive /-- A continuous function with compact support is bounded below. -/] theorem Continuous.bddBelow_range_of_hasCompactMulSupport [ClosedIicTopology α] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddBelow (range f) := (h.isCompact_range hf).bddBelow /-- A continuous function with compact support is bounded above. -/ @[to_additive /-- A continuous function with compact support is bounded above. -/] theorem Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology α] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f) := Continuous.bddBelow_range_of_hasCompactMulSupport (α := αᵒᵈ) hf h end LinearOrder section ConditionallyCompleteLinearOrder variable {α β γ : Type} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] theorem IsCompact.sSup_lt_iff_of_continuous [ClosedIciTopology α] {f : β → α} {K : Set β} (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y := by refine ⟨fun h x hx => (le_csSup (hK.bddAbove_image hf) <\| mem_image_of_mem f hx).trans_lt h, fun h => ?_⟩ obtain ⟨x, hx, h2x⟩ := hK.exists_isMaxOn h0K hf refine (csSup_le (h0K.image f) ?_).trans_lt (h x hx) rintro _ ⟨x', hx', rfl⟩; exact h2x hx' theorem IsCompact.lt_sInf_iff_of_continuous [ClosedIicTopology α] {f : β → α} {K : Set β} (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) : y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x := IsCompact.sSup_lt_iff_of_continuous (α := αᵒᵈ) hK h0K hf y end ConditionallyCompleteLinearOrder /-! ### Min and max elements of a compact set -/ section InfSup variable {α β : Type} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] theorem IsCompact.sInf_mem [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sInf s ∈ s := let ⟨_a, ha⟩ := hs.exists_isLeast ne_s ha.csInf_mem theorem IsCompact.sSup_mem [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : sSup s ∈ s := IsCompact.sInf_mem (α := αᵒᵈ) hs ne_s theorem IsCompact.isGLB_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : IsGLB s (sInf s) := isGLB_csInf ne_s hs.bddBelow theorem IsCompact.isLUB_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : IsLUB s (sSup s) := IsCompact.isGLB_sInf (α := αᵒᵈ) hs ne_s theorem IsCompact.isLeast_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : IsLeast s (sInf s) := ⟨hs.sInf_mem ne_s, (hs.isGLB_sInf ne_s).1⟩ theorem IsCompact.isGreatest_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty) : IsGreatest s (sSup s) := IsCompact.isLeast_sInf (α := αᵒᵈ) hs ne_s theorem IsCompact.exists_sInf_image_eq_and_le [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y := let ⟨x, hxs, hx⟩ := (hs.image_of_continuousOn hf).sInf_mem (ne_s.image f) ⟨x, hxs, hx.symm, fun _y hy => hx.trans_le <\| csInf_le (hs.image_of_continuousOn hf).bddBelow <\| mem_image_of_mem f hy⟩ theorem IsCompact.exists_sSup_image_eq_and_ge [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x := IsCompact.exists_sInf_image_eq_and_le (α := αᵒᵈ) hs ne_s hf theorem IsCompact.exists_sInf_image_eq [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x := let ⟨x, hxs, hx, _⟩ := hs.exists_sInf_image_eq_and_le ne_s hf ⟨x, hxs, hx⟩ theorem IsCompact.exists_sSup_image_eq [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) : ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x := IsCompact.exists_sInf_image_eq (α := αᵒᵈ) hs ne_s end InfSup section ExistsExtr variable {α β : Type} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] theorem IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x := let ⟨x, hxt, hfx⟩ := ht.exists_isMinOn ⟨z, hz⟩ hf ⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_ge (hfx hz), hfx⟩ theorem IsCompact.exists_isMaxOn_mem_subset [ClosedIciTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z' < f z) : ∃ x ∈ s, IsMaxOn f t x := let ⟨x, hxt, hfx⟩ := ht.exists_isMaxOn ⟨z, hz⟩ hf ⟨x, by_contra fun hxs => (hfz x ⟨hxt, hxs⟩).not_ge (hfx hz), hfx⟩ -- TODO: we could assume `t ∈ 𝓝ˢ s` (a.k.a. `s ⊆ interior t`) instead of `s ⊆ t` and `IsOpen s`. theorem IsCompact.exists_isLocalMin_mem_open [ClosedIicTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x := let ⟨x, hxs, h⟩ := ht.exists_isMinOn_mem_subset hf hz hfz ⟨x, hxs, h.isLocalMin <\| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩ theorem IsCompact.exists_isLocalMax_mem_open [ClosedIciTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z' < f z) (hs : IsOpen s) : ∃ x ∈ s, IsLocalMax f x := let ⟨x, hxs, h⟩ := ht.exists_isMaxOn_mem_subset hf hz hfz ⟨x, hxs, h.isLocalMax <\| mem_nhds_iff.2 ⟨s, hst, hs, hxs⟩⟩ end ExistsExtr variable {α β γ : Type} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ] theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) : s = Icc (sInf s) (sSup s) := eq_Icc_csInf_csSup_of_connected_bdd_closed h₁ h₂.bddBelow h₂.bddAbove h₂.isClosed /-- If `f : γ → β → α` is a function that is continuous as a function on `γ × β`, `α` is a conditionally complete linear order, and `K : Set β` is a compact set, then `fun x ↦ sSup (f x '' K)` is a continuous function. -/ /- TODO: generalize. The following version seems to be true: ``` theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ} (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ˢ 𝓝[K] y) (𝓝 (g y))) (hgc : ContinuousOn g K) : Tendsto (fun x => sSup (f x '' K)) l (𝓝 (sSup (g '' K))) := _ ``` Moreover, it seems that `hgc` follows from `hf` (Yury Kudryashov). -/ theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K) (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) := by rcases eq_empty_or_nonempty K with (rfl \| h0K) · simp_rw [image_empty] exact continuous_const rw [continuous_iff_continuousAt] intro x obtain ⟨y, hyK, h2y, hy⟩ := hK.exists_sSup_image_eq_and_ge h0K (show Continuous (f x) from hf.comp <\| .prodMk_right x).continuousOn rw [ContinuousAt, h2y, tendsto_order] have := tendsto_order.mp ((show Continuous fun x => f x y from hf.comp <\| .prodMk_left _).tendsto x) refine ⟨fun z hz => ?_, fun z hz => ?_⟩ · refine (this.1 z hz).mono fun x' hx' => hx'.trans_le <\| le_csSup ?_ <\| mem_image_of_mem (f x') hyK exact hK.bddAbove_image (hf.comp <\| .prodMk_right x').continuousOn · have h : ({x} : Set γ) ×ˢ K ⊆ ↿f ⁻¹' Iio z := by rintro ⟨x', y'⟩ ⟨(rfl : x' = x), hy'⟩ exact (hy y' hy').trans_lt hz obtain ⟨u, v, hu, _, hxu, hKv, huv⟩ := generalized_tube_lemma isCompact_singleton hK (isOpen_Iio.preimage hf) h refine eventually_of_mem (hu.mem_nhds (singleton_subset_iff.mp hxu)) fun x' hx' => ?_ rw [hK.sSup_lt_iff_of_continuous h0K (show Continuous (f x') from hf.comp <\| .prodMk_right x').continuousOn] exact fun y' hy' => huv (mk_mem_prod hx' (hKv hy')) theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K) (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) := IsCompact.continuous_sSup (α := αᵒᵈ) hK hf namespace ContinuousOn /-! ### Image of a closed interval -/ variable [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a b c : α} open scoped Interval theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <\| Icc a b) : f '' Icc a b = Icc (sInf <\| f '' Icc a b) (sSup <\| f '' Icc a b) := eq_Icc_of_connected_compact ⟨(nonempty_Icc.2 hab).image f, isPreconnected_Icc.image f h⟩ (isCompact_Icc.image_of_continuousOn h) theorem image_uIcc_eq_Icc (h : ContinuousOn f [[a, b]]) : f '' [[a, b]] = Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) := image_Icc min_le_max h theorem image_uIcc (h : ContinuousOn f <\| [[a, b]]) : f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] := by refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm refine csInf_le_csSup ?_ ?_ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc] exacts [bddBelow_Icc, bddAbove_Icc] theorem sInf_image_Icc_le (h : ContinuousOn f <\| Icc a b) (hc : c ∈ Icc a b) : sInf (f '' Icc a b) ≤ f c := by have := mem_image_of_mem f hc rw [h.image_Icc (hc.1.trans hc.2)] at this exact this.1 theorem le_sSup_image_Icc (h : ContinuousOn f <\| Icc a b) (hc : c ∈ Icc a b) : f c ≤ sSup (f '' Icc a b) := by have := mem_image_of_mem f hc rw [h.image_Icc (hc.1.trans hc.2)] at this exact this.2 theorem image_Icc_of_monotoneOn (hab : a ≤ b) (h : ContinuousOn f <\| Icc a b) (h' : MonotoneOn f <\| Icc a b) : f '' Icc a b = Icc (f a) (f b) := by rw [h.image_Icc hab] congr! · exact h'.sInf_image_Icc hab · exact h'.sSup_image_Icc hab theorem image_Icc_of_antitoneOn (hab : a ≤ b) (h : ContinuousOn f <\| Icc a b) (h' : AntitoneOn f <\| Icc a b) : f '' Icc a b = Icc (f b) (f a) := by have : Icc (f b) (f a) = Icc (toDual (f a)) (toDual (f b)) := by rw [Icc_toDual]; rfl rw [this] exact image_Icc_of_monotoneOn (β := βᵒᵈ) hab h h'.dual_right end ContinuousOn
.lake/packages/mathlib/Mathlib/Topology/Order/ScottTopology.lean	import Mathlib.Order.ScottContinuity import Mathlib.Topology.Order.UpperLowerSetTopology /-! # Scott topology This file introduces the Scott topology on a preorder. ## Main definitions - `DirSupInacc` - a set `u` is said to be inaccessible by directed joins if, when the least upper bound of a directed set `d` lies in `u` then `d` has non-empty intersection with `u`. - `DirSupClosed` - a set `s` is said to be closed under directed joins if, whenever a directed set `d` has a least upper bound `a` and is a subset of `s` then `a` also lies in `s`. - `Topology.scott` - the Scott topology is defined as the join of the topology of upper sets and the Scott-Hausdorff topology (the topological space where a set `u` is open if, when the least upper bound of a directed set `d` lies in `u` then there is a tail of `d` which is a subset of `u`). ## Main statements - `Topology.IsScott.isUpperSet_of_isOpen`: Scott open sets are upper. - `Topology.IsScott.isLowerSet_of_isClosed`: Scott closed sets are lower. - `Topology.IsScott.monotone_of_continuous`: Functions continuous w.r.t. the Scott topology are monotone. - `Topology.IsScott.scottContinuousOn_iff_continuous` - a function is Scott continuous (preserves least upper bounds of directed sets) if and only if it is continuous w.r.t. the Scott topology. - `Topology.IsScott.instT0Space` - the Scott topology on a partial order is T₀. ## Implementation notes A type synonym `WithScott` is introduced and for a preorder `α`, `WithScott α` is made an instance of `TopologicalSpace` by the `scott` topology. We define a mixin class `IsScott` for the class of types which are both a preorder and a topology and where the topology is the `scott` topology. It is shown that `WithScott α` is an instance of `IsScott`. A class `Scott` is defined in `Topology/OmegaCompletePartialOrder` and made an instance of a topological space by defining the open sets to be those which have characteristic functions which are monotone and preserve limits of countable chains (`OmegaCompletePartialOrder.Continuous'`). A Scott continuous function between `OmegaCompletePartialOrder`s is always `OmegaCompletePartialOrder.Continuous'` (`OmegaCompletePartialOrder.ScottContinuous.continuous'`). The converse is true in some special cases, but not in general ([Domain Theory, 2.2.4][abramsky_gabbay_maibaum_1994]). ## References * [Abramsky and Jung, Domain Theory][abramsky_gabbay_maibaum_1994] * [Gierz et al, A Compendium of Continuous Lattices][GierzEtAl1980] * [Karner, Continuous monoids and semirings][Karner2004] ## Tags Scott topology, preorder -/ open Set variable {α β : Type} /-! ### Prerequisite order properties -/ section Preorder variable [Preorder α] {s t : Set α} /-- A set `s` is said to be inaccessible by directed joins on `D` if, when the least upper bound of a directed set `d` in `D` lies in `s` then `d` has non-empty intersection with `s`. -/ def DirSupInaccOn (D : Set (Set α)) (s : Set α) : Prop := ∀ ⦃d⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → a ∈ s → (d ∩ s).Nonempty /-- A set `s` is said to be inaccessible by directed joins if, when the least upper bound of a directed set `d` lies in `s` then `d` has non-empty intersection with `s`. -/ def DirSupInacc (s : Set α) : Prop := ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → a ∈ s → (d ∩ s).Nonempty @[simp] lemma dirSupInaccOn_univ : DirSupInaccOn univ s ↔ DirSupInacc s := by simp [DirSupInaccOn, DirSupInacc] @[simp] lemma DirSupInacc.dirSupInaccOn {D : Set (Set α)} : DirSupInacc s → DirSupInaccOn D s := fun h _ _ d₂ d₃ _ hda => h d₂ d₃ hda lemma DirSupInaccOn.mono {D₁ D₂ : Set (Set α)} (hD : D₁ ⊆ D₂) (hf : DirSupInaccOn D₂ s) : DirSupInaccOn D₁ s := fun ⦃_⦄ a ↦ hf (hD a) /-- A set `s` is said to be closed under directed joins if, whenever a directed set `d` has a least upper bound `a` and is a subset of `s` then `a` also lies in `s`. -/ def DirSupClosed (s : Set α) : Prop := ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → d ⊆ s → a ∈ s @[simp] lemma dirSupInacc_compl : DirSupInacc sᶜ ↔ DirSupClosed s := by simp [DirSupInacc, DirSupClosed, ← not_disjoint_iff_nonempty_inter, not_imp_not, disjoint_compl_right_iff] @[simp] lemma dirSupClosed_compl : DirSupClosed sᶜ ↔ DirSupInacc s := by rw [← dirSupInacc_compl, compl_compl] alias ⟨DirSupInacc.of_compl, DirSupClosed.compl⟩ := dirSupInacc_compl alias ⟨DirSupClosed.of_compl, DirSupInacc.compl⟩ := dirSupClosed_compl lemma DirSupClosed.inter (hs : DirSupClosed s) (ht : DirSupClosed t) : DirSupClosed (s ∩ t) := fun _d hd hd' _a ha hds ↦ ⟨hs hd hd' ha <\| hds.trans inter_subset_left, ht hd hd' ha <\| hds.trans inter_subset_right⟩ lemma DirSupInacc.union (hs : DirSupInacc s) (ht : DirSupInacc t) : DirSupInacc (s ∪ t) := by rw [← dirSupClosed_compl, compl_union]; exact hs.compl.inter ht.compl lemma IsUpperSet.dirSupClosed (hs : IsUpperSet s) : DirSupClosed s := fun _d ⟨_b, hb⟩ _ _a ha hds ↦ hs (ha.1 hb) <\| hds hb lemma IsLowerSet.dirSupInacc (hs : IsLowerSet s) : DirSupInacc s := hs.compl.dirSupClosed.of_compl lemma dirSupClosed_Iic (a : α) : DirSupClosed (Iic a) := fun _d _ _ _a ha ↦ (isLUB_le_iff ha).2 end Preorder section CompleteLattice variable [CompleteLattice α] {s : Set α} lemma dirSupInacc_iff_forall_sSup : DirSupInacc s ↔ ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → sSup d ∈ s → (d ∩ s).Nonempty := by simp [DirSupInacc, isLUB_iff_sSup_eq] lemma dirSupClosed_iff_forall_sSup : DirSupClosed s ↔ ∀ ⦃d⦄, d.Nonempty → DirectedOn (· ≤ ·) d → d ⊆ s → sSup d ∈ s := by simp [DirSupClosed, isLUB_iff_sSup_eq] end CompleteLattice namespace Topology /-! ### Scott-Hausdorff topology -/ section ScottHausdorff /-- The Scott-Hausdorff topology. A set `u` is open in the Scott-Hausdorff topology iff when the least upper bound of a directed set `d` lies in `u` then there is a tail of `d` which is a subset of `u`. -/ def scottHausdorff (α : Type) (D : Set (Set α)) [Preorder α] : TopologicalSpace α where IsOpen u := ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ u → ∃ b ∈ d, Ici b ∩ d ⊆ u isOpen_univ := fun d _ ⟨b, hb⟩ _ _ _ _ ↦ ⟨b, hb, (Ici b ∩ d).subset_univ⟩ isOpen_inter s t hs ht d hd₀ hd₁ hd₂ a hd₃ ha := by obtain ⟨b₁, hb₁d, hb₁ds⟩ := hs hd₀ hd₁ hd₂ hd₃ ha.1 obtain ⟨b₂, hb₂d, hb₂dt⟩ := ht hd₀ hd₁ hd₂ hd₃ ha.2 obtain ⟨c, hcd, hc⟩ := hd₂ b₁ hb₁d b₂ hb₂d exact ⟨c, hcd, fun e ⟨hce, hed⟩ ↦ ⟨hb₁ds ⟨hc.1.trans hce, hed⟩, hb₂dt ⟨hc.2.trans hce, hed⟩⟩⟩ isOpen_sUnion := fun s h d hd₀ hd₁ hd₂ a hd₃ ⟨s₀, hs₀s, has₀⟩ ↦ by obtain ⟨b, hbd, hbds₀⟩ := h s₀ hs₀s hd₀ hd₁ hd₂ hd₃ has₀ exact ⟨b, hbd, Set.subset_sUnion_of_subset s s₀ hbds₀ hs₀s⟩ variable (α) (D : Set (Set α)) [Preorder α] [TopologicalSpace α] /-- Predicate for an ordered topological space to be equipped with its Scott-Hausdorff topology. A set `u` is open in the Scott-Hausdorff topology iff when the least upper bound of a directed set `d` lies in `u` then there is a tail of `d` which is a subset of `u`. -/ class IsScottHausdorff : Prop where topology_eq_scottHausdorff : ‹TopologicalSpace α› = scottHausdorff α D instance : @IsScottHausdorff α D _ (scottHausdorff α D) := @IsScottHausdorff.mk _ _ _ (scottHausdorff α D) rfl namespace IsScottHausdorff variable {s : Set α} lemma topology_eq [IsScottHausdorff α D] : ‹_› = scottHausdorff α D := topology_eq_scottHausdorff variable {α D} lemma isOpen_iff [IsScottHausdorff α D] : IsOpen s ↔ ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s → ∃ b ∈ d, Ici b ∩ d ⊆ s := by simp [topology_eq_scottHausdorff (α := α) (D := D), IsOpen, scottHausdorff] lemma dirSupInaccOn_of_isOpen [IsScottHausdorff α D] (h : IsOpen s) : DirSupInaccOn D s := fun d hd₀ hd₁ hd₂ a hda hd₃ ↦ by obtain ⟨b, hbd, hb⟩ := isOpen_iff.mp h hd₀ hd₁ hd₂ hda hd₃; exact ⟨b, hbd, hb ⟨le_rfl, hbd⟩⟩ lemma dirSupClosed_of_isClosed [IsScottHausdorff α univ] (h : IsClosed s) : DirSupClosed s := by apply DirSupInacc.of_compl rw [← dirSupInaccOn_univ] exact (dirSupInaccOn_of_isOpen h.isOpen_compl) end IsScottHausdorff end ScottHausdorff section ScottHausdorff namespace IsScottHausdorff variable {s : Set α} [Preorder α] {t : TopologicalSpace α} [IsScottHausdorff α univ] lemma isOpen_of_isLowerSet (h : IsLowerSet s) : IsOpen s := (isOpen_iff (D := univ)).2 fun _d _ ⟨b, hb⟩ _ _ hda ha ↦ ⟨b, hb, fun _ hc ↦ h (mem_upperBounds.1 hda.1 _ hc.2) ha⟩ lemma isClosed_of_isUpperSet (h : IsUpperSet s) : IsClosed s := isOpen_compl_iff.1 <\| isOpen_of_isLowerSet h.compl end IsScottHausdorff end ScottHausdorff /-! ### Scott topology -/ section Scott section Preorder /-- The Scott topology. It is defined as the join of the topology of upper sets and the Scott-Hausdorff topology. -/ def scott (α : Type) (D : Set (Set α)) [Preorder α] : TopologicalSpace α := upperSet α ⊔ scottHausdorff α D lemma upperSet_le_scott [Preorder α] : upperSet α ≤ scott α univ := le_sup_left lemma scottHausdorff_le_scott [Preorder α] : scottHausdorff α univ ≤ scott α univ := le_sup_right variable (α) (D) [Preorder α] [TopologicalSpace α] /-- Predicate for an ordered topological space to be equipped with its Scott topology. The Scott topology is defined as the join of the topology of upper sets and the Scott Hausdorff topology. -/ class IsScott : Prop where topology_eq_scott : ‹TopologicalSpace α› = scott α D end Preorder namespace IsScott section Preorder variable (α) (D) [Preorder α] [TopologicalSpace α] lemma topology_eq [IsScott α D] : ‹_› = scott α D := topology_eq_scott variable {α} {D} {s : Set α} {a : α} lemma isOpen_iff_isUpperSet_and_scottHausdorff_open [IsScott α D] : IsOpen s ↔ IsUpperSet s ∧ IsOpen[scottHausdorff α D] s := by rw [topology_eq α D]; rfl lemma isOpen_iff_isUpperSet_and_dirSupInaccOn [IsScott α D] : IsOpen s ↔ IsUpperSet s ∧ DirSupInaccOn D s := by rw [isOpen_iff_isUpperSet_and_scottHausdorff_open (D := D)] refine and_congr_right fun h ↦ ⟨@IsScottHausdorff.dirSupInaccOn_of_isOpen _ _ _ (scottHausdorff α D) _ _, fun h' d d₀ d₁ d₂ _ d₃ ha ↦ ?_⟩ obtain ⟨b, hbd, hbu⟩ := h' d₀ d₁ d₂ d₃ ha exact ⟨b, hbd, Subset.trans inter_subset_left (h.Ici_subset hbu)⟩ lemma isClosed_iff_isLowerSet_and_dirSupClosed [IsScott α univ] : IsClosed s ↔ IsLowerSet s ∧ DirSupClosed s := by rw [← isOpen_compl_iff, isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ), isUpperSet_compl, dirSupInaccOn_univ, dirSupInacc_compl] lemma isUpperSet_of_isOpen [IsScott α D] : IsOpen s → IsUpperSet s := fun h ↦ (isOpen_iff_isUpperSet_and_scottHausdorff_open (D := D).mp h).left lemma isLowerSet_of_isClosed [IsScott α univ] : IsClosed s → IsLowerSet s := fun h ↦ (isClosed_iff_isLowerSet_and_dirSupClosed.mp h).left lemma dirSupClosed_of_isClosed [IsScott α univ] : IsClosed s → DirSupClosed s := fun h ↦ (isClosed_iff_isLowerSet_and_dirSupClosed.mp h).right lemma lowerClosure_subset_closure [IsScott α univ] : ↑(lowerClosure s) ⊆ closure s := by convert closure.mono (@upperSet_le_scott α _) · rw [@IsUpperSet.closure_eq_lowerClosure α _ (upperSet α) ?_ s] infer_instance · exact topology_eq α univ instance [IsScott α univ] : ClosedIicTopology α where isClosed_Iic _ := isClosed_iff_isLowerSet_and_dirSupClosed.2 ⟨isLowerSet_Iic _, dirSupClosed_Iic _⟩ @[deprecated (since := "2025-07-02")] protected alias isClosed_Iic := isClosed_Iic /-- The closure of a singleton `{a}` in the Scott topology is the right-closed left-infinite interval `(-∞,a]`. -/ @[simp] lemma closure_singleton [IsScott α univ] : closure {a} = Iic a := le_antisymm (closure_minimal (by rw [singleton_subset_iff, mem_Iic]) isClosed_Iic) <\| by rw [← LowerSet.coe_Iic, ← lowerClosure_singleton] apply lowerClosure_subset_closure variable [Preorder β] [TopologicalSpace β] [IsScott β univ] {f : α → β} lemma monotone_of_continuous [IsScott α D] (hf : Continuous f) : Monotone f := fun _ b hab ↦ by by_contra h simpa only [mem_compl_iff, mem_preimage, mem_Iic, le_refl, not_true] using isUpperSet_of_isOpen (D := D) ((isOpen_compl_iff.2 isClosed_Iic).preimage hf) hab h @[simp] lemma scottContinuousOn_iff_continuous {D : Set (Set α)} [Topology.IsScott α D] (hD : ∀ a b : α, a ≤ b → {a, b} ∈ D) : ScottContinuousOn D f ↔ Continuous f := by refine ⟨fun h ↦ continuous_def.2 fun u hu ↦ ?_, ?_⟩ · rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)] exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD), fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <\| (isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp hu).2 trivial (Nonempty.image f hd₁) (directedOn_image.mpr (hd₂.mono @(h.monotone D hD))) (h h₀ hd₁ hd₂ hd₃) ha⟩ · refine fun hf t h₀ d₁ d₂ a d₃ ↦ ⟨(monotone_of_continuous (D := D) hf).mem_upperBounds_image d₃.1, fun b hb ↦ ?_⟩ by_contra h let u := (Iic b)ᶜ have hu : IsOpen (f ⁻¹' u) := isClosed_Iic.isOpen_compl.preimage hf rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)] at hu obtain ⟨c, hcd, hfcb⟩ := hu.2 h₀ d₁ d₂ d₃ h simp only [upperBounds, mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_setOf] at hb exact hfcb <\| hb _ hcd @[deprecated (since := "2025-07-02")] alias scottContinuous_iff_continuous := scottContinuousOn_iff_continuous end Preorder section PartialOrder variable [PartialOrder α] [TopologicalSpace α] [IsScott α univ] /-- The Scott topology on a partial order is T₀. -/ -- see Note [lower instance priority] instance (priority := 90) : T0Space α := (t0Space_iff_inseparable α).2 fun x y h ↦ Iic_injective <\| by simpa only [inseparable_iff_closure_eq, IsScott.closure_singleton] using h end PartialOrder section CompleteLinearOrder variable [CompleteLinearOrder α] lemma isOpen_iff_Iic_compl_or_univ [TopologicalSpace α] [Topology.IsScott α univ] (U : Set α) : IsOpen U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U := by constructor · intro hU rcases eq_empty_or_nonempty Uᶜ with eUc \| neUc · exact Or.inl (compl_empty_iff.mp eUc) · apply Or.inr use sSup Uᶜ rw [compl_eq_comm, le_antisymm_iff] exact ⟨fun _ ha ↦ le_sSup ha, (isLowerSet_of_isClosed hU.isClosed_compl).Iic_subset (dirSupClosed_iff_forall_sSup.mp (dirSupClosed_of_isClosed hU.isClosed_compl) neUc (isChain_of_trichotomous Uᶜ).directedOn le_rfl)⟩ · rintro (rfl \| ⟨a, rfl⟩) · exact isOpen_univ · exact isClosed_Iic.isOpen_compl -- N.B. A number of conditions equivalent to `scott α = upper α` are given in Gierz _et al_, -- Chapter III, Exercise 3.23. lemma scott_eq_upper_of_completeLinearOrder : scott α univ = upper α := by letI := upper α ext U rw [@Topology.IsUpper.isTopologicalSpace_basis _ _ (upper α) ({ topology_eq_upperTopology := rfl }) U] letI := scott α univ rw [@isOpen_iff_Iic_compl_or_univ _ _ (scott α univ) ({ topology_eq_scott := rfl }) U] /- The upper topology on a complete linear order is the Scott topology -/ instance [TopologicalSpace α] [IsUpper α] : IsScott α univ where topology_eq_scott := by rw [scott_eq_upper_of_completeLinearOrder] exact IsUpper.topology_eq α end CompleteLinearOrder lemma isOpen_iff_scottContinuous_mem [Preorder α] {s : Set α} [TopologicalSpace α] [IsScott α univ] : IsOpen s ↔ ScottContinuous fun x ↦ x ∈ s := by rw [← scottContinuousOn_univ, scottContinuousOn_iff_continuous (fun _ _ _ ↦ by trivial)] exact isOpen_iff_continuous_mem end IsScott /-- Type synonym for a preorder equipped with the Scott topology -/ def WithScott (α : Type) := α namespace WithScott /-- `toScott` is the identity function to the `WithScott` of a type. -/ @[match_pattern] def toScott : α ≃ WithScott α := Equiv.refl _ /-- `ofScott` is the identity function from the `WithScott` of a type. -/ @[match_pattern] def ofScott : WithScott α ≃ α := Equiv.refl _ @[simp] lemma toScott_symm_eq : (@toScott α).symm = ofScott := rfl @[simp] lemma ofScott_symm_eq : (@ofScott α).symm = toScott := rfl @[simp] lemma toScott_ofScott (a : WithScott α) : toScott (ofScott a) = a := rfl @[simp] lemma ofScott_toScott (a : α) : ofScott (toScott a) = a := rfl lemma toScott_inj {a b : α} : toScott a = toScott b ↔ a = b := Iff.rfl lemma ofScott_inj {a b : WithScott α} : ofScott a = ofScott b ↔ a = b := Iff.rfl /-- A recursor for `WithScott`. Use as `induction x`. -/ @[elab_as_elim, cases_eliminator, induction_eliminator] protected def rec {β : WithScott α → Sort _} (h : ∀ a, β (toScott a)) : ∀ a, β a := fun a ↦ h (ofScott a) instance [Nonempty α] : Nonempty (WithScott α) := ‹Nonempty α› instance [Inhabited α] : Inhabited (WithScott α) := ‹Inhabited α› variable [Preorder α] instance : Preorder (WithScott α) := ‹Preorder α› instance : TopologicalSpace (WithScott α) := scott α univ instance : IsScott (WithScott α) univ := ⟨rfl⟩ lemma isOpen_iff_isUpperSet_and_scottHausdorff_open' {u : Set α} : IsOpen (WithScott.ofScott ⁻¹' u) ↔ IsUpperSet u ∧ (scottHausdorff α univ).IsOpen u := Iff.rfl end WithScott end Scott variable [Preorder α] lemma scottHausdorff_le_lower : scottHausdorff α univ ≤ lower α := fun s h => IsScottHausdorff.isOpen_of_isLowerSet (t := scottHausdorff α univ) <\| (@IsLower.isLowerSet_of_isOpen (Topology.WithLower α) _ _ _ s h) variable [TopologicalSpace α] /-- If `α` is equipped with the Scott topology, then it is homeomorphic to `WithScott α`. -/ def IsScott.withScottHomeomorph [IsScott α univ] : WithScott α ≃ₜ α := WithScott.ofScott.toHomeomorphOfIsInducing ⟨IsScott.topology_eq α univ ▸ induced_id.symm⟩ lemma IsScott.scottHausdorff_le [IsScott α univ] : scottHausdorff α univ ≤ ‹TopologicalSpace α› := by rw [IsScott.topology_eq α univ, scott]; exact le_sup_right lemma IsLower.scottHausdorff_le [IsLower α] : scottHausdorff α univ ≤ ‹TopologicalSpace α› := fun _ h ↦ IsScottHausdorff.isOpen_of_isLowerSet (t := scottHausdorff α univ) <\| IsLower.isLowerSet_of_isOpen h end Topology
.lake/packages/mathlib/Mathlib/Topology/Order/CountableSeparating.lean	import Mathlib.Topology.Order.Basic import Mathlib.Order.Filter.CountableSeparatingOn /-! # Countably many infinite intervals separate points In this file we prove that in a linear order with second countable order topology, the points can be separated by countably many infinite intervals. We prove 4 versions of this statement (one for each of the infinite intervals), as well as provide convenience corollaries about `Filter.EventuallyEq`. -/ open Set variable {X : Type} [TopologicalSpace X] [LinearOrder X] [OrderTopology X] [SecondCountableTopology X] namespace HasCountableSeparatingOn variable {s : Set X} instance range_Iio : HasCountableSeparatingOn X (· ∈ range Iio) s := by constructor rcases TopologicalSpace.exists_countable_dense X with ⟨s, hsc, hsd⟩ set t := s ∪ {x \| ∃ y, y ⋖ x} refine ⟨Iio '' t, .image ?_ _, ?_, ?_⟩ · exact hsc.union countable_setOf_covBy_left · exact image_subset_range _ _ · rintro x - y - h by_contra! hne wlog hlt : x < y generalizing x y · refine this y x ?_ hne.symm (hne.lt_or_gt.resolve_left hlt) simpa only [iff_comm] using h cases (Ioo x y).eq_empty_or_nonempty with \| inl he => specialize h (Iio y) (mem_image_of_mem _ (.inr ⟨x, hlt, by simpa using Set.ext_iff.mp he⟩)) simp [hlt.not_ge] at h \| inr hne => rcases hsd.inter_open_nonempty _ isOpen_Ioo hne with ⟨z, ⟨hxz, hzy⟩, hzs⟩ simpa [hxz, hzy.not_gt] using h (Iio z) (mem_image_of_mem _ (.inl hzs)) instance range_Ioi : HasCountableSeparatingOn X (· ∈ range Ioi) s := .range_Iio (X := Xᵒᵈ) instance range_Iic : HasCountableSeparatingOn X (· ∈ range Iic) s := let ⟨t, htc, ht_sub, ht⟩ := (range_Ioi (X := X) (s := s)).1 ⟨compl '' t, htc.image _, by simpa [← compl_inj_iff (x := Ioi _)] using ht_sub, by simpa [not_iff_not]⟩ instance range_Ici : HasCountableSeparatingOn X (· ∈ range Ici) s := range_Iic (X := Xᵒᵈ) end HasCountableSeparatingOn namespace Filter.EventuallyEq variable {α : Type} {l : Filter α} [CountableInterFilter l] {f g : α → X} lemma of_forall_eventually_lt_iff (h : ∀ x, ∀ᶠ a in l, f a < x ↔ g a < x) : f =ᶠ[l] g := of_forall_separating_preimage (· ∈ range Iio) <\| forall_mem_range.2 <\| fun x ↦ .set_eq (h x) lemma of_forall_eventually_le_iff (h : ∀ x, ∀ᶠ a in l, f a ≤ x ↔ g a ≤ x) : f =ᶠ[l] g := of_forall_separating_preimage (· ∈ range Iic) <\| forall_mem_range.2 <\| fun x ↦ .set_eq (h x) lemma of_forall_eventually_gt_iff (h : ∀ x, ∀ᶠ a in l, x < f a ↔ x < g a) : f =ᶠ[l] g := of_forall_eventually_lt_iff (X := Xᵒᵈ) h lemma of_forall_eventually_ge_iff (h : ∀ x, ∀ᶠ a in l, x ≤ f a ↔ x ≤ g a) : f =ᶠ[l] g := of_forall_eventually_le_iff (X := Xᵒᵈ) h end Filter.EventuallyEq
.lake/packages/mathlib/Mathlib/Topology/Order/T5.lean	import Mathlib.Order.Interval.Set.OrdConnectedComponent import Mathlib.Topology.Order.Basic import Mathlib.Topology.Separation.Regular /-! # Linear order is a completely normal Hausdorff topological space In this file we prove that a linear order with order topology is a completely normal Hausdorff topological space. -/ open Filter Set Function OrderDual Topology Interval variable {X : Type*} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {a : X} {s t : Set X} namespace Set @[simp] theorem ordConnectedComponent_mem_nhds : ordConnectedComponent s a ∈ 𝓝 a ↔ s ∈ 𝓝 a := by refine ⟨fun h => mem_of_superset h ordConnectedComponent_subset, fun h => ?_⟩ rcases exists_Icc_mem_subset_of_mem_nhds h with ⟨b, c, ha, ha', hs⟩ exact mem_of_superset ha' (subset_ordConnectedComponent ha hs) theorem compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE (hd : Disjoint s (closure t)) (ha : a ∈ s) : (ordConnectedSection (ordSeparatingSet s t))ᶜ ∈ 𝓝[≥] a := by have hmem : tᶜ ∈ 𝓝[≥] a := by refine mem_nhdsWithin_of_mem_nhds ?_ rw [← mem_interior_iff_mem_nhds, interior_compl] exact disjoint_left.1 hd ha rcases exists_Icc_mem_subset_of_mem_nhdsGE hmem with ⟨b, hab, hmem', hsub⟩ by_cases H : Disjoint (Icc a b) (ordConnectedSection <\| ordSeparatingSet s t) · exact mem_of_superset hmem' (disjoint_left.1 H) · simp only [Set.disjoint_left, not_forall, Classical.not_not] at H rcases H with ⟨c, ⟨hac, hcb⟩, hc⟩ have hsub' : Icc a b ⊆ ordConnectedComponent tᶜ a := subset_ordConnectedComponent (left_mem_Icc.2 hab) hsub have hd : Disjoint s (ordConnectedSection (ordSeparatingSet s t)) := disjoint_left_ordSeparatingSet.mono_right ordConnectedSection_subset replace hac : a < c := hac.lt_of_ne <\| Ne.symm <\| ne_of_mem_of_not_mem hc <\| disjoint_left.1 hd ha filter_upwards [Ico_mem_nhdsGE hac] with x hx hx' refine hx.2.ne (eq_of_mem_ordConnectedSection_of_uIcc_subset hx' hc ?_) refine subset_inter (subset_iUnion₂_of_subset a ha ?_) ?_ · exact OrdConnected.uIcc_subset inferInstance (hsub' ⟨hx.1, hx.2.le.trans hcb⟩) (hsub' ⟨hac.le, hcb⟩) · rcases mem_iUnion₂.1 (ordConnectedSection_subset hx').2 with ⟨y, hyt, hxy⟩ refine subset_iUnion₂_of_subset y hyt (OrdConnected.uIcc_subset inferInstance hxy ?_) refine subset_ordConnectedComponent left_mem_uIcc hxy ?_ suffices c < y by rw [uIcc_of_ge (hx.2.trans this).le] exact ⟨hx.2.le, this.le⟩ refine lt_of_not_ge fun hyc => ?_ have hya : y < a := not_le.1 fun hay => hsub ⟨hay, hyc.trans hcb⟩ hyt exact hxy (Icc_subset_uIcc ⟨hya.le, hx.1⟩) ha theorem compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE (hd : Disjoint s (closure t)) (ha : a ∈ s) : (ordConnectedSection <\| ordSeparatingSet s t)ᶜ ∈ 𝓝[≤] a := by have hd' : Disjoint (ofDual ⁻¹' s) (closure <\| ofDual ⁻¹' t) := hd have ha' : toDual a ∈ ofDual ⁻¹' s := ha simpa only [dual_ordSeparatingSet, dual_ordConnectedSection] using compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE hd' ha' theorem compl_ordConnectedSection_ordSeparatingSet_mem_nhds (hd : Disjoint s (closure t)) (ha : a ∈ s) : (ordConnectedSection <\| ordSeparatingSet s t)ᶜ ∈ 𝓝 a := by rw [← nhdsLE_sup_nhdsGE, mem_sup] exact ⟨compl_ordConnectedSection_ordSeparatingSet_mem_nhdsLE hd ha, compl_ordConnectedSection_ordSeparatingSet_mem_nhdsGE hd ha⟩ theorem ordT5Nhd_mem_nhdsSet (hd : Disjoint s (closure t)) : ordT5Nhd s t ∈ 𝓝ˢ s := bUnion_mem_nhdsSet fun x hx => ordConnectedComponent_mem_nhds.2 <\| inter_mem (by rw [← mem_interior_iff_mem_nhds, interior_compl] exact disjoint_left.1 hd hx) (compl_ordConnectedSection_ordSeparatingSet_mem_nhds hd hx) end Set open Set /-- A linear order with order topology is a completely normal Hausdorff topological space. -/ instance (priority := 100) OrderTopology.completelyNormalSpace : CompletelyNormalSpace X := ⟨fun s t h₁ h₂ => Filter.disjoint_iff.2 ⟨ordT5Nhd s t, ordT5Nhd_mem_nhdsSet h₂, ordT5Nhd t s, ordT5Nhd_mem_nhdsSet h₁.symm, disjoint_ordT5Nhd⟩⟩ instance (priority := 100) OrderTopology.t5Space : T5Space X := T5Space.mk
.lake/packages/mathlib/Mathlib/Topology/Order/ExtrClosure.lean	import Mathlib.Topology.Order.OrderClosed import Mathlib.Topology.Order.LocalExtr /-! # Maximum/minimum on the closure of a set In this file we prove several versions of the following statement: if `f : X → Y` has a (local or not) maximum (or minimum) on a set `s` at a point `a` and is continuous on the closure of `s`, then `f` has an extremum of the same type on `Closure s` at `a`. -/ open Filter Set open Topology variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [Preorder Y] [OrderClosedTopology Y] {f : X → Y} {s : Set X} {a : X} protected theorem IsMaxOn.closure (h : IsMaxOn f s a) (hc : ContinuousOn f (closure s)) : IsMaxOn f (closure s) a := fun x hx => ContinuousWithinAt.closure_le hx ((hc x hx).mono subset_closure) continuousWithinAt_const h protected theorem IsMinOn.closure (h : IsMinOn f s a) (hc : ContinuousOn f (closure s)) : IsMinOn f (closure s) a := h.dual.closure hc protected theorem IsExtrOn.closure (h : IsExtrOn f s a) (hc : ContinuousOn f (closure s)) : IsExtrOn f (closure s) a := h.elim (fun h => Or.inl <\| h.closure hc) fun h => Or.inr <\| h.closure hc protected theorem IsLocalMaxOn.closure (h : IsLocalMaxOn f s a) (hc : ContinuousOn f (closure s)) : IsLocalMaxOn f (closure s) a := by rcases mem_nhdsWithin.1 h with ⟨U, Uo, aU, hU⟩ refine mem_nhdsWithin.2 ⟨U, Uo, aU, ?_⟩ rintro x ⟨hxU, hxs⟩ refine ContinuousWithinAt.closure_le ?_ ?_ continuousWithinAt_const hU · rwa [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_inter_of_mem, ← mem_closure_iff_nhdsWithin_neBot] exact nhdsWithin_le_nhds (Uo.mem_nhds hxU) · exact (hc _ hxs).mono (inter_subset_right.trans subset_closure) protected theorem IsLocalMinOn.closure (h : IsLocalMinOn f s a) (hc : ContinuousOn f (closure s)) : IsLocalMinOn f (closure s) a := IsLocalMaxOn.closure h.dual hc protected theorem IsLocalExtrOn.closure (h : IsLocalExtrOn f s a) (hc : ContinuousOn f (closure s)) : IsLocalExtrOn f (closure s) a := h.elim (fun h => Or.inl <\| h.closure hc) fun h => Or.inr <\| h.closure hc
.lake/packages/mathlib/Mathlib/Topology/Order/LiminfLimsup.lean	import Mathlib.Order.Filter.CountableInter import Mathlib.Order.LiminfLimsup import Mathlib.Topology.Order.Monotone /-! # Lemmas about liminf and limsup in an order topology. ## Main declarations * `BoundedLENhdsClass`: Typeclass stating that neighborhoods are eventually bounded above. * `BoundedGENhdsClass`: Typeclass stating that neighborhoods are eventually bounded below. ## Implementation notes The same lemmas are true in `ℝ`, `ℝ × ℝ`, `ι → ℝ`, `EuclideanSpace ι ℝ`. To avoid code duplication, we provide an ad hoc axiomatisation of the properties we need. -/ open Filter TopologicalSpace open scoped Topology universe u v variable {ι α β R S : Type} {π : ι → Type} /-- Ad hoc typeclass stating that neighborhoods are eventually bounded above. -/ class BoundedLENhdsClass (α : Type) [Preorder α] [TopologicalSpace α] : Prop where isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) /-- Ad hoc typeclass stating that neighborhoods are eventually bounded below. -/ class BoundedGENhdsClass (α : Type) [Preorder α] [TopologicalSpace α] : Prop where isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) section Preorder variable [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] section BoundedLENhdsClass variable [BoundedLENhdsClass α] [BoundedLENhdsClass β] {f : Filter ι} {u : ι → α} {a : α} theorem isBounded_le_nhds (a : α) : (𝓝 a).IsBounded (· ≤ ·) := BoundedLENhdsClass.isBounded_le_nhds _ theorem Filter.Tendsto.isBoundedUnder_le (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≤ ·) u := (isBounded_le_nhds a).mono h theorem Filter.Tendsto.bddAbove_range_of_cofinite [IsDirected α (· ≤ ·)] (h : Tendsto u cofinite (𝓝 a)) : BddAbove (Set.range u) := h.isBoundedUnder_le.bddAbove_range_of_cofinite theorem Filter.Tendsto.bddAbove_range [IsDirected α (· ≤ ·)] {u : ℕ → α} (h : Tendsto u atTop (𝓝 a)) : BddAbove (Set.range u) := h.isBoundedUnder_le.bddAbove_range theorem isCobounded_ge_nhds (a : α) : (𝓝 a).IsCobounded (· ≥ ·) := (isBounded_le_nhds a).isCobounded_flip theorem Filter.Tendsto.isCoboundedUnder_ge [NeBot f] (h : Tendsto u f (𝓝 a)) : f.IsCoboundedUnder (· ≥ ·) u := h.isBoundedUnder_le.isCobounded_flip instance : BoundedGENhdsClass αᵒᵈ := ⟨@isBounded_le_nhds α _ _ _⟩ instance Prod.instBoundedLENhdsClass : BoundedLENhdsClass (α × β) := by refine ⟨fun x ↦ ?_⟩ obtain ⟨a, ha⟩ := isBounded_le_nhds x.1 obtain ⟨b, hb⟩ := isBounded_le_nhds x.2 rw [← @Prod.mk.eta _ _ x, nhds_prod_eq] exact ⟨(a, b), ha.prod_mk hb⟩ instance Pi.instBoundedLENhdsClass [Finite ι] [∀ i, Preorder (π i)] [∀ i, TopologicalSpace (π i)] [∀ i, BoundedLENhdsClass (π i)] : BoundedLENhdsClass (∀ i, π i) := by refine ⟨fun x ↦ ?_⟩ rw [nhds_pi] choose f hf using fun i ↦ isBounded_le_nhds (x i) exact ⟨f, eventually_pi hf⟩ end BoundedLENhdsClass section BoundedGENhdsClass variable [BoundedGENhdsClass α] [BoundedGENhdsClass β] {f : Filter ι} {u : ι → α} {a : α} theorem isBounded_ge_nhds (a : α) : (𝓝 a).IsBounded (· ≥ ·) := BoundedGENhdsClass.isBounded_ge_nhds _ theorem Filter.Tendsto.isBoundedUnder_ge (h : Tendsto u f (𝓝 a)) : f.IsBoundedUnder (· ≥ ·) u := (isBounded_ge_nhds a).mono h theorem Filter.Tendsto.bddBelow_range_of_cofinite [IsDirected α (· ≥ ·)] (h : Tendsto u cofinite (𝓝 a)) : BddBelow (Set.range u) := h.isBoundedUnder_ge.bddBelow_range_of_cofinite theorem Filter.Tendsto.bddBelow_range [IsDirected α (· ≥ ·)] {u : ℕ → α} (h : Tendsto u atTop (𝓝 a)) : BddBelow (Set.range u) := h.isBoundedUnder_ge.bddBelow_range theorem isCobounded_le_nhds (a : α) : (𝓝 a).IsCobounded (· ≤ ·) := (isBounded_ge_nhds a).isCobounded_flip theorem Filter.Tendsto.isCoboundedUnder_le [NeBot f] (h : Tendsto u f (𝓝 a)) : f.IsCoboundedUnder (· ≤ ·) u := h.isBoundedUnder_ge.isCobounded_flip instance : BoundedLENhdsClass αᵒᵈ := ⟨@isBounded_ge_nhds α _ _ _⟩ instance Prod.instBoundedGENhdsClass : BoundedGENhdsClass (α × β) := ⟨(Prod.instBoundedLENhdsClass (α := αᵒᵈ) (β := βᵒᵈ)).isBounded_le_nhds⟩ instance Pi.instBoundedGENhdsClass [Finite ι] [∀ i, Preorder (π i)] [∀ i, TopologicalSpace (π i)] [∀ i, BoundedGENhdsClass (π i)] : BoundedGENhdsClass (∀ i, π i) := ⟨(Pi.instBoundedLENhdsClass (π := fun i ↦ (π i)ᵒᵈ)).isBounded_le_nhds⟩ end BoundedGENhdsClass -- See note [lower instance priority] instance (priority := 100) OrderTop.to_BoundedLENhdsClass [OrderTop α] : BoundedLENhdsClass α := ⟨fun _a ↦ isBounded_le_of_top⟩ -- See note [lower instance priority] instance (priority := 100) OrderBot.to_BoundedGENhdsClass [OrderBot α] : BoundedGENhdsClass α := ⟨fun _a ↦ isBounded_ge_of_bot⟩ end Preorder -- See note [lower instance priority] instance (priority := 100) BoundedLENhdsClass.of_closedIciTopology [LinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] : BoundedLENhdsClass α := ⟨fun a ↦ ((isTop_or_exists_gt a).elim fun h ↦ ⟨a, Eventually.of_forall h⟩) <\| Exists.imp fun _b ↦ eventually_le_nhds⟩ -- See note [lower instance priority] instance (priority := 100) BoundedGENhdsClass.of_closedIicTopology [LinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] : BoundedGENhdsClass α := inferInstanceAs <\| BoundedGENhdsClass αᵒᵈᵒᵈ section LiminfLimsup section ConditionallyCompleteLinearOrder variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] /-- If the liminf and the limsup of a filter coincide, then this filter converges to their common value, at least if the filter is eventually bounded above and below. -/ theorem le_nhds_of_limsSup_eq_limsInf {f : Filter α} {a : α} (hl : f.IsBounded (· ≤ ·)) (hg : f.IsBounded (· ≥ ·)) (hs : f.limsSup = a) (hi : f.limsInf = a) : f ≤ 𝓝 a := tendsto_order.2 ⟨fun _ hb ↦ gt_mem_sets_of_limsInf_gt hg <\| hi.symm ▸ hb, fun _ hb ↦ lt_mem_sets_of_limsSup_lt hl <\| hs.symm ▸ hb⟩ theorem limsSup_nhds (a : α) : limsSup (𝓝 a) = a := csInf_eq_of_forall_ge_of_forall_gt_exists_lt (isBounded_le_nhds a) (fun a' (h : { n : α \| n ≤ a' } ∈ 𝓝 a) ↦ show a ≤ a' from @mem_of_mem_nhds _ _ a _ h) fun b (hba : a < b) ↦ show ∃ c, { n : α \| n ≤ c } ∈ 𝓝 a ∧ c < b from match dense_or_discrete a b with \| Or.inl ⟨c, hac, hcb⟩ => ⟨c, ge_mem_nhds hac, hcb⟩ \| Or.inr ⟨_, h⟩ => ⟨a, (𝓝 a).sets_of_superset (gt_mem_nhds hba) h, hba⟩ theorem limsInf_nhds (a : α) : limsInf (𝓝 a) = a := limsSup_nhds (α := αᵒᵈ) a /-- If a filter is converging, its limsup coincides with its limit. -/ theorem limsInf_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsInf = a := have hb_ge : IsBounded (· ≥ ·) f := (isBounded_ge_nhds a).mono h have hb_le : IsBounded (· ≤ ·) f := (isBounded_le_nhds a).mono h le_antisymm (calc f.limsInf ≤ f.limsSup := limsInf_le_limsSup hb_le hb_ge _ ≤ (𝓝 a).limsSup := limsSup_le_limsSup_of_le h hb_ge.isCobounded_flip (isBounded_le_nhds a) _ = a := limsSup_nhds a) (calc a = (𝓝 a).limsInf := (limsInf_nhds a).symm _ ≤ f.limsInf := limsInf_le_limsInf_of_le h (isBounded_ge_nhds a) hb_le.isCobounded_flip) /-- If a filter is converging, its liminf coincides with its limit. -/ theorem limsSup_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsSup = a := limsInf_eq_of_le_nhds (α := αᵒᵈ) h /-- If a function has a limit, then its limsup coincides with its limit. -/ theorem Filter.Tendsto.limsup_eq {f : Filter β} {u : β → α} {a : α} [NeBot f] (h : Tendsto u f (𝓝 a)) : limsup u f = a := limsSup_eq_of_le_nhds h /-- If a function has a limit, then its liminf coincides with its limit. -/ theorem Filter.Tendsto.liminf_eq {f : Filter β} {u : β → α} {a : α} [NeBot f] (h : Tendsto u f (𝓝 a)) : liminf u f = a := limsInf_eq_of_le_nhds h /-- If the liminf and the limsup of a function coincide, then the limit of the function exists and has the same value. -/ theorem tendsto_of_liminf_eq_limsup {f : Filter β} {u : β → α} {a : α} (hinf : liminf u f = a) (hsup : limsup u f = a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) := le_nhds_of_limsSup_eq_limsInf h h' hsup hinf /-- If a number `a` is less than or equal to the `liminf` of a function `f` at some filter and is greater than or equal to the `limsup` of `f`, then `f` tends to `a` along this filter. -/ theorem tendsto_of_le_liminf_of_limsup_le {f : Filter β} {u : β → α} {a : α} (hinf : a ≤ liminf u f) (hsup : limsup u f ≤ a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) := by rcases f.eq_or_neBot with rfl \| _ · exact tendsto_bot · exact tendsto_of_liminf_eq_limsup (le_antisymm (le_trans (liminf_le_limsup h h') hsup) hinf) (le_antisymm hsup (le_trans hinf (liminf_le_limsup h h'))) h h' /-- Assume that, for any `a < b`, a sequence cannot be infinitely many times below `a` and above `b`. If it is also ultimately bounded above and below, then it has to converge. This even works if `a` and `b` are restricted to a dense subset. -/ theorem tendsto_of_no_upcrossings [DenselyOrdered α] {f : Filter β} {u : β → α} {s : Set α} (hs : Dense s) (H : ∀ a ∈ s, ∀ b ∈ s, a < b → ¬((∃ᶠ n in f, u n < a) ∧ ∃ᶠ n in f, b < u n)) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : ∃ c : α, Tendsto u f (𝓝 c) := by rcases f.eq_or_neBot with rfl \| hbot · exact ⟨sInf ∅, tendsto_bot⟩ refine ⟨limsup u f, ?_⟩ apply tendsto_of_le_liminf_of_limsup_le _ le_rfl h h' by_contra! hlt obtain ⟨a, ⟨⟨la, au⟩, as⟩⟩ : ∃ a, (f.liminf u < a ∧ a < f.limsup u) ∧ a ∈ s := dense_iff_inter_open.1 hs (Set.Ioo (f.liminf u) (f.limsup u)) isOpen_Ioo (Set.nonempty_Ioo.2 hlt) obtain ⟨b, ⟨⟨ab, bu⟩, bs⟩⟩ : ∃ b, (a < b ∧ b < f.limsup u) ∧ b ∈ s := dense_iff_inter_open.1 hs (Set.Ioo a (f.limsup u)) isOpen_Ioo (Set.nonempty_Ioo.2 au) have A : ∃ᶠ n in f, u n < a := frequently_lt_of_liminf_lt (IsBounded.isCobounded_ge h) la have B : ∃ᶠ n in f, b < u n := frequently_lt_of_lt_limsup (IsBounded.isCobounded_le h') bu exact H a as b bs ab ⟨A, B⟩ variable [FirstCountableTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α} theorem eventually_le_limsup (hf : IsBoundedUnder (· ≤ ·) f u := by isBoundedDefault) : ∀ᶠ b in f, u b ≤ f.limsup u := by obtain ha \| ha := isTop_or_exists_gt (f.limsup u) · exact Eventually.of_forall fun _ => ha _ by_cases H : IsGLB (Set.Ioi (f.limsup u)) (f.limsup u) · obtain ⟨u, -, -, hua, hu⟩ := H.exists_seq_antitone_tendsto ha have := fun n => eventually_lt_of_limsup_lt (hu n) hf exact (eventually_countable_forall.2 this).mono fun b hb => ge_of_tendsto hua <\| Eventually.of_forall fun n => (hb _).le · obtain ⟨x, hx, xa⟩ : ∃ x, (∀ ⦃b⦄, f.limsup u < b → x ≤ b) ∧ f.limsup u < x := by simp only [IsGLB, IsGreatest, lowerBounds, upperBounds, Set.mem_Ioi, Set.mem_setOf_eq, not_and, not_forall, not_le, exists_prop] at H exact H fun x => le_of_lt filter_upwards [eventually_lt_of_limsup_lt xa hf] with y hy contrapose! hy exact hx hy theorem eventually_liminf_le (hf : IsBoundedUnder (· ≥ ·) f u := by isBoundedDefault) : ∀ᶠ b in f, f.liminf u ≤ u b := eventually_le_limsup (α := αᵒᵈ) hf end ConditionallyCompleteLinearOrder section CompleteLinearOrder variable [CompleteLinearOrder α] [TopologicalSpace α] [FirstCountableTopology α] [OrderTopology α] {f : Filter β} [CountableInterFilter f] {u : β → α} @[simp] theorem limsup_eq_bot : f.limsup u = ⊥ ↔ u =ᶠ[f] ⊥ := ⟨fun h => (EventuallyLE.trans eventually_le_limsup <\| Eventually.of_forall fun _ => h.le).mono fun _ hx => le_antisymm hx bot_le, fun h => by rw [limsup_congr h] exact limsup_const_bot⟩ @[simp] theorem liminf_eq_top : f.liminf u = ⊤ ↔ u =ᶠ[f] ⊤ := limsup_eq_bot (α := αᵒᵈ) end CompleteLinearOrder end LiminfLimsup section Monotone variable {F : Filter ι} [NeBot F] [ConditionallyCompleteLinearOrder R] [TopologicalSpace R] [OrderTopology R] [ConditionallyCompleteLinearOrder S] [TopologicalSpace S] [OrderTopology S] /-- An antitone function between (conditionally) complete linear ordered spaces sends a `Filter.limsSup` to the `Filter.liminf` of the image if the function is continuous at the `limsSup` (and the filter is bounded from above and frequently bounded from below). -/ theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S} (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup) (bdd_above : F.IsBounded (· ≤ ·) := by isBoundedDefault) (cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) : f F.limsSup = F.liminf f := by apply le_antisymm · rw [limsSup, f_decr.map_csInf_of_continuousAt f_cont bdd_above cobdd] apply le_of_forall_lt intro c hc simp only [liminf, limsInf, eventually_map] at hc ⊢ obtain ⟨d, hd, h'd⟩ := exists_lt_of_lt_csSup (bdd_above.recOn fun x hx ↦ ⟨f x, Set.mem_image_of_mem f hx⟩) hc apply lt_csSup_of_lt ?_ ?_ h'd · simpa only [BddAbove, upperBounds] using Antitone.isCoboundedUnder_ge_of_isCobounded f_decr cobdd · rcases hd with ⟨e, ⟨he, fe_eq_d⟩⟩ filter_upwards [he] with x hx using (fe_eq_d.symm ▸ f_decr hx) · by_cases! h' : ∃ c, c < F.limsSup ∧ Set.Ioo c F.limsSup = ∅ · rcases h' with ⟨c, c_lt, hc⟩ have B : ∃ᶠ n in F, F.limsSup ≤ n := by apply (frequently_lt_of_lt_limsSup cobdd c_lt).mono intro x hx by_contra! have : (Set.Ioo c F.limsSup).Nonempty := ⟨x, ⟨hx, this⟩⟩ simp only [hc, Set.not_nonempty_empty] at this apply liminf_le_of_frequently_le _ (bdd_above.isBoundedUnder f_decr) exact B.mono fun x hx ↦ f_decr hx by_contra! H have not_bot : ¬ IsBot F.limsSup := fun maybe_bot ↦ lt_irrefl (F.liminf f) <\| lt_of_le_of_lt (liminf_le_of_frequently_le (Frequently.of_forall (fun r ↦ f_decr (maybe_bot r))) (bdd_above.isBoundedUnder f_decr)) H obtain ⟨l, l_lt, h'l⟩ : ∃ l < F.limsSup, Set.Ioc l F.limsSup ⊆ { x : R \| f x < F.liminf f } := by apply exists_Ioc_subset_of_mem_nhds ((tendsto_order.1 f_cont.tendsto).2 _ H) simpa [IsBot] using not_bot obtain ⟨m, l_m, m_lt⟩ : (Set.Ioo l F.limsSup).Nonempty := by contrapose! h' exact ⟨l, l_lt, h'⟩ have B : F.liminf f ≤ f m := by apply liminf_le_of_frequently_le _ _ · apply (frequently_lt_of_lt_limsSup cobdd m_lt).mono exact fun x hx ↦ f_decr hx.le · exact IsBounded.isBoundedUnder f_decr bdd_above have I : f m < F.liminf f := h'l ⟨l_m, m_lt.le⟩ exact lt_irrefl _ (B.trans_lt I) /-- A continuous antitone function between (conditionally) complete linear ordered spaces sends a `Filter.limsup` to the `Filter.liminf` of the images (if the filter is bounded from above and frequently bounded from below). -/ theorem Antitone.map_limsup_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R) (f_cont : ContinuousAt f (F.limsup a)) (bdd_above : F.IsBoundedUnder (· ≤ ·) a := by isBoundedDefault) (cobdd : F.IsCoboundedUnder (· ≤ ·) a := by isBoundedDefault) : f (F.limsup a) = F.liminf (f ∘ a) := f_decr.map_limsSup_of_continuousAt f_cont bdd_above cobdd /-- An antitone function between (conditionally) complete linear ordered spaces sends a `Filter.limsInf` to the `Filter.limsup` of the image if the function is continuous at the `limsInf` (and the filter is bounded from below and frequently bounded from above). -/ theorem Antitone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R → S} (f_decr : Antitone f) (f_cont : ContinuousAt f F.limsInf) (cobdd : F.IsCobounded (· ≥ ·) := by isBoundedDefault) (bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) : f F.limsInf = F.limsup f := Antitone.map_limsSup_of_continuousAt (R := Rᵒᵈ) (S := Sᵒᵈ) f_decr.dual f_cont bdd_below cobdd /-- A continuous antitone function between (conditionally) complete linear ordered spaces sends a `Filter.liminf` to the `Filter.limsup` of the images (if the filter is bounded from below and frequently bounded from above). -/ theorem Antitone.map_liminf_of_continuousAt {f : R → S} (f_decr : Antitone f) (a : ι → R) (f_cont : ContinuousAt f (F.liminf a)) (cobdd : F.IsCoboundedUnder (· ≥ ·) a := by isBoundedDefault) (bdd_below : F.IsBoundedUnder (· ≥ ·) a := by isBoundedDefault) : f (F.liminf a) = F.limsup (f ∘ a) := f_decr.map_limsInf_of_continuousAt f_cont cobdd bdd_below /-- A monotone function between (conditionally) complete linear ordered spaces sends a `Filter.limsSup` to the `Filter.limsup` of the image if the function is continuous at the `limsSup` (and the filter is bounded from above and frequently bounded from below). -/ theorem Monotone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S} (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsSup) (bdd_above : F.IsBounded (· ≤ ·) := by isBoundedDefault) (cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) : f F.limsSup = F.limsup f := Antitone.map_limsSup_of_continuousAt (S := Sᵒᵈ) f_incr f_cont bdd_above cobdd /-- A continuous monotone function between (conditionally) complete linear ordered spaces sends a `Filter.limsup` to the `Filter.limsup` of the images (if the filter is bounded from above and frequently bounded from below). -/ theorem Monotone.map_limsup_of_continuousAt {f : R → S} (f_incr : Monotone f) (a : ι → R) (f_cont : ContinuousAt f (F.limsup a)) (bdd_above : F.IsBoundedUnder (· ≤ ·) a := by isBoundedDefault) (cobdd : F.IsCoboundedUnder (· ≤ ·) a := by isBoundedDefault) : f (F.limsup a) = F.limsup (f ∘ a) := f_incr.map_limsSup_of_continuousAt f_cont bdd_above cobdd /-- A monotone function between (conditionally) complete linear ordered spaces sends a `Filter.limsInf` to the `Filter.liminf` of the image if the function is continuous at the `limsInf` (and the filter is bounded from below and frequently bounded from above). -/ theorem Monotone.map_limsInf_of_continuousAt {F : Filter R} [NeBot F] {f : R → S} (f_incr : Monotone f) (f_cont : ContinuousAt f F.limsInf) (cobdd : F.IsCobounded (· ≥ ·) := by isBoundedDefault) (bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) : f F.limsInf = F.liminf f := Antitone.map_limsSup_of_continuousAt (R := Rᵒᵈ) f_incr.dual f_cont bdd_below cobdd /-- A continuous monotone function between (conditionally) complete linear ordered spaces sends a `Filter.liminf` to the `Filter.liminf` of the images (if the filter is bounded from below and frequently bounded from above). -/ theorem Monotone.map_liminf_of_continuousAt {f : R → S} (f_incr : Monotone f) (a : ι → R) (f_cont : ContinuousAt f (F.liminf a)) (cobdd : F.IsCoboundedUnder (· ≥ ·) a := by isBoundedDefault) (bdd_below : F.IsBoundedUnder (· ≥ ·) a := by isBoundedDefault) : f (F.liminf a) = F.liminf (f ∘ a) := f_incr.map_limsInf_of_continuousAt f_cont cobdd bdd_below end Monotone
.lake/packages/mathlib/Mathlib/Topology/Order/IntermediateValue.lean	import Mathlib.Order.Interval.Set.Image import Mathlib.Order.CompleteLatticeIntervals import Mathlib.Topology.Order.DenselyOrdered import Mathlib.Topology.Order.Monotone import Mathlib.Topology.Connected.TotallyDisconnected /-! # Intermediate Value Theorem In this file we prove the Intermediate Value Theorem: if `f : α → β` is a function defined on a connected set `s` that takes both values `≤ a` and values `≥ a` on `s`, then it is equal to `a` at some point of `s`. We also prove that intervals in a dense conditionally complete order are preconnected and any preconnected set is an interval. Then we specialize IVT to functions continuous on intervals. ## Main results * `IsPreconnected_I??` : all intervals `I??` are preconnected, * `IsPreconnected.intermediate_value`, `intermediate_value_univ` : Intermediate Value Theorem for connected sets and connected spaces, respectively; * `intermediate_value_Icc`, `intermediate_value_Icc'`: Intermediate Value Theorem for functions on closed intervals. ### Miscellaneous facts * `IsClosed.Icc_subset_of_forall_mem_nhdsWithin` : “Continuous induction” principle; if `s ∩ [a, b]` is closed, `a ∈ s`, and for each `x ∈ [a, b) ∩ s` some of its right neighborhoods is included `s`, then `[a, b] ⊆ s`. * `IsClosed.Icc_subset_of_forall_exists_gt`, `IsClosed.mem_of_ge_of_forall_exists_gt` : two other versions of the “continuous induction” principle. * `ContinuousOn.StrictMonoOn_of_InjOn_Ioo` : Every continuous injective `f : (a, b) → δ` is strictly monotone or antitone (increasing or decreasing). ## Tags intermediate value theorem, connected space, connected set -/ open Filter OrderDual TopologicalSpace Function Set open scoped Topology Filter Interval universe u v /-! ### Intermediate value theorem on a (pre)connected space In this section we prove the following theorem (see `IsPreconnected.intermediate_value₂`): if `f` and `g` are two functions continuous on a preconnected set `s`, `f a ≤ g a` at some `a ∈ s` and `g b ≤ f b` at some `b ∈ s`, then `f c = g c` at some `c ∈ s`. We prove several versions of this statement, including the classical IVT that corresponds to a constant function `g`. -/ section variable {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] /-- Intermediate value theorem for two functions: if `f` and `g` are two continuous functions on a preconnected space and `f a ≤ g a` and `g b ≤ f b`, then for some `x` we have `f x = g x`. -/ theorem intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x := by obtain ⟨x, _, hfg, hgf⟩ : (univ ∩ { x \| f x ≤ g x ∧ g x ≤ f x }).Nonempty := isPreconnected_closed_iff.1 PreconnectedSpace.isPreconnected_univ _ _ (isClosed_le hf hg) (isClosed_le hg hf) (fun _ _ => le_total _ _) ⟨a, trivial, ha⟩ ⟨b, trivial, hb⟩ exact ⟨x, le_antisymm hfg hgf⟩ theorem intermediate_value_univ₂_eventually₁ [PreconnectedSpace X] {a : X} {l : Filter X} [NeBot l] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x, f x = g x := let ⟨_, h⟩ := he.exists; intermediate_value_univ₂ hf hg ha h theorem intermediate_value_univ₂_eventually₂ [PreconnectedSpace X] {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) : ∃ x, f x = g x := let ⟨_, h₁⟩ := he₁.exists let ⟨_, h₂⟩ := he₂.exists intermediate_value_univ₂ hf hg h₁ h₂ /-- Intermediate value theorem for two functions: if `f` and `g` are two functions continuous on a preconnected set `s` and for some `a b ∈ s` we have `f a ≤ g a` and `g b ≤ f b`, then for some `x ∈ s` we have `f x = g x`. -/ theorem IsPreconnected.intermediate_value₂ {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a) (hb' : g b ≤ f b) : ∃ x ∈ s, f x = g x := let ⟨x, hx⟩ := @intermediate_value_univ₂ s α _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ ⟨b, hb⟩ _ _ (continuousOn_iff_continuous_restrict.1 hf) (continuousOn_iff_continuous_restrict.1 hg) ha' hb' ⟨x, x.2, hx⟩ theorem IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x := by rw [continuousOn_iff_continuous_restrict] at hf hg obtain ⟨b, h⟩ := @intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _ (comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _) exact ⟨b, b.prop, h⟩ theorem IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) : ∃ x ∈ s, f x = g x := by rw [continuousOn_iff_continuous_restrict] at hf hg obtain ⟨b, h⟩ := @intermediate_value_univ₂_eventually₂ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) _ _ (comap_coe_neBot_of_le_principal hl₁) (comap_coe_neBot_of_le_principal hl₂) _ _ hf hg (he₁.comap _) (he₂.comap _) exact ⟨b, b.prop, h⟩ /-- Intermediate Value Theorem for continuous functions on connected sets. -/ theorem IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s) (hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s) : Icc (f a) (f b) ⊆ f '' s := fun _x hx => hs.intermediate_value₂ ha hb hf continuousOn_const hx.1 hx.2 theorem IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s := fun _ h => hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1 (ht.eventually_const_le h.2) theorem IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s := fun _ h => (hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h.2 (ht.eventually_le_const h.1)).imp fun _ h => h.imp_right Eq.symm theorem IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) : Ioo v₁ v₂ ⊆ f '' s := fun _ h => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_const h.1) (ht₂.eventually_const_le h.2) theorem IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht : Tendsto f l atTop) : Ici (f a) ⊆ f '' s := fun y h => hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h (tendsto_atTop.1 ht y) theorem IsPreconnected.intermediate_value_Iic {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht : Tendsto f l atBot) : Iic (f a) ⊆ f '' s := fun y h => (hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h (tendsto_atBot.1 ht y)).imp fun _ h => h.imp_right Eq.symm theorem IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s := fun y h => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_const h) (ht₂.eventually_ge_atTop y) theorem IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s := fun y h => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y) (ht₂.eventually_const_le h) theorem IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s := fun y _ => hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y) (ht₂.eventually_ge_atTop y) /-- Intermediate Value Theorem for continuous functions on connected spaces. -/ theorem intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) : Icc (f a) (f b) ⊆ range f := fun _ hx => intermediate_value_univ₂ hf continuous_const hx.1 hx.2 /-- Intermediate Value Theorem for continuous functions on connected spaces. -/ theorem mem_range_of_exists_le_of_exists_ge [PreconnectedSpace X] {c : α} {f : X → α} (hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) : c ∈ range f := let ⟨a, ha⟩ := h₁; let ⟨b, hb⟩ := h₂; intermediate_value_univ a b hf ⟨ha, hb⟩ /-! ### (Pre)connected sets in a linear order In this section we prove the following results: * `IsPreconnected.ordConnected`: any preconnected set `s` in a linear order is `OrdConnected`, i.e. `a ∈ s` and `b ∈ s` imply `Icc a b ⊆ s`; * `IsPreconnected.mem_intervals`: any preconnected set `s` in a conditionally complete linear order is one of the intervals `Set.Icc`, `set.`Ico`, `set.Ioc`, `set.Ioo`, ``Set.Ici`, `Set.Iic`, `Set.Ioi`, `Set.Iio`; note that this is false for non-complete orders: e.g., in `ℝ \ {0}`, the set of positive numbers cannot be represented as `Set.Ioi _`. -/ /-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/ theorem IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : Icc a b ⊆ s := by simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id theorem IsPreconnected.ordConnected {s : Set α} (h : IsPreconnected s) : OrdConnected s := ⟨fun _ hx _ hy => h.Icc_subset hx hy⟩ /-- If a preconnected set contains endpoints of an interval, then it includes the whole interval. -/ theorem IsConnected.Icc_subset {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : Icc a b ⊆ s := hs.2.Icc_subset ha hb /-- If preconnected set in a linear order space is unbounded below and above, then it is the whole space. -/ theorem IsPreconnected.eq_univ_of_unbounded {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s) (ha : ¬BddAbove s) : s = univ := by refine eq_univ_of_forall fun x => ?_ obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, y < x := not_bddBelow_iff.1 hb x obtain ⟨z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x exact hs.Icc_subset ys zs ⟨le_of_lt hy, le_of_lt hz⟩ end variable {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] /-- A bounded connected subset of a conditionally complete linear order includes the open interval `(Inf s, Sup s)`. -/ theorem IsConnected.Ioo_csInf_csSup_subset {s : Set α} (hs : IsConnected s) (hb : BddBelow s) (ha : BddAbove s) : Ioo (sInf s) (sSup s) ⊆ s := fun _x hx => let ⟨_y, ys, hy⟩ := (isGLB_lt_iff (isGLB_csInf hs.nonempty hb)).1 hx.1 let ⟨_z, zs, hz⟩ := (lt_isLUB_iff (isLUB_csSup hs.nonempty ha)).1 hx.2 hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ theorem eq_Icc_csInf_csSup_of_connected_bdd_closed {s : Set α} (hc : IsConnected s) (hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s) : s = Icc (sInf s) (sSup s) := (subset_Icc_csInf_csSup hb ha).antisymm <\| hc.Icc_subset (hcl.csInf_mem hc.nonempty hb) (hcl.csSup_mem hc.nonempty ha) theorem IsPreconnected.Ioi_csInf_subset {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s) (ha : ¬BddAbove s) : Ioi (sInf s) ⊆ s := fun x hx => have sne : s.Nonempty := nonempty_of_not_bddAbove ha let ⟨_y, ys, hy⟩ : ∃ y ∈ s, y < x := (isGLB_lt_iff (isGLB_csInf sne hb)).1 hx let ⟨_z, zs, hz⟩ : ∃ z ∈ s, x < z := not_bddAbove_iff.1 ha x hs.Icc_subset ys zs ⟨hy.le, hz.le⟩ theorem IsPreconnected.Iio_csSup_subset {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s) (ha : BddAbove s) : Iio (sSup s) ⊆ s := IsPreconnected.Ioi_csInf_subset (α := αᵒᵈ) hs ha hb /-- A preconnected set in a conditionally complete linear order is either one of the intervals `[Inf s, Sup s]`, `[Inf s, Sup s)`, `(Inf s, Sup s]`, `(Inf s, Sup s)`, `[Inf s, +∞)`, `(Inf s, +∞)`, `(-∞, Sup s]`, `(-∞, Sup s)`, `(-∞, +∞)`, or `∅`. The converse statement requires `α` to be densely ordered. -/ theorem IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) : s ∈ ({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s), Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) := by rcases s.eq_empty_or_nonempty with (rfl \| hne) · apply_rules [Or.inr, mem_singleton] have hs' : IsConnected s := ⟨hne, hs⟩ by_cases hb : BddBelow s <;> by_cases ha : BddAbove s · refine mem_of_subset_of_mem ?_ <\| mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset (hs'.Ioo_csInf_csSup_subset hb ha) (subset_Icc_csInf_csSup hb ha) simp only [insert_subset_iff, mem_insert_iff, mem_singleton_iff, true_or, or_true, singleton_subset_iff, and_self] · refine Or.inr <\| Or.inr <\| Or.inr <\| Or.inr ?_ rcases mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_csInf_subset hb ha) fun x hx ↦ csInf_le hb hx with hs \| hs · exact Or.inl hs · exact Or.inr (Or.inl hs) · iterate 6 apply Or.inr rcases mem_Iic_Iio_of_subset_of_subset (hs.Iio_csSup_subset hb ha) fun x hx ↦ le_csSup ha hx with hs \| hs · exact Or.inl hs · exact Or.inr (Or.inl hs) · iterate 8 apply Or.inr exact Or.inl (hs.eq_univ_of_unbounded hb ha) /-- A preconnected set is either one of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, or `univ`, or `∅`. The converse statement requires `α` to be densely ordered. Though one can represent `∅` as `(Inf ∅, Inf ∅)`, we include it into the list of possible cases to improve readability. -/ theorem setOf_isPreconnected_subset_of_ordered : { s : Set α \| IsPreconnected s } ⊆ -- bounded intervals (range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪ -- unbounded intervals and `univ` (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by intro s hs rcases hs.mem_intervals with (hs \| hs \| hs \| hs \| hs \| hs \| hs \| hs \| hs \| hs) <;> rw [hs] <;> simp only [union_insert, union_singleton, mem_insert_iff, mem_union, mem_range, Prod.exists, uncurry_apply_pair, exists_apply_eq_apply, true_or, or_true, exists_apply_eq_apply2] /-! ### Intervals are connected In this section we prove that a closed interval (hence, any `OrdConnected` set) in a dense conditionally complete linear order is preconnected. -/ /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and the set `s ∩ [a, b)` has no maximal point, then `b ∈ s`. -/ theorem IsClosed.mem_of_ge_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty) : b ∈ s := by let S := s ∩ Icc a b replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩ have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩ let c := sSup (s ∩ Icc a b) have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2 rcases eq_or_lt_of_le c_le with hc \| hc · exact hc ▸ c_mem.1 exfalso rcases hgt c ⟨c_mem.1, c_mem.2.1, hc⟩ with ⟨x, xs, cx, xb⟩ exact not_lt_of_ge (le_csSup Sbd ⟨xs, le_trans (le_csSup Sbd ha) (le_of_lt cx), xb⟩) cx /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and for any `a ≤ x < y ≤ b`, `x ∈ s`, the set `s ∩ (x, y]` is not empty, then `[a, b] ⊆ s`. -/ theorem IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s := by intro y hy have : IsClosed (s ∩ Icc a y) := by suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y from this ▸ hs.inter isClosed_Icc grind [inter_assoc, inter_eq_self_of_subset_right, Icc_subset_Icc_right] exact IsClosed.mem_of_ge_of_forall_exists_gt this ha hy.1 fun x hx ↦ hgt x ⟨hx.1, Ico_subset_Ico_right hy.2 hx.2⟩ y hx.2.2 /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `b`, and the set `s ∩ (a, b]` has no minimal point, then `a ∈ s`. -/ theorem IsClosed.mem_of_ge_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (hb : b ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty) : a ∈ s := by suffices OrderDual.toDual a ∈ ofDual ⁻¹' s by aesop have : IsClosed (OrderDual.ofDual ⁻¹' (s ∩ Icc a b)) := hs rw [preimage_inter, ← Icc_toDual] at this apply this.mem_of_ge_of_forall_exists_gt (by aesop) (by aesop) (fun x hx ↦ ?_) rw [Ico_toDual, ← preimage_inter, ← Equiv.image_symm_eq_preimage, mem_image] at hx aesop /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `b`, and for any `a ≤ y < x ≤ b`, `x ∈ s`, the set `s ∩ [y, x)` is not empty, then `[a, b] ⊆ s`. -/ theorem IsClosed.Icc_subset_of_forall_exists_lt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (hb : b ∈ s) (hgt : ∀ x ∈ s ∩ Ioc a b, ∀ y ∈ Iio x, (s ∩ Ico y x).Nonempty) : Icc a b ⊆ s := by intro y hy have : IsClosed (s ∩ Icc y b) := by suffices s ∩ Icc y b = s ∩ Icc a b ∩ Icc y b from this ▸ hs.inter isClosed_Icc grind [Icc_subset_Icc_left, inter_eq_self_of_subset_right, inter_assoc] exact IsClosed.mem_of_ge_of_forall_exists_lt this hb hy.2 fun x hx ↦ hgt x ⟨hx.1, Ioc_subset_Ioc_left hy.1 hx.2⟩ y hx.2.1 variable [DenselyOrdered α] {a b : α} /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and for any `x ∈ [a, b)` such that `[a, x]` is included in `s`, the set `s` includes some open neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. -/ lemma IsClosed.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t) : Icc a b ⊆ s := by rcases lt_or_ge b a with hab \| hab · simp_all set A := {t ∈ Icc a b \| Icc a t ⊆ s} have a_mem : a ∈ A := ⟨left_mem_Icc.mpr hab, by simp [ha]⟩ have bdd_A : BddAbove A := ⟨b, fun t ht ↦ ht.1.2⟩ set t₁ := sSup A have t₁_mem : t₁ ∈ Icc a b := ⟨le_csSup bdd_A a_mem, csSup_le ⟨a, a_mem⟩ (fun t ht ↦ ht.1.2)⟩ obtain ⟨⟨t₁a, t₁b⟩, ht₁⟩ : t₁ ∈ A := by refine ⟨t₁_mem, fun t ht ↦ ?_⟩ rcases ht.2.eq_or_lt with rfl \| h · have : closure A ⊆ s ∩ Icc a b := by apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩) have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩ exact ht.2 this apply this.trans inter_subset_left exact csSup_mem_closure ⟨a, a_mem⟩ bdd_A · obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h apply cA.2 exact ⟨ht.1, tc.le⟩ suffices t₁ = b by simpa [this] using ht₁ apply eq_of_le_of_not_lt t₁b fun t₁b' ↦ ?_ obtain ⟨m, t₁m, H⟩ : ∃ m > t₁, Ioo t₁ m ⊆ s := (mem_nhdsGT_iff_exists_Ioo_subset' t₁b').mp (h t₁ ⟨t₁a, t₁b'⟩ (fun s hs ↦ ht₁ hs)) obtain ⟨t, hat, ht⟩ : ∃ t, t₁ < t ∧ t < min m b := exists_between (lt_min t₁m t₁b') have : t ∈ A := by refine ⟨⟨by order, ht.le.trans (min_le_right _ _)⟩, fun t' ht' ↦ ?_⟩ rcases le_or_gt t' t₁ with h't' \| h't' · exact ht₁ ⟨ht'.1, h't'⟩ · exact H ⟨h't', ht'.2.trans_lt <\| ht.trans_le <\| min_le_left ..⟩ have : t ≤ t₁ := le_csSup bdd_A this order /-- A "continuous induction principle" for a closed interval: if a set `s` meets `[a, b]` on a closed subset, contains `a`, and for any `x ∈ s ∩ [a, b)` the set `s` includes some open neighborhood of `x` within `(x, +∞)`, then `[a, b] ⊆ s`. -/ theorem IsClosed.Icc_subset_of_forall_mem_nhdsWithin {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) : Icc a b ⊆ s := hs.Icc_subset_of_forall_mem_nhdsGT_of_Icc_subset ha (fun _t ht h't ↦ hgt _ ⟨h't ⟨ht.1, le_rfl⟩, ht⟩) theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s) (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) : (Icc a b ∩ (s ∩ t)).Nonempty := by have xyab : Icc x y ⊆ Icc a b := Icc_subset_Icc hx.1.1 hy.1.2 by_contra hst suffices Icc x y ⊆ s from hst ⟨y, xyab <\| right_mem_Icc.2 hxy, this <\| right_mem_Icc.2 hxy, hy.2⟩ apply (IsClosed.inter hs isClosed_Icc).Icc_subset_of_forall_mem_nhdsWithin hx.2 rintro z ⟨zs, hz⟩ have zt : z ∈ tᶜ := fun zt => hst ⟨z, xyab <\| Ico_subset_Icc_self hz, zs, zt⟩ have : tᶜ ∩ Ioc z y ∈ 𝓝[>] z := by rw [← nhdsWithin_Ioc_eq_nhdsGT hz.2] exact mem_nhdsWithin.2 ⟨tᶜ, ht.isOpen_compl, zt, Subset.rfl⟩ apply mem_of_superset this have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩) exact fun w ⟨wt, wzy⟩ => (this wzy).elim id fun h => (wt h).elim /-- A closed interval in a densely ordered conditionally complete linear order is preconnected. -/ theorem isPreconnected_Icc : IsPreconnected (Icc a b) := isPreconnected_closed_iff.2 (by rintro s t hs ht hab ⟨x, hx⟩ ⟨y, hy⟩ -- This used to use `wlog`, but it was causing timeouts. rcases le_total x y with h \| h · exact isPreconnected_Icc_aux x y s t h hs ht hab hx hy · rw [inter_comm s t] rw [union_comm s t] at hab exact isPreconnected_Icc_aux y x t s h ht hs hab hy hx) theorem isPreconnected_uIcc : IsPreconnected ([[a, b]]) := isPreconnected_Icc theorem Set.OrdConnected.isPreconnected {s : Set α} (h : s.OrdConnected) : IsPreconnected s := isPreconnected_of_forall_pair fun x hx y hy => ⟨[[x, y]], h.uIcc_subset hx hy, left_mem_uIcc, right_mem_uIcc, isPreconnected_uIcc⟩ theorem isPreconnected_iff_ordConnected {s : Set α} : IsPreconnected s ↔ OrdConnected s := ⟨IsPreconnected.ordConnected, Set.OrdConnected.isPreconnected⟩ theorem isPreconnected_Ici : IsPreconnected (Ici a) := ordConnected_Ici.isPreconnected theorem isPreconnected_Iic : IsPreconnected (Iic a) := ordConnected_Iic.isPreconnected theorem isPreconnected_Iio : IsPreconnected (Iio a) := ordConnected_Iio.isPreconnected theorem isPreconnected_Ioi : IsPreconnected (Ioi a) := ordConnected_Ioi.isPreconnected theorem isPreconnected_Ioo : IsPreconnected (Ioo a b) := ordConnected_Ioo.isPreconnected theorem isPreconnected_Ioc : IsPreconnected (Ioc a b) := ordConnected_Ioc.isPreconnected theorem isPreconnected_Ico : IsPreconnected (Ico a b) := ordConnected_Ico.isPreconnected theorem isConnected_Ici : IsConnected (Ici a) := ⟨nonempty_Ici, isPreconnected_Ici⟩ theorem isConnected_Iic : IsConnected (Iic a) := ⟨nonempty_Iic, isPreconnected_Iic⟩ theorem isConnected_Ioi [NoMaxOrder α] : IsConnected (Ioi a) := ⟨nonempty_Ioi, isPreconnected_Ioi⟩ theorem isConnected_Iio [NoMinOrder α] : IsConnected (Iio a) := ⟨nonempty_Iio, isPreconnected_Iio⟩ theorem isConnected_Icc (h : a ≤ b) : IsConnected (Icc a b) := ⟨nonempty_Icc.2 h, isPreconnected_Icc⟩ theorem isConnected_Ioo (h : a < b) : IsConnected (Ioo a b) := ⟨nonempty_Ioo.2 h, isPreconnected_Ioo⟩ theorem isConnected_Ioc (h : a < b) : IsConnected (Ioc a b) := ⟨nonempty_Ioc.2 h, isPreconnected_Ioc⟩ theorem isConnected_Ico (h : a < b) : IsConnected (Ico a b) := ⟨nonempty_Ico.2 h, isPreconnected_Ico⟩ instance (priority := 100) ordered_connected_space : PreconnectedSpace α := ⟨ordConnected_univ.isPreconnected⟩ /-- In a dense conditionally complete linear order, the set of preconnected sets is exactly the set of the intervals `Icc`, `Ico`, `Ioc`, `Ioo`, `Ici`, `Ioi`, `Iic`, `Iio`, `(-∞, +∞)`, or `∅`. Though one can represent `∅` as `(sInf s, sInf s)`, we include it into the list of possible cases to improve readability. -/ theorem setOf_isPreconnected_eq_of_ordered : { s : Set α \| IsPreconnected s } = -- bounded intervals range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪ -- unbounded intervals and `univ` (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by refine Subset.antisymm setOf_isPreconnected_subset_of_ordered ?_ simp only [subset_def, forall_mem_range, uncurry, or_imp, forall_and, mem_union, mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true, isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Ioo, isPreconnected_Ioi, isPreconnected_Iio, isPreconnected_Ici, isPreconnected_Iic, isPreconnected_univ, isPreconnected_empty] /-- This lemmas characterizes when a subset `s` of a densely ordered conditionally complete linear order is totally disconnected with respect to the order topology: between any two distinct points of `s` must lie a point not in `s`. -/ lemma isTotallyDisconnected_iff_lt {s : Set α} : IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y := by simp only [IsTotallyDisconnected, isPreconnected_iff_ordConnected, ← not_nontrivial_iff, nontrivial_iff_exists_lt, not_exists, not_and] refine ⟨fun h x hx y hy hxy ↦ ?_, fun h t hts ht x hx y hy hxy ↦ ?_⟩ · simp_rw [← not_ordConnected_inter_Icc_iff hx hy] exact fun hs ↦ h _ inter_subset_left hs _ ⟨hx, le_rfl, hxy.le⟩ _ ⟨hy, hxy.le, le_rfl⟩ hxy · obtain ⟨z, h1z, h2z⟩ := h x (hts hx) y (hts hy) hxy exact h1z <\| hts <\| ht.1 hx hy ⟨h2z.1.le, h2z.2.le⟩ /-! ### Intermediate Value Theorem on an interval In this section we prove several versions of the Intermediate Value Theorem for a function continuous on an interval. -/ variable {δ : Type} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] /-- Intermediate Value Theorem* for continuous functions on closed intervals, case `f a ≤ t ≤ f b`. -/ theorem intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Icc (f a) (f b) ⊆ f '' Icc a b := isPreconnected_Icc.intermediate_value (left_mem_Icc.2 hab) (right_mem_Icc.2 hab) hf /-- Intermediate Value Theorem for continuous functions on closed intervals, case `f a ≥ t ≥ f b`. -/ theorem intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Icc (f b) (f a) ⊆ f '' Icc a b := isPreconnected_Icc.intermediate_value (right_mem_Icc.2 hab) (left_mem_Icc.2 hab) hf /-- Intermediate Value Theorem for continuous functions on closed intervals, unordered case. -/ theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f [[a, b]]) : [[f a, f b]] ⊆ f '' uIcc a b := by cases le_total (f a) (f b) <;> simp [, isPreconnected_uIcc.intermediate_value] /-- If `f : α → α` is continuous on `[[a, b]]`, `a ≤ f a`, and `f b ≤ b`, then `f` has a fixed point on `[[a, b]]`. -/ theorem exists_mem_uIcc_isFixedPt {a b : α} {f : α → α} (hf : ContinuousOn f (uIcc a b)) (ha : a ≤ f a) (hb : f b ≤ b) : ∃ c ∈ [[a, b]], IsFixedPt f c := isPreconnected_uIcc.intermediate_value₂ right_mem_uIcc left_mem_uIcc hf continuousOn_id hb ha /-- If `f : α → α` is continuous on `[a, b]`, `a ≤ b`, `a ≤ f a`, and `f b ≤ b`, then `f` has a fixed point on `[a, b]`. In particular, if `[a, b]` is forward-invariant under `f`, then `f` has a fixed point on `[a, b]`, see `exists_mem_Icc_isFixedPt_of_mapsTo`. -/ theorem exists_mem_Icc_isFixedPt {a b : α} {f : α → α} (hf : ContinuousOn f (Icc a b)) (hle : a ≤ b) (ha : a ≤ f a) (hb : f b ≤ b) : ∃ c ∈ Icc a b, IsFixedPt f c := isPreconnected_Icc.intermediate_value₂ (right_mem_Icc.2 hle) (left_mem_Icc.2 hle) hf continuousOn_id hb ha /-- If a closed interval is forward-invariant under a continuous map `f : α → α`, then this map has a fixed point on this interval. -/ theorem exists_mem_Icc_isFixedPt_of_mapsTo {a b : α} {f : α → α} (hf : ContinuousOn f (Icc a b)) (hle : a ≤ b) (hmaps : MapsTo f (Icc a b) (Icc a b)) : ∃ c ∈ Icc a b, IsFixedPt f c := exists_mem_Icc_isFixedPt hf hle (hmaps <\| left_mem_Icc.2 hle).1 (hmaps <\| right_mem_Icc.2 hle).2 theorem intermediate_value_Ico {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ico (f a) (f b) ⊆ f '' Ico a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_ge (he ▸ h.1))) fun hlt => @IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩ (right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _ ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ico_subset_Icc_self) theorem intermediate_value_Ico' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioc (f b) (f a) ⊆ f '' Ico a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_ge (he ▸ h.2))) fun hlt => @IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ico _ _ ⟨refl a, hlt⟩ (right_nhdsWithin_Ico_neBot hlt) inf_le_right _ (hf.mono Ico_subset_Icc_self) _ ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ico_subset_Icc_self) theorem intermediate_value_Ioc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioc (f a) (f b) ⊆ f '' Ioc a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_le_of_gt (he ▸ h.1))) fun hlt => @IsPreconnected.intermediate_value_Ioc _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩ (left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _ ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioc_subset_Icc_self) theorem intermediate_value_Ioc' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ico (f b) (f a) ⊆ f '' Ioc a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_le_of_gt (he ▸ h.2))) fun hlt => @IsPreconnected.intermediate_value_Ico _ _ _ _ _ _ _ isPreconnected_Ioc _ _ ⟨hlt, refl b⟩ (left_nhdsWithin_Ioc_neBot hlt) inf_le_right _ (hf.mono Ioc_subset_Icc_self) _ ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioc_subset_Icc_self) theorem intermediate_value_Ioo {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioo (f a) (f b) ⊆ f '' Ioo a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.2 (not_lt_of_gt (he ▸ h.1))) fun hlt => @IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _ (left_nhdsWithin_Ioo_neBot hlt) (right_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _ (hf.mono Ioo_subset_Icc_self) _ _ ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioo_subset_Icc_self) ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ioo_subset_Icc_self) theorem intermediate_value_Ioo' {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) : Ioo (f b) (f a) ⊆ f '' Ioo a b := Or.elim (eq_or_lt_of_le hab) (fun he _ h => absurd h.1 (not_lt_of_gt (he ▸ h.2))) fun hlt => @IsPreconnected.intermediate_value_Ioo _ _ _ _ _ _ _ isPreconnected_Ioo _ _ (right_nhdsWithin_Ioo_neBot hlt) (left_nhdsWithin_Ioo_neBot hlt) inf_le_right inf_le_right _ (hf.mono Ioo_subset_Icc_self) _ _ ((hf.continuousWithinAt ⟨hab, refl b⟩).mono Ioo_subset_Icc_self) ((hf.continuousWithinAt ⟨refl a, hab⟩).mono Ioo_subset_Icc_self) /-- Intermediate value theorem: if `f` is continuous on an order-connected set `s` and `a`, `b` are two points of this set, then `f` sends `s` to a superset of `Icc (f x) (f y)`. -/ theorem ContinuousOn.surjOn_Icc {s : Set α} [hs : OrdConnected s] {f : α → δ} (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (Icc (f a) (f b)) := hs.isPreconnected.intermediate_value ha hb hf /-- Intermediate value theorem: if `f` is continuous on an order-connected set `s` and `a`, `b` are two points of this set, then `f` sends `s` to a superset of `[f x, f y]`. -/ theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ} (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (uIcc (f a) (f b)) := by rcases le_total (f a) (f b) with hab \| hab <;> simp [hf.surjOn_Icc, ] /-- A continuous function which tendsto `Filter.atTop` along `Filter.atTop` and to `atBot` along `at_bot` is surjective. -/ theorem Continuous.surjective {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atTop atTop) (h_bot : Tendsto f atBot atBot) : Function.Surjective f := fun p => mem_range_of_exists_le_of_exists_ge hf (h_bot.eventually (eventually_le_atBot p)).exists (h_top.eventually (eventually_ge_atTop p)).exists /-- A continuous function which tendsto `Filter.atBot` along `Filter.atTop` and to `Filter.atTop` along `atBot` is surjective. -/ theorem Continuous.surjective' {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atBot atTop) (h_bot : Tendsto f atTop atBot) : Function.Surjective f := Continuous.surjective (α := αᵒᵈ) hf h_top h_bot /-- If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s` tends to `at_bot : Filter β` along `at_bot : Filter ↥s` and tends to `Filter.atTop : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the conclusion as `Function.surjOn f s Set.univ`. -/ theorem ContinuousOn.surjOn_of_tendsto {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty) (hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atBot) (htop : Tendsto (fun x : s => f x) atTop atTop) : SurjOn f s univ := haveI := Classical.inhabited_of_nonempty hs.to_subtype surjOn_iff_surjective.2 <\| hf.restrict.surjective htop hbot /-- If a function `f : α → β` is continuous on a nonempty interval `s`, its restriction to `s` tends to `Filter.atTop : Filter β` along `Filter.atBot : Filter ↥s` and tends to `Filter.atBot : Filter β` along `Filter.atTop : Filter ↥s`, then the restriction of `f` to `s` is surjective. We formulate the conclusion as `Function.surjOn f s Set.univ`. -/ theorem ContinuousOn.surjOn_of_tendsto' {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty) (hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atTop) (htop : Tendsto (fun x : s => f x) atTop atBot) : SurjOn f s univ := ContinuousOn.surjOn_of_tendsto (δ := δᵒᵈ) hs hf hbot htop theorem Continuous.strictMono_of_inj_boundedOrder [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf : f ⊥ ≤ f ⊤) (hf_i : Injective f) : StrictMono f := by intro a b hab by_contra! h have H : f b < f a := lt_of_le_of_ne h <\| hf_i.ne hab.ne' by_cases! ha : f a ≤ f ⊥ · obtain ⟨u, hu⟩ := intermediate_value_Ioc le_top hf_c.continuousOn ⟨H.trans_le ha, hf⟩ have : u = ⊥ := hf_i hu.2 simp_all · by_cases! hb : f ⊥ < f b · obtain ⟨u, hu⟩ := intermediate_value_Ioo bot_le hf_c.continuousOn ⟨hb, H⟩ rw [hf_i hu.2] at hu exact (hab.trans hu.1.2).false · replace hb : f b < f ⊥ := lt_of_le_of_ne hb <\| hf_i.ne (lt_of_lt_of_le' hab bot_le).ne' obtain ⟨u, hu⟩ := intermediate_value_Ioo' hab.le hf_c.continuousOn ⟨hb, ha⟩ have : u = ⊥ := hf_i hu.2 simp_all theorem Continuous.strictAnti_of_inj_boundedOrder [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf : f ⊤ ≤ f ⊥) (hf_i : Injective f) : StrictAnti f := hf_c.strictMono_of_inj_boundedOrder (δ := δᵒᵈ) hf hf_i theorem Continuous.strictMono_of_inj_boundedOrder' [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f := (le_total (f ⊥) (f ⊤)).imp (hf_c.strictMono_of_inj_boundedOrder · hf_i) (hf_c.strictAnti_of_inj_boundedOrder · hf_i) /-- Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is continuous and injective. Then `f` is strictly monotone (increasing) if it is strictly monotone (increasing) on some closed interval `[a, b]`. -/ theorem Continuous.strictMonoOn_of_inj_rigidity {f : α → δ} (hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b) (hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f := by intro x y hxy let s := min a x let t := max b y have hsa : s ≤ a := min_le_left a x have hbt : b ≤ t := le_max_left b y have hf_mono_st : StrictMonoOn f (Icc s t) ∨ StrictAntiOn f (Icc s t) := by have : Fact (s ≤ t) := ⟨hsa.trans <\| hbt.trans' hab.le⟩ have := Continuous.strictMono_of_inj_boundedOrder' (f := Set.restrict (Icc s t) f) hf_c.continuousOn.restrict hf_i.injOn.injective exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr have (h : StrictAntiOn f (Icc s t)) : False := by have : Icc a b ⊆ Icc s t := Icc_subset_Icc hsa hbt replace : StrictAntiOn f (Icc a b) := StrictAntiOn.mono h this replace : IsAntichain (· ≤ ·) (Icc a b) := IsAntichain.of_strictMonoOn_antitoneOn hf_mono this.antitoneOn exact this.not_lt (left_mem_Icc.mpr (le_of_lt hab)) (right_mem_Icc.mpr (le_of_lt hab)) hab replace hf_mono_st : StrictMonoOn f (Icc s t) := hf_mono_st.resolve_right this have hsx : s ≤ x := min_le_right a x have hyt : y ≤ t := le_max_right b y replace : Icc x y ⊆ Icc s t := Icc_subset_Icc hsx hyt replace : StrictMonoOn f (Icc x y) := StrictMonoOn.mono hf_mono_st this exact this (left_mem_Icc.mpr (le_of_lt hxy)) (right_mem_Icc.mpr (le_of_lt hxy)) hxy /-- Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone (increasing) if `f(a) ≤ f(b)`. -/ theorem ContinuousOn.strictMonoOn_of_injOn_Icc {a b : α} {f : α → δ} (hab : a ≤ b) (hfab : f a ≤ f b) (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) : StrictMonoOn f (Icc a b) := by have : Fact (a ≤ b) := ⟨hab⟩ refine StrictMono.of_restrict ?_ set g : Icc a b → δ := Set.restrict (Icc a b) f have hgab : g ⊥ ≤ g ⊤ := by aesop exact Continuous.strictMono_of_inj_boundedOrder (f := g) hf_c.restrict hgab hf_i.injective /-- Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly antitone (decreasing) if `f(b) ≤ f(a)`. -/ theorem ContinuousOn.strictAntiOn_of_injOn_Icc {a b : α} {f : α → δ} (hab : a ≤ b) (hfab : f b ≤ f a) (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) : StrictAntiOn f (Icc a b) := ContinuousOn.strictMonoOn_of_injOn_Icc (δ := δᵒᵈ) hab hfab hf_c hf_i /-- Suppose `f : [a, b] → δ` is continuous and injective. Then `f` is strictly monotone or antitone (increasing or decreasing). -/ theorem ContinuousOn.strictMonoOn_of_injOn_Icc' {a b : α} {f : α → δ} (hab : a ≤ b) (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) : StrictMonoOn f (Icc a b) ∨ StrictAntiOn f (Icc a b) := (le_total (f a) (f b)).imp (ContinuousOn.strictMonoOn_of_injOn_Icc hab · hf_c hf_i) (ContinuousOn.strictAntiOn_of_injOn_Icc hab · hf_c hf_i) /-- Suppose `α` is equipped with a conditionally complete linear dense order and `f : α → δ` is continuous and injective. Then `f` is strictly monotone or antitone (increasing or decreasing). -/ theorem Continuous.strictMono_of_inj {f : α → δ} (hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f := by have H {c d : α} (hcd : c < d) : StrictMono f ∨ StrictAnti f := (hf_c.continuousOn.strictMonoOn_of_injOn_Icc' hcd.le hf_i.injOn).imp (hf_c.strictMonoOn_of_inj_rigidity hf_i hcd) (hf_c.strictMonoOn_of_inj_rigidity (δ := δᵒᵈ) hf_i hcd) cases subsingleton_or_nontrivial α with \| inl h => exact Or.inl <\| Subsingleton.strictMono f \| inr h => obtain ⟨a, b, hab⟩ := exists_pair_lt α exact H hab /-- Every continuous injective `f : (a, b) → δ` is strictly monotone or antitone (increasing or decreasing). -/ theorem ContinuousOn.strictMonoOn_of_injOn_Ioo {a b : α} {f : α → δ} (hab : a < b) (hf_c : ContinuousOn f (Ioo a b)) (hf_i : InjOn f (Ioo a b)) : StrictMonoOn f (Ioo a b) ∨ StrictAntiOn f (Ioo a b) := by haveI : Inhabited (Ioo a b) := Classical.inhabited_of_nonempty (nonempty_Ioo_subtype hab) let g : Ioo a b → δ := Set.restrict (Ioo a b) f have : StrictMono g ∨ StrictAnti g := Continuous.strictMono_of_inj hf_c.restrict hf_i.injective exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr
.lake/packages/mathlib/Mathlib/Topology/Order/Basic.lean	import Mathlib.Order.Filter.Interval import Mathlib.Order.Interval.Set.Pi import Mathlib.Order.OrdContinuous import Mathlib.Tactic.TFAE import Mathlib.Tactic.NormNum import Mathlib.Topology.Order.LeftRight import Mathlib.Topology.Order.OrderClosed /-! # Theory of topology on ordered spaces ## Main definitions The order topology on an ordered space is the topology generated by all open intervals (or equivalently by those of the form `(-∞, a)` and `(b, +∞)`). We define it as `Preorder.topology α`. However, we do not register it as an instance (as many existing ordered types already have topologies, which would be equal but not definitionally equal to `Preorder.topology α`). Instead, we introduce a class `OrderTopology α` (which is a `Prop`, also known as a mixin) saying that on the type `α` having already a topological space structure and a preorder structure, the topological structure is equal to the order topology. We prove many basic properties of such topologies. ## Main statements This file contains the proofs of the following facts. For exact requirements (`OrderClosedTopology` vs `OrderTopology`, `Preorder` vs `PartialOrder` vs `LinearOrder` etc) see their statements. * `exists_Ioc_subset_of_mem_nhds`, `exists_Ico_subset_of_mem_nhds` : if `x < y`, then any neighborhood of `x` includes an interval `[x, z)` for some `z ∈ (x, y]`, and any neighborhood of `y` includes an interval `(z, y]` for some `z ∈ [x, y)`. * `tendsto_of_tendsto_of_tendsto_of_le_of_le` : theorem known as squeeze theorem, sandwich theorem, theorem of Carabinieri, and two policemen (and a drunk) theorem; if `g` and `h` both converge to `a`, and eventually `g x ≤ f x ≤ h x`, then `f` converges to `a`. ## Implementation notes We do _not_ register the order topology as an instance on a preorder (or even on a linear order). Indeed, on many such spaces, a topology has already been constructed in a different way (think of the discrete spaces `ℕ` or `ℤ`, or `ℝ` that could inherit a topology as the completion of `ℚ`), and is in general not defeq to the one generated by the intervals. We make it available as a definition `Preorder.topology α` though, that can be registered as an instance when necessary, or for specific types. -/ open Set Filter TopologicalSpace Topology Function open OrderDual (toDual ofDual) universe u v w variable {α : Type u} {β : Type v} {γ : Type w} -- TODO: define `Preorder.topology` before `OrderTopology` and reuse the def /-- The order topology on an ordered type is the topology generated by open intervals. We register it on a preorder, but it is mostly interesting in linear orders, where it is also order-closed. We define it as a mixin. If you want to introduce the order topology on a preorder, use `Preorder.topology`. -/ class OrderTopology (α : Type) [t : TopologicalSpace α] [Preorder α] : Prop where /-- The topology is generated by open intervals `Set.Ioi _` and `Set.Iio _`. -/ topology_eq_generate_intervals : t = generateFrom { s \| ∃ a, s = Ioi a ∨ s = Iio a } /-- (Order) topology on a partial order `α` generated by the subbase of open intervals `(a, ∞) = { x ∣ a < x }, (-∞, b) = {x ∣ x < b}` for all `a, b` in `α`. We do not register it as an instance as many ordered sets are already endowed with the same topology, most often in a non-defeq way though. Register as a local instance when necessary. -/ def Preorder.topology (α : Type) [Preorder α] : TopologicalSpace α := generateFrom { s : Set α \| ∃ a : α, s = { b : α \| a < b } ∨ s = { b : α \| b < a } } section Preorder variable [ts : TopologicalSpace α] [Preorder α] instance [t : OrderTopology α] : OrderTopology αᵒᵈ := ⟨by convert OrderTopology.topology_eq_generate_intervals (α := α) using 6 apply or_comm⟩ protected theorem OrderTopology.continuous_iff [OrderTopology α] [TopologicalSpace β] {f : β → α} : Continuous f ↔ ∀ a, IsOpen (f ⁻¹' Ioi a) ∧ IsOpen (f ⁻¹' Iio a) := by simp_rw [OrderTopology.topology_eq_generate_intervals, continuous_generateFrom_iff] aesop theorem isOpen_iff_generate_intervals [t : OrderTopology α] {s : Set α} : IsOpen s ↔ GenerateOpen { s \| ∃ a, s = Ioi a ∨ s = Iio a } s := by rw [t.topology_eq_generate_intervals]; rfl theorem isOpen_lt' [OrderTopology α] (a : α) : IsOpen { b : α \| a < b } := isOpen_iff_generate_intervals.2 <\| .basic _ ⟨a, .inl rfl⟩ theorem isOpen_Ioi' [OrderTopology α] (a : α) : IsOpen (Ioi a) := isOpen_lt' a theorem isOpen_gt' [OrderTopology α] (a : α) : IsOpen { b : α \| b < a } := isOpen_iff_generate_intervals.2 <\| .basic _ ⟨a, .inr rfl⟩ theorem isOpen_Iio' [OrderTopology α] (a : α) : IsOpen (Iio a) := isOpen_gt' a theorem lt_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a < x := (isOpen_lt' _).mem_nhds h theorem le_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 b, a ≤ x := (lt_mem_nhds h).mono fun _ => le_of_lt theorem gt_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x < b := (isOpen_gt' _).mem_nhds h theorem ge_mem_nhds [OrderTopology α] {a b : α} (h : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := (gt_mem_nhds h).mono fun _ => le_of_lt theorem nhds_eq_order [OrderTopology α] (a : α) : 𝓝 a = (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b) := by rw [OrderTopology.topology_eq_generate_intervals (α := α), nhds_generateFrom] simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists, iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio] theorem tendsto_order [OrderTopology α] {f : β → α} {a : α} {x : Filter β} : Tendsto f x (𝓝 a) ↔ (∀ a' < a, ∀ᶠ b in x, a' < f b) ∧ ∀ a' > a, ∀ᶠ b in x, f b < a' := by simp only [nhds_eq_order a, tendsto_inf, tendsto_iInf, tendsto_principal]; rfl instance tendstoIccClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Icc (𝓝 a) (𝓝 a) := by simp only [nhds_eq_order, iInf_subtype'] refine ((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass fun s _ => ?_ refine ((ordConnected_biInter ?_).inter (ordConnected_biInter ?_)).out <;> intro _ _ exacts [ordConnected_Ioi, ordConnected_Iio] instance tendstoIcoClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Ico (𝓝 a) (𝓝 a) := tendstoIxxClass_of_subset fun _ _ => Ico_subset_Icc_self instance tendstoIocClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Ioc (𝓝 a) (𝓝 a) := tendstoIxxClass_of_subset fun _ _ => Ioc_subset_Icc_self instance tendstoIooClassNhds [OrderTopology α] (a : α) : TendstoIxxClass Ioo (𝓝 a) (𝓝 a) := tendstoIxxClass_of_subset fun _ _ => Ioo_subset_Icc_self variable (α) in /-- In a second countable topological space with the order topology, the topology is generated by half-infinite open intervals with endpoints in a countable set. -/ lemma exists_countable_generateFrom_Ioi_Iio [OrderTopology α] [SecondCountableTopology α] : ∃ (c : Set α), c.Countable ∧ ts = generateFrom { s \| ∃ a ∈ c, s = Ioi a ∨ s = Iio a } := by rcases isEmpty_or_nonempty α with hα \| hα · exact ⟨∅, by simp, Subsingleton.elim _ _⟩ obtain ⟨t, t_subs, t_count, ht⟩ : ∃ t ⊆ { s \| ∃ a, s = Ioi a ∨ s = Iio a }, t.Countable ∧ ts = generateFrom t := exists_countable_of_generateFrom OrderTopology.topology_eq_generate_intervals have A : ∀ s ∈ t, ∃ a, s = Ioi a ∨ s = Iio a := t_subs choose! a ha using A refine ⟨a '' t, t_count.image _, ?_⟩ apply le_antisymm · apply le_generateFrom_iff_subset_isOpen.2 simp only [mem_image, exists_exists_and_eq_and, setOf_subset_setOf, forall_exists_index, and_imp] grind [isOpen_Iio', isOpen_Ioi'] · rw [ht] apply generateFrom_anti grind /-- If a topology is generated by half-open intervals with endpoints in a set `c`, then the sets formed by intersecting finitely many of these intervals form a topological basis. -/ lemma isTopologicalBasis_biInter_Ioi_Iio_of_generateFrom (c : Set α) (h : ts = generateFrom { s \| ∃ a ∈ c, s = Ioi a ∨ s = Iio a }) : IsTopologicalBasis {s \| ∃ (f g : Set α), f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ (⋂ a ∈ g, Iio a)} := by refine IsTopologicalBasis.of_isOpen_of_subset ?_ (isTopologicalBasis_of_subbasis h) ?_ · rintro u ⟨f, g, hfc, hgc, hf_fin, hg_fin, rfl⟩ apply IsOpen.inter · apply hf_fin.isOpen_biInter (fun i hi ↦ ?_) rw [h] exact isOpen_generateFrom_of_mem ⟨i, hfc hi, Or.inl rfl⟩ · apply hg_fin.isOpen_biInter (fun i hi ↦ ?_) rw [h] exact isOpen_generateFrom_of_mem ⟨i, hgc hi, Or.inr rfl⟩ simp only [exists_and_left, image_subset_iff, preimage_setOf_eq, setOf_subset_setOf, and_imp] intro k k_fin hk let kl := {s ∈ k \| ∃ a ∈ c, s = Ioi a} let kr := {s ∈ k \| ∃ a ∈ c, s = Iio a} have k_eq : k = kl ∪ kr := by have : ∀ s ∈ k, ∃ a ∈ c, s = Ioi a ∨ s = Iio a := hk ext simp only [mem_union, mem_setOf_eq, kl, kr] grind have : Finite kl := k_fin.subset (by simp [k_eq]) have : Finite kr := k_fin.subset (by simp [k_eq]) have Al : ∀ s : kl, ∃ a ∈ c, s = Ioi a := fun s ↦ s.2.2 choose al alc hal using Al have Ar : ∀ s : kr, ∃ a ∈ c, s = Iio a := fun s ↦ s.2.2 choose ar arc har using Ar refine ⟨range al, by simp [range_subset_iff, alc], range ar, by simp [range_subset_iff, arc], finite_range _, finite_range _, ?_⟩ rw [k_eq, sInter_eq_biInter, biInter_union, biInter_range, biInter_range, biInter_eq_iInter, biInter_eq_iInter] simp [hal, har] /-- Squeeze theorem (also known as sandwich theorem). This version assumes that inequalities hold eventually for the filter. -/ theorem tendsto_of_tendsto_of_tendsto_of_le_of_le' [OrderTopology α] {f g h : β → α} {b : Filter β} {a : α} (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : ∀ᶠ b in b, g b ≤ f b) (hfh : ∀ᶠ b in b, f b ≤ h b) : Tendsto f b (𝓝 a) := (hg.Icc hh).of_smallSets <\| hgf.and hfh alias Filter.Tendsto.squeeze' := tendsto_of_tendsto_of_tendsto_of_le_of_le' /-- Squeeze theorem (also known as sandwich theorem). This version assumes that inequalities hold everywhere. -/ theorem tendsto_of_tendsto_of_tendsto_of_le_of_le [OrderTopology α] {f g h : β → α} {b : Filter β} {a : α} (hg : Tendsto g b (𝓝 a)) (hh : Tendsto h b (𝓝 a)) (hgf : g ≤ f) (hfh : f ≤ h) : Tendsto f b (𝓝 a) := tendsto_of_tendsto_of_tendsto_of_le_of_le' hg hh (Eventually.of_forall hgf) (Eventually.of_forall hfh) alias Filter.Tendsto.squeeze := tendsto_of_tendsto_of_tendsto_of_le_of_le theorem nhds_order_unbounded [OrderTopology α] {a : α} (hu : ∃ u, a < u) (hl : ∃ l, l < a) : 𝓝 a = ⨅ (l) (_ : l < a) (u) (_ : a < u), 𝓟 (Ioo l u) := by simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, , ← inf_principal, ← Ioi_inter_Iio]; rfl theorem tendsto_order_unbounded [OrderTopology α] {f : β → α} {a : α} {x : Filter β} (hu : ∃ u, a < u) (hl : ∃ l, l < a) (h : ∀ l u, l < a → a < u → ∀ᶠ b in x, l < f b ∧ f b < u) : Tendsto f x (𝓝 a) := by simp only [nhds_order_unbounded hu hl, tendsto_iInf, tendsto_principal] exact fun l hl u => h l u hl end Preorder instance tendstoIxxNhdsWithin {α : Type} [TopologicalSpace α] (a : α) {s t : Set α} {Ixx} [TendstoIxxClass Ixx (𝓝 a) (𝓝 a)] [TendstoIxxClass Ixx (𝓟 s) (𝓟 t)] : TendstoIxxClass Ixx (𝓝[s] a) (𝓝[t] a) := Filter.tendstoIxxClass_inf instance tendstoIccClassNhdsPi {ι : Type} {α : ι → Type} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)] [∀ i, OrderTopology (α i)] (f : ∀ i, α i) : TendstoIxxClass Icc (𝓝 f) (𝓝 f) := by constructor conv in (𝓝 f).smallSets => rw [nhds_pi, Filter.pi] simp only [smallSets_iInf, smallSets_comap_eq_comap_image, tendsto_iInf, tendsto_comap_iff] intro i have : Tendsto (fun g : ∀ i, α i => g i) (𝓝 f) (𝓝 (f i)) := (continuous_apply i).tendsto f refine (this.comp tendsto_fst).Icc (this.comp tendsto_snd) \|>.smallSets_mono ?_ filter_upwards [] using fun ⟨f, g⟩ ↦ image_subset_iff.mpr fun p hp ↦ ⟨hp.1 i, hp.2 i⟩ theorem induced_topology_le_preorder [Preorder α] [Preorder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) : induced f ‹TopologicalSpace β› ≤ Preorder.topology α := by let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩ refine le_of_nhds_le_nhds fun x => ?_ simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal, Ioi, Iio, ← hf] refine inf_le_inf (le_iInf₂ fun a ha => ?_) (le_iInf₂ fun a ha => ?_) exacts [iInf₂_le (f a) ha, iInf₂_le (f a) ha] theorem induced_topology_eq_preorder [Preorder α] [Preorder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : ∀ {x y}, f x < f y ↔ x < y) (H₁ : ∀ {a b x}, b < f a → ¬(b < f x) → ∃ y, y < a ∧ b ≤ f y) (H₂ : ∀ {a b x}, f a < b → ¬(f x < b) → ∃ y, a < y ∧ f y ≤ b) : induced f ‹TopologicalSpace β› = Preorder.topology α := by let _ := Preorder.topology α; have : OrderTopology α := ⟨rfl⟩ refine le_antisymm (induced_topology_le_preorder hf) ?_ refine le_of_nhds_le_nhds fun a => ?_ simp only [nhds_eq_order, nhds_induced, comap_inf, comap_iInf, comap_principal] refine inf_le_inf (le_iInf₂ fun b hb => ?_) (le_iInf₂ fun b hb => ?_) · by_cases! h : ∃ x, ¬(b < f x) · rcases h with ⟨x, hx⟩ rcases H₁ hb hx with ⟨y, hya, hyb⟩ exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => hyb.trans_lt (hf.2 hz)) · exact le_principal_iff.2 (univ_mem' h) · by_cases! h : ∃ x, ¬(f x < b) · rcases h with ⟨x, hx⟩ rcases H₂ hb hx with ⟨y, hya, hyb⟩ exact iInf₂_le_of_le y hya (principal_mono.2 fun z hz => (hf.2 hz).trans_le hyb) · exact le_principal_iff.2 (univ_mem' h) theorem induced_orderTopology' {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β] [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y) (H₁ : ∀ {a x}, x < f a → ∃ b < a, x ≤ f b) (H₂ : ∀ {a x}, f a < x → ∃ b > a, f b ≤ x) : @OrderTopology _ (induced f ta) _ := let _ := induced f ta ⟨induced_topology_eq_preorder hf (fun h _ => H₁ h) (fun h _ => H₂ h)⟩ theorem induced_orderTopology {α : Type u} {β : Type v} [Preorder α] [ta : TopologicalSpace β] [Preorder β] [OrderTopology β] (f : α → β) (hf : ∀ {x y}, f x < f y ↔ x < y) (H : ∀ {x y}, x < y → ∃ a, x < f a ∧ f a < y) : @OrderTopology _ (induced f ta) _ := induced_orderTopology' f (hf) (fun xa => let ⟨b, xb, ba⟩ := H xa; ⟨b, hf.1 ba, le_of_lt xb⟩) fun ax => let ⟨b, ab, bx⟩ := H ax; ⟨b, hf.1 ab, le_of_lt bx⟩ /-- The topology induced by a strictly monotone function with order-connected range is the preorder topology. -/ nonrec theorem StrictMono.induced_topology_eq_preorder {α β : Type} [LinearOrder α] [LinearOrder β] [t : TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : StrictMono f) (hc : OrdConnected (range f)) : t.induced f = Preorder.topology α := by refine induced_topology_eq_preorder hf.lt_iff_lt (fun h₁ h₂ => ?_) fun h₁ h₂ => ?_ · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨not_lt.1 h₂, h₁.le⟩ with ⟨y, rfl⟩ exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩ · rcases hc.out (mem_range_self _) (mem_range_self _) ⟨h₁.le, not_lt.1 h₂⟩ with ⟨y, rfl⟩ exact ⟨y, hf.lt_iff_lt.1 h₁, le_rfl⟩ /-- A strictly monotone function between linear orders with order topology is a topological embedding provided that the range of `f` is order-connected. -/ theorem StrictMono.isEmbedding_of_ordConnected {α β : Type} [LinearOrder α] [LinearOrder β] [TopologicalSpace α] [h : OrderTopology α] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : StrictMono f) (hc : OrdConnected (range f)) : IsEmbedding f := ⟨⟨h.1.trans <\| Eq.symm <\| hf.induced_topology_eq_preorder hc⟩, hf.injective⟩ /-- An `OrderEmbedding` is a topological embedding provided that the range of `f` is order-connected -/ lemma OrderEmbedding.isEmbedding_of_ordConnected {α β : Type} [LinearOrder α] [LinearOrder β] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [OrderTopology β] (f : α ↪o β) (hc : OrdConnected (range f)) : Topology.IsEmbedding f := f.strictMono.isEmbedding_of_ordConnected hc /-- On a `Set.OrdConnected` subset of a linear order, the order topology for the restriction of the order is the same as the restriction to the subset of the order topology. -/ instance orderTopology_of_ordConnected {α : Type u} [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {t : Set α} [ht : OrdConnected t] : OrderTopology t := ⟨(Subtype.strictMono_coe t).induced_topology_eq_preorder <\| by rwa [← @Subtype.range_val _ t] at ht⟩ theorem nhdsGE_eq_iInf_inf_principal [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) : 𝓝[≥] a = (⨅ (u) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) := by rw [nhdsWithin, nhds_eq_order] refine le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf ?_ inf_le_left) inf_le_right) exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <\| Ici_subset_Ioi.2 hl) theorem nhdsLE_eq_iInf_inf_principal [TopologicalSpace α] [Preorder α] [OrderTopology α] (a : α) : 𝓝[≤] a = (⨅ l < a, 𝓟 (Ioi l)) ⊓ 𝓟 (Iic a) := nhdsGE_eq_iInf_inf_principal (toDual a) theorem nhdsGE_eq_iInf_principal [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α} (ha : ∃ u, a < u) : 𝓝[≥] a = ⨅ (u) (_ : a < u), 𝓟 (Ico a u) := by simp only [nhdsGE_eq_iInf_inf_principal, biInf_inf ha, inf_principal, Iio_inter_Ici] theorem nhdsLE_eq_iInf_principal [TopologicalSpace α] [Preorder α] [OrderTopology α] {a : α} (ha : ∃ l, l < a) : 𝓝[≤] a = ⨅ l < a, 𝓟 (Ioc l a) := by simp only [nhdsLE_eq_iInf_inf_principal, biInf_inf ha, inf_principal, Ioi_inter_Iic] theorem nhdsGE_basis_of_exists_gt [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (ha : ∃ u, a < u) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u := (nhdsGE_eq_iInf_principal ha).symm ▸ hasBasis_biInf_principal (fun b hb c hc => ⟨min b c, lt_min hb hc, Ico_subset_Ico_right (min_le_left _ _), Ico_subset_Ico_right (min_le_right _ _)⟩) ha theorem nhdsLE_basis_of_exists_lt [TopologicalSpace α] [LinearOrder α] [OrderTopology α] {a : α} (ha : ∃ l, l < a) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := by convert nhdsGE_basis_of_exists_gt (α := αᵒᵈ) ha using 2 exact Ico_toDual.symm theorem nhdsGE_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMaxOrder α] (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) fun u => Ico a u := nhdsGE_basis_of_exists_gt (exists_gt a) theorem nhdsLE_basis [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [NoMinOrder α] (a : α) : (𝓝[≤] a).HasBasis (fun l => l < a) fun l => Ioc l a := nhdsLE_basis_of_exists_lt (exists_lt a) theorem nhds_top_order [TopologicalSpace α] [Preorder α] [OrderTop α] [OrderTopology α] : 𝓝 (⊤ : α) = ⨅ (l) (_ : l < ⊤), 𝓟 (Ioi l) := by simp [nhds_eq_order (⊤ : α)] theorem nhds_bot_order [TopologicalSpace α] [Preorder α] [OrderBot α] [OrderTopology α] : 𝓝 (⊥ : α) = ⨅ (l) (_ : ⊥ < l), 𝓟 (Iio l) := by simp [nhds_eq_order (⊥ : α)] theorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α] [Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsLE_basis_of_exists_lt this theorem nhds_bot_basis [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α] [Nontrivial α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) fun a : α => Iio a := nhds_top_basis (α := αᵒᵈ) theorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α] [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici := nhds_top_basis.to_hasBasis (fun _a ha => let ⟨b, hab, hb⟩ := exists_between ha; ⟨b, hb, Ici_subset_Ioi.mpr hab⟩) fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩ theorem nhds_bot_basis_Iic [TopologicalSpace α] [LinearOrder α] [OrderBot α] [OrderTopology α] [Nontrivial α] [DenselyOrdered α] : (𝓝 ⊥).HasBasis (fun a : α => ⊥ < a) Iic := nhds_top_basis_Ici (α := αᵒᵈ) theorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢ intro x hx filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le theorem tendsto_nhds_bot_mono [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ᶠ[l] f) : Tendsto g l (𝓝 ⊥) := tendsto_nhds_top_mono (β := βᵒᵈ) hf hg theorem tendsto_nhds_top_mono' [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ g) : Tendsto g l (𝓝 ⊤) := tendsto_nhds_top_mono hf (Eventually.of_forall hg) theorem tendsto_nhds_bot_mono' [TopologicalSpace β] [Preorder β] [OrderBot β] [OrderTopology β] {l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊥)) (hg : g ≤ f) : Tendsto g l (𝓝 ⊥) := tendsto_nhds_bot_mono hf (Eventually.of_forall hg) section LinearOrder variable [TopologicalSpace α] [LinearOrder α] section OrderTopology theorem order_separated [OrderTopology α] {a₁ a₂ : α} (h : a₁ < a₂) : ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ a₁ ∈ u ∧ a₂ ∈ v ∧ ∀ b₁ ∈ u, ∀ b₂ ∈ v, b₁ < b₂ := let ⟨x, hx, y, hy, h⟩ := h.exists_disjoint_Iio_Ioi ⟨Iio x, Ioi y, isOpen_gt' _, isOpen_lt' _, hx, hy, h⟩ -- see Note [lower instance priority] instance (priority := 100) OrderTopology.to_orderClosedTopology [OrderTopology α] : OrderClosedTopology α where isClosed_le' := isOpen_compl_iff.1 <\| isOpen_prod_iff.mpr fun a₁ a₂ (h : ¬a₁ ≤ a₂) => have h : a₂ < a₁ := lt_of_not_ge h let ⟨u, v, hu, hv, ha₁, ha₂, h⟩ := order_separated h ⟨v, u, hv, hu, ha₂, ha₁, fun ⟨b₁, b₂⟩ ⟨h₁, h₂⟩ => not_le_of_gt <\| h b₂ h₂ b₁ h₁⟩ theorem exists_Ioc_subset_of_mem_nhds [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ l, l < a) : ∃ l < a, Ioc l a ⊆ s := (nhdsLE_basis_of_exists_lt h).mem_iff.mp (nhdsWithin_le_nhds hs) theorem exists_Ioc_subset_of_mem_nhds' [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {l : α} (hl : l < a) : ∃ l' ∈ Ico l a, Ioc l' a ⊆ s := let ⟨l', hl'a, hl's⟩ := exists_Ioc_subset_of_mem_nhds hs ⟨l, hl⟩ ⟨max l l', ⟨le_max_left _ _, max_lt hl hl'a⟩, (Ioc_subset_Ioc_left <\| le_max_right _ _).trans hl's⟩ theorem exists_Ico_subset_of_mem_nhds' [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) {u : α} (hu : a < u) : ∃ u' ∈ Ioc a u, Ico a u' ⊆ s := by simpa only [OrderDual.exists, exists_prop, Ico_toDual, Ioc_toDual] using exists_Ioc_subset_of_mem_nhds' (show ofDual ⁻¹' s ∈ 𝓝 (toDual a) from hs) hu.dual theorem exists_Ico_subset_of_mem_nhds [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) (h : ∃ u, a < u) : ∃ u, a < u ∧ Ico a u ⊆ s := let ⟨_l', hl'⟩ := h let ⟨l, hl⟩ := exists_Ico_subset_of_mem_nhds' hs hl' ⟨l, hl.1.1, hl.2⟩ theorem exists_Icc_mem_subset_of_mem_nhdsGE [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝[≥] a) : ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s := by rcases (em (IsMax a)).imp_right not_isMax_iff.mp with (ha \| ha) · use a simpa [ha.Ici_eq] using hs · rcases(nhdsGE_basis_of_exists_gt ha).mem_iff.mp hs with ⟨b, hab, hbs⟩ rcases eq_empty_or_nonempty (Ioo a b) with (H \| ⟨c, hac, hcb⟩) · have : Ico a b = Icc a a := by rw [← Icc_union_Ioo_eq_Ico le_rfl hab, H, union_empty] exact ⟨a, le_rfl, this ▸ ⟨Ico_mem_nhdsGE hab, hbs⟩⟩ · refine ⟨c, hac.le, Icc_mem_nhdsGE hac, ?_⟩ exact (Icc_subset_Ico_right hcb).trans hbs theorem exists_Icc_mem_subset_of_mem_nhdsLE [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝[≤] a) : ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by simpa only [Icc_toDual, toDual.surjective.exists] using exists_Icc_mem_subset_of_mem_nhdsGE (α := αᵒᵈ) (a := toDual a) hs theorem exists_Icc_mem_subset_of_mem_nhds [OrderTopology α] {a : α} {s : Set α} (hs : s ∈ 𝓝 a) : ∃ b c, a ∈ Icc b c ∧ Icc b c ∈ 𝓝 a ∧ Icc b c ⊆ s := by rcases exists_Icc_mem_subset_of_mem_nhdsLE (nhdsWithin_le_nhds hs) with ⟨b, hba, hb_nhds, hbs⟩ rcases exists_Icc_mem_subset_of_mem_nhdsGE (nhdsWithin_le_nhds hs) with ⟨c, hac, hc_nhds, hcs⟩ refine ⟨b, c, ⟨hba, hac⟩, ?_⟩ rw [← Icc_union_Icc_eq_Icc hba hac, ← nhdsLE_sup_nhdsGE] exact ⟨union_mem_sup hb_nhds hc_nhds, union_subset hbs hcs⟩ theorem IsOpen.exists_Ioo_subset [OrderTopology α] [Nontrivial α] {s : Set α} (hs : IsOpen s) (h : s.Nonempty) : ∃ a b, a < b ∧ Ioo a b ⊆ s := by obtain ⟨x, hx⟩ : ∃ x, x ∈ s := h obtain ⟨y, hy⟩ : ∃ y, y ≠ x := exists_ne x rcases lt_trichotomy x y with (H \| rfl \| H) · obtain ⟨u, xu, hu⟩ : ∃ u, x < u ∧ Ico x u ⊆ s := exists_Ico_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩ exact ⟨x, u, xu, Ioo_subset_Ico_self.trans hu⟩ · exact (hy rfl).elim · obtain ⟨l, lx, hl⟩ : ∃ l, l < x ∧ Ioc l x ⊆ s := exists_Ioc_subset_of_mem_nhds (hs.mem_nhds hx) ⟨y, H⟩ exact ⟨l, x, lx, Ioo_subset_Ioc_self.trans hl⟩ theorem dense_of_exists_between [OrderTopology α] [Nontrivial α] {s : Set α} (h : ∀ ⦃a b⦄, a < b → ∃ c ∈ s, a < c ∧ c < b) : Dense s := by refine dense_iff_inter_open.2 fun U U_open U_nonempty => ?_ obtain ⟨a, b, hab, H⟩ : ∃ a b : α, a < b ∧ Ioo a b ⊆ U := U_open.exists_Ioo_subset U_nonempty obtain ⟨x, xs, hx⟩ : ∃ x ∈ s, a < x ∧ x < b := h hab exact ⟨x, ⟨H hx, xs⟩⟩ theorem IsUpperSet.isClosed [OrderTopology α] [WellFoundedLT α] {s : Set α} (h : IsUpperSet s) : IsClosed s := by obtain rfl \| ⟨a, rfl⟩ := h.eq_empty_or_Ici exacts [isClosed_empty, isClosed_Ici] theorem IsLowerSet.isClosed [OrderTopology α] [WellFoundedGT α] {s : Set α} (h : IsLowerSet s) : IsClosed s := h.toDual.isClosed theorem IsLowerSet.isOpen [OrderTopology α] [WellFoundedLT α] {s : Set α} (h : IsLowerSet s) : IsOpen s := by simpa using h.compl.isClosed theorem IsUpperSet.isOpen [OrderTopology α] [WellFoundedGT α] {s : Set α} (h : IsUpperSet s) : IsOpen s := h.toDual.isOpen /-- A set in a nontrivial densely linear ordered type is dense in the sense of topology if and only if for any `a < b` there exists `c ∈ s`, `a < c < b`. Each implication requires less typeclass assumptions. -/ theorem dense_iff_exists_between [OrderTopology α] [DenselyOrdered α] [Nontrivial α] {s : Set α} : Dense s ↔ ∀ a b, a < b → ∃ c ∈ s, a < c ∧ c < b := ⟨fun h _ _ hab => h.exists_between hab, dense_of_exists_between⟩ /-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`, provided `a` is neither a bottom element nor a top element. -/ theorem mem_nhds_iff_exists_Ioo_subset' [OrderTopology α] {a : α} {s : Set α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := by constructor · intro h rcases exists_Ico_subset_of_mem_nhds h hu with ⟨u, au, hu⟩ rcases exists_Ioc_subset_of_mem_nhds h hl with ⟨l, la, hl⟩ exact ⟨l, u, ⟨la, au⟩, Ioc_union_Ico_eq_Ioo la au ▸ union_subset hl hu⟩ · rintro ⟨l, u, ha, h⟩ apply mem_of_superset (Ioo_mem_nhds ha.1 ha.2) h /-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`. -/ theorem mem_nhds_iff_exists_Ioo_subset [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α} {s : Set α} : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s := mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt a) theorem nhds_basis_Ioo' [OrderTopology α] {a : α} (hl : ∃ l, l < a) (hu : ∃ u, a < u) : (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 := ⟨fun s => (mem_nhds_iff_exists_Ioo_subset' hl hu).trans <\| by simp⟩ theorem nhds_basis_Ioo [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] (a : α) : (𝓝 a).HasBasis (fun b : α × α => b.1 < a ∧ a < b.2) fun b => Ioo b.1 b.2 := nhds_basis_Ioo' (exists_lt a) (exists_gt a) theorem Filter.Eventually.exists_Ioo_subset [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α} {p : α → Prop} (hp : ∀ᶠ x in 𝓝 a, p x) : ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ { x \| p x } := mem_nhds_iff_exists_Ioo_subset.1 hp theorem Dense.topology_eq_generateFrom [OrderTopology α] [DenselyOrdered α] {s : Set α} (hs : Dense s) : ‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_ refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_) · simp only [union_subset_iff, image_subset_iff] exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩ · rintro _ ⟨a, rfl \| rfl⟩ · rw [hs.Ioi_eq_biUnion] let _ := generateFrom (Ioi '' s ∪ Iio '' s) exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <\| .inl <\| mem_image_of_mem _ h.1 · rw [hs.Iio_eq_biUnion] let _ := generateFrom (Ioi '' s ∪ Iio '' s) exact isOpen_iUnion fun x ↦ isOpen_iUnion fun h ↦ .basic _ <\| .inr <\| mem_image_of_mem _ h.1 theorem PredOrder.hasBasis_nhds_Ioc_of_exists_gt [OrderTopology α] [PredOrder α] {a : α} (ha : ∃ u, a < u) : (𝓝 a).HasBasis (a < ·) (Set.Ico a ·) := PredOrder.nhdsGE_eq_nhds a ▸ nhdsGE_basis_of_exists_gt ha theorem PredOrder.hasBasis_nhds_Ioc [OrderTopology α] [PredOrder α] [NoMaxOrder α] {a : α} : (𝓝 a).HasBasis (a < ·) (Set.Ico a ·) := PredOrder.hasBasis_nhds_Ioc_of_exists_gt (exists_gt a) theorem SuccOrder.hasBasis_nhds_Ioc_of_exists_lt [OrderTopology α] [SuccOrder α] {a : α} (ha : ∃ l, l < a) : (𝓝 a).HasBasis (· < a) (Set.Ioc · a) := SuccOrder.nhdsLE_eq_nhds a ▸ nhdsLE_basis_of_exists_lt ha theorem SuccOrder.hasBasis_nhds_Ioc [OrderTopology α] [SuccOrder α] {a : α} [NoMinOrder α] : (𝓝 a).HasBasis (· < a) (Set.Ioc · a) := SuccOrder.hasBasis_nhds_Ioc_of_exists_lt (exists_lt a) variable (α) in /-- Let `α` be a densely ordered linear order with order topology. If `α` is a separable space, then it has second countable topology. Note that the "densely ordered" assumption cannot be dropped, see [double arrow space](https://topology.pi-base.org/spaces/S000093) for a counterexample. -/ theorem SecondCountableTopology.of_separableSpace_orderTopology [OrderTopology α] [DenselyOrdered α] [SeparableSpace α] : SecondCountableTopology α := by rcases exists_countable_dense α with ⟨s, hc, hd⟩ refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩ exact (hc.image _).union (hc.image _) /-- The set of points which are isolated on the right is countable when the space is second-countable. -/ theorem countable_setOf_covBy_right [OrderTopology α] [SecondCountableTopology α] : Set.Countable { x : α \| ∃ y, x ⋖ y } := by nontriviality α let s := { x : α \| ∃ y, x ⋖ y } have : ∀ x ∈ s, ∃ y, x ⋖ y := fun x => id choose! y hy using this have Hy : ∀ x z, x ∈ s → z < y x → z ≤ x := fun x z hx => (hy x hx).le_of_lt suffices H : ∀ a : Set α, IsOpen a → Set.Countable { x \| x ∈ s ∧ x ∈ a ∧ y x ∉ a } by have : s ⊆ ⋃ a ∈ countableBasis α, { x \| x ∈ s ∧ x ∈ a ∧ y x ∉ a } := fun x hx => by rcases (isBasis_countableBasis α).exists_mem_of_ne (hy x hx).ne with ⟨a, ab, xa, ya⟩ exact mem_iUnion₂.2 ⟨a, ab, hx, xa, ya⟩ refine Set.Countable.mono this ?_ refine Countable.biUnion (countable_countableBasis α) fun a ha => H _ ?_ exact isOpen_of_mem_countableBasis ha intro a ha suffices H : Set.Countable { x \| (x ∈ s ∧ x ∈ a ∧ y x ∉ a) ∧ ¬IsBot x } from H.of_diff (subsingleton_isBot α).countable simp only [and_assoc] let t := { x \| x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x } have : ∀ x ∈ t, ∃ z < x, Ioc z x ⊆ a := by intro x hx apply exists_Ioc_subset_of_mem_nhds (ha.mem_nhds hx.2.1) simpa only [IsBot, not_forall, not_le] using hx.right.right.right choose! z hz h'z using this have : PairwiseDisjoint t fun x => Ioc (z x) x := fun x xt x' x't hxx' => by rcases hxx'.lt_or_gt with (h' \| h') · refine disjoint_left.2 fun u ux ux' => xt.2.2.1 ?_ refine h'z x' x't ⟨ux'.1.trans_le (ux.2.trans (hy x xt.1).le), ?_⟩ by_contra! H exact lt_irrefl _ ((Hy _ _ xt.1 H).trans_lt h') · refine disjoint_left.2 fun u ux ux' => x't.2.2.1 ?_ refine h'z x xt ⟨ux.1.trans_le (ux'.2.trans (hy x' x't.1).le), ?_⟩ by_contra! H exact lt_irrefl _ ((Hy _ _ x't.1 H).trans_lt h') refine this.countable_of_isOpen (fun x hx => ?_) fun x hx => ⟨x, hz x hx, le_rfl⟩ suffices H : Ioc (z x) x = Ioo (z x) (y x) by rw [H] exact isOpen_Ioo exact Subset.antisymm (Ioc_subset_Ioo_right (hy x hx.1).lt) fun u hu => ⟨hu.1, Hy _ _ hx.1 hu.2⟩ /-- The set of points which are isolated on the left is countable when the space is second-countable. -/ theorem countable_setOf_covBy_left [OrderTopology α] [SecondCountableTopology α] : Set.Countable { x : α \| ∃ y, y ⋖ x } := by convert countable_setOf_covBy_right (α := αᵒᵈ) using 5 exact toDual_covBy_toDual_iff.symm /-- The set of points which are isolated on the left is countable when the space is second-countable. -/ theorem countable_of_isolated_left' [OrderTopology α] [SecondCountableTopology α] : Set.Countable { x : α \| ∃ y, y < x ∧ Ioo y x = ∅ } := by simpa only [← covBy_iff_Ioo_eq] using countable_setOf_covBy_left /-- Consider a disjoint family of intervals `(x, y)` with `x < y` in a second-countable space. Then the family is countable. This is not a straightforward consequence of second-countability as some of these intervals might be empty (but in fact this can happen only for countably many of them). -/ theorem Set.PairwiseDisjoint.countable_of_Ioo [OrderTopology α] [SecondCountableTopology α] {y : α → α} {s : Set α} (h : PairwiseDisjoint s fun x => Ioo x (y x)) (h' : ∀ x ∈ s, x < y x) : s.Countable := have : (s \ { x \| ∃ y, x ⋖ y }).Countable := (h.subset diff_subset).countable_of_isOpen (fun _ _ => isOpen_Ioo) fun x hx => (h' _ hx.1).exists_lt_lt (mt (Exists.intro (y x)) hx.2) this.of_diff countable_setOf_covBy_right /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. We give here a version relative to a set `t`. -/ theorem countable_image_lt_image_Ioi_within [OrderTopology α] [LinearOrder β] [SecondCountableTopology α] (t : Set β) (f : β → α) : Set.Countable {x ∈ t \| ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y} := by /- If the values of `f` are separated above on the right of `x`, there is an interval `(f x, z x)` which is not reached by `f`. This gives a family of disjoint open intervals in `α`. Such a family can only be countable as `α` is second-countable. -/ nontriviality β have : Nonempty α := Nonempty.map f (by infer_instance) let s := {x ∈ t \| ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y} have : ∀ x, x ∈ s → ∃ z, f x < z ∧ ∀ y ∈ t, x < y → z ≤ f y := fun x hx ↦ hx.2 -- choose `z x` such that `f` does not take the values in `(f x, z x)`. choose! z hz using this have I : InjOn f s := by apply StrictMonoOn.injOn intro x hx y hy hxy calc f x < z x := (hz x hx).1 _ ≤ f y := (hz x hx).2 y hy.1 hxy -- show that `f s` is countable by arguing that a disjoint family of disjoint open intervals -- (the intervals `(f x, z x)`) is at most countable. have fs_count : (f '' s).Countable := by have A : (f '' s).PairwiseDisjoint fun x => Ioo x (z (invFunOn f s x)) := by rintro _ ⟨u, us, rfl⟩ _ ⟨v, vs, rfl⟩ huv wlog hle : u ≤ v generalizing u v · exact (this v vs u us huv.symm (le_of_not_ge hle)).symm have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv) apply disjoint_iff_forall_ne.2 rintro a ha b hb rfl simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v vs.1 hlt)).trans hb.1) apply Set.PairwiseDisjoint.countable_of_Ioo A rintro _ ⟨y, ys, rfl⟩ simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1 exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(x, ∞)` is separated above from `f x` is countable. -/ theorem countable_image_lt_image_Ioi [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α] : Set.Countable {x \| ∃ z, f x < z ∧ ∀ y, x < y → z ≤ f y} := by simpa using countable_image_lt_image_Ioi_within univ f /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(x, ∞)` is separated below from `f x` is countable. We give here a version relative to a set `t`. -/ theorem countable_image_gt_image_Ioi_within [OrderTopology α] [LinearOrder β] [SecondCountableTopology α] (t : Set β) (f : β → α) : Set.Countable {x ∈ t \| ∃ z, z < f x ∧ ∀ y ∈ t, x < y → f y ≤ z} := countable_image_lt_image_Ioi_within (α := αᵒᵈ) t f /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(x, ∞)` is separated below from `f x` is countable. -/ theorem countable_image_gt_image_Ioi [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α] : Set.Countable {x \| ∃ z, z < f x ∧ ∀ y, x < y → f y ≤ z} := countable_image_lt_image_Ioi (α := αᵒᵈ) f /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(-∞, x)` is separated above from `f x` is countable. We give here a version relative to a set `t`. -/ theorem countable_image_lt_image_Iio_within [OrderTopology α] [LinearOrder β] [SecondCountableTopology α] (t : Set β) (f : β → α) : Set.Countable {x ∈ t \| ∃ z, f x < z ∧ ∀ y ∈ t, y < x → z ≤ f y} := countable_image_lt_image_Ioi_within (β := βᵒᵈ) t f /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(-∞, x)` is separated above from `f x` is countable. -/ theorem countable_image_lt_image_Iio [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α] : Set.Countable {x \| ∃ z, f x < z ∧ ∀ y, y < x → z ≤ f y} := countable_image_lt_image_Ioi (β := βᵒᵈ) f /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(-∞, x)` is separated below from `f x` is countable. We give here a version relative to a set `t`. -/ theorem countable_image_gt_image_Iio_within [OrderTopology α] [LinearOrder β] [SecondCountableTopology α] (t : Set β) (f : β → α) : Set.Countable {x ∈ t \| ∃ z, z < f x ∧ ∀ y ∈ t, y < x → f y ≤ z} := countable_image_lt_image_Ioi_within (α := αᵒᵈ) (β := βᵒᵈ) t f /-- For a function taking values in a second countable space, the set of points `x` for which the image under `f` of `(-∞, x)` is separated below from `f x` is countable. -/ theorem countable_image_gt_image_Iio [OrderTopology α] [LinearOrder β] (f : β → α) [SecondCountableTopology α] : Set.Countable {x \| ∃ z, z < f x ∧ ∀ y, y < x → f y ≤ z} := countable_image_lt_image_Ioi (α := αᵒᵈ) (β := βᵒᵈ) f instance instIsCountablyGenerated_atTop [OrderTopology α] [SecondCountableTopology α] : IsCountablyGenerated (atTop : Filter α) := by by_cases h : ∃ (x : α), IsTop x · rcases h with ⟨x, hx⟩ rw [atTop_eq_pure_of_isTop hx] exact isCountablyGenerated_pure x · rcases exists_countable_basis α with ⟨b, b_count, b_ne, hb⟩ have : Countable b := by exact Iff.mpr countable_coe_iff b_count have A : ∀ (s : b), ∃ (x : α), x ∈ (s : Set α) := by intro s have : (s : Set α) ≠ ∅ := by intro H apply b_ne convert s.2 exact H.symm exact Iff.mp notMem_singleton_empty this choose a ha using A have : (atTop : Filter α) = (generate (Ici '' (range a))) := by apply atTop_eq_generate_of_not_bddAbove intro ⟨x, hx⟩ simp only [IsTop, not_exists, not_forall, not_le] at h rcases h x with ⟨y, hy⟩ obtain ⟨s, sb, -, hs⟩ : ∃ s, s ∈ b ∧ y ∈ s ∧ s ⊆ Ioi x := hb.exists_subset_of_mem_open hy isOpen_Ioi have I : a ⟨s, sb⟩ ≤ x := hx (mem_range_self _) have J : x < a ⟨s, sb⟩ := hs (ha ⟨s, sb⟩) exact lt_irrefl _ (I.trans_lt J) rw [this] exact ⟨_, (countable_range _).image _, rfl⟩ instance instIsCountablyGenerated_atBot [OrderTopology α] [SecondCountableTopology α] : IsCountablyGenerated (atBot : Filter α) := @instIsCountablyGenerated_atTop αᵒᵈ _ _ _ _ section Pi /-! ### Intervals in `Π i, X i` belong to `𝓝 x` For each lemma `pi_Ixx_mem_nhds` we add a non-dependent version `pi_Ixx_mem_nhds'` because sometimes Lean fails to unify different instances while trying to apply the dependent version to, e.g., `ι → ℝ`. -/ variable [OrderTopology α] {ι : Type} {X : ι → Type} [Finite ι] [∀ i, LinearOrder (X i)] [∀ i, TopologicalSpace (X i)] [∀ i, OrderTopology (X i)] {a b x : ∀ i, X i} {a' b' x' : ι → α} theorem pi_Iic_mem_nhds (ha : ∀ i, x i < a i) : Iic a ∈ 𝓝 x := pi_univ_Iic a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Iic_mem_nhds (ha _) theorem pi_Iic_mem_nhds' (ha : ∀ i, x' i < a' i) : Iic a' ∈ 𝓝 x' := pi_Iic_mem_nhds ha theorem pi_Ici_mem_nhds (ha : ∀ i, a i < x i) : Ici a ∈ 𝓝 x := pi_univ_Ici a ▸ set_pi_mem_nhds (Set.toFinite _) fun _ _ => Ici_mem_nhds (ha _) theorem pi_Ici_mem_nhds' (ha : ∀ i, a' i < x' i) : Ici a' ∈ 𝓝 x' := pi_Ici_mem_nhds ha theorem pi_Icc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Icc a b ∈ 𝓝 x := pi_univ_Icc a b ▸ set_pi_mem_nhds finite_univ fun _ _ => Icc_mem_nhds (ha _) (hb _) theorem pi_Icc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Icc a' b' ∈ 𝓝 x' := pi_Icc_mem_nhds ha hb variable [Nonempty ι] theorem pi_Iio_mem_nhds (ha : ∀ i, x i < a i) : Iio a ∈ 𝓝 x := mem_of_superset (set_pi_mem_nhds finite_univ fun i _ ↦ Iio_mem_nhds (ha i)) (pi_univ_Iio_subset a) theorem pi_Iio_mem_nhds' (ha : ∀ i, x' i < a' i) : Iio a' ∈ 𝓝 x' := pi_Iio_mem_nhds ha theorem pi_Ioi_mem_nhds (ha : ∀ i, a i < x i) : Ioi a ∈ 𝓝 x := pi_Iio_mem_nhds (X := fun i => (X i)ᵒᵈ) ha theorem pi_Ioi_mem_nhds' (ha : ∀ i, a' i < x' i) : Ioi a' ∈ 𝓝 x' := pi_Ioi_mem_nhds ha theorem pi_Ioc_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioc a b ∈ 𝓝 x := by refine mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => ?_) (pi_univ_Ioc_subset a b) exact Ioc_mem_nhds (ha i) (hb i) theorem pi_Ioc_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioc a' b' ∈ 𝓝 x' := pi_Ioc_mem_nhds ha hb theorem pi_Ico_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ico a b ∈ 𝓝 x := by refine mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => ?_) (pi_univ_Ico_subset a b) exact Ico_mem_nhds (ha i) (hb i) theorem pi_Ico_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ico a' b' ∈ 𝓝 x' := pi_Ico_mem_nhds ha hb theorem pi_Ioo_mem_nhds (ha : ∀ i, a i < x i) (hb : ∀ i, x i < b i) : Ioo a b ∈ 𝓝 x := by refine mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => ?_) (pi_univ_Ioo_subset a b) exact Ioo_mem_nhds (ha i) (hb i) theorem pi_Ioo_mem_nhds' (ha : ∀ i, a' i < x' i) (hb : ∀ i, x' i < b' i) : Ioo a' b' ∈ 𝓝 x' := pi_Ioo_mem_nhds ha hb end Pi end OrderTopology end LinearOrder section ConditionallyCompleteLinearOrder variable {X : Type} [ConditionallyCompleteLinearOrder X] [TopologicalSpace X] [OrderTopology X] variable {Y : Type*} [ConditionallyCompleteLinearOrder Y] [TopologicalSpace Y] [OrderTopology Y] variable [DenselyOrdered X] {f : X → Y} {x : X} /-- An order-theoretically left-continuous function is topologically left-continuous, assuming the function is between conditionally complete linear orders with order topologies, and the domain is densely ordered. -/ lemma LeftOrdContinuous.continuousWithinAt_Iic (hf : LeftOrdContinuous f) : ContinuousWithinAt f (Iic x) x := by rw [ContinuousWithinAt, OrderTopology.topology_eq_generate_intervals (α := Y)] simp_rw [TopologicalSpace.tendsto_nhds_generateFrom_iff, mem_nhdsWithin] rintro V ⟨z, rfl \| rfl⟩ hxz -- The case `V = Ioi z`. · obtain ⟨_, ⟨a, hax, rfl⟩, hza⟩ := (lt_isLUB_iff <\| hf isLUB_Iio).mp hxz exact ⟨Ioi a, isOpen_Ioi, hax, fun b hab ↦ hza.trans_le <\| hf.mono hab.1.le⟩ -- The case `V = Iio z`. · exact ⟨univ, isOpen_univ, trivial, fun a ha ↦ (hf.mono ha.2).trans_lt hxz⟩ /-- An order-theoretically right-continuous function is topologically right-continuous, assuming the function is between conditionally complete linear orders with order topologies, and the domain is densely ordered. -/ lemma RightOrdContinuous.continuousWithinAt_Ici (hf : RightOrdContinuous f) : ContinuousWithinAt f (Ici x) x := hf.orderDual.continuousWithinAt_Iic end ConditionallyCompleteLinearOrder
.lake/packages/mathlib/Mathlib/Topology/Order/Priestley.lean	import Mathlib.Order.UpperLower.Basic import Mathlib.Topology.Connected.TotallyDisconnected /-! # Priestley spaces This file defines Priestley spaces. A Priestley space is an ordered compact topological space such that any two distinct points can be separated by a clopen upper set. ## Main declarations * `PriestleySpace`: Prop-valued mixin stating the Priestley separation axiom: Any two distinct points can be separated by a clopen upper set. ## Implementation notes We do not include compactness in the definition, so a Priestley space is to be declared as follows: `[Preorder α] [TopologicalSpace α] [CompactSpace α] [PriestleySpace α]` ## References * [Wikipedia, Priestley space](https://en.wikipedia.org/wiki/Priestley_space) * [Davey, Priestley Introduction to Lattices and Order][davey_priestley] -/ open Set variable {α : Type} /-- A Priestley space is an ordered topological space such that any two distinct points can be separated by a clopen upper set. Compactness is often assumed, but we do not include it here. -/ class PriestleySpace (α : Type) [Preorder α] [TopologicalSpace α] : Prop where priestley {x y : α} : ¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U variable [TopologicalSpace α] section Preorder variable [Preorder α] [PriestleySpace α] {x y : α} theorem exists_isClopen_upper_of_not_le : ¬x ≤ y → ∃ U : Set α, IsClopen U ∧ IsUpperSet U ∧ x ∈ U ∧ y ∉ U := PriestleySpace.priestley theorem exists_isClopen_lower_of_not_le (h : ¬x ≤ y) : ∃ U : Set α, IsClopen U ∧ IsLowerSet U ∧ x ∉ U ∧ y ∈ U := let ⟨U, hU, hU', hx, hy⟩ := exists_isClopen_upper_of_not_le h ⟨Uᶜ, hU.compl, hU'.compl, Classical.not_not.2 hx, hy⟩ end Preorder section PartialOrder variable [PartialOrder α] [PriestleySpace α] {x y : α} theorem exists_isClopen_upper_or_lower_of_ne (h : x ≠ y) : ∃ U : Set α, IsClopen U ∧ (IsUpperSet U ∨ IsLowerSet U) ∧ x ∈ U ∧ y ∉ U := by obtain h \| h := h.not_le_or_not_ge · exact (exists_isClopen_upper_of_not_le h).imp fun _ ↦ And.imp_right <\| And.imp_left Or.inl · obtain ⟨U, hU, hU', hy, hx⟩ := exists_isClopen_lower_of_not_le h exact ⟨U, hU, Or.inr hU', hx, hy⟩ -- See note [lower instance priority] instance (priority := 100) PriestleySpace.toTotallySeparatedSpace : TotallySeparatedSpace α where isTotallySeparated_univ _ _ _ _ h := (exists_isClopen_upper_or_lower_of_ne h).elim fun U ⟨hU, _, hx, hy⟩ => ⟨U, Uᶜ, hU.isOpen, hU.compl.isOpen, hx, hy, union_compl_self U ▸ subset_rfl, disjoint_compl_right⟩ end PartialOrder
.lake/packages/mathlib/Mathlib/Topology/Order/ProjIcc.lean	import Mathlib.Order.Interval.Set.ProjIcc import Mathlib.Topology.Order.Basic /-! # Projection onto a closed interval In this file we prove that the projection `Set.projIcc f a b h` is a quotient map, and use it to show that `Set.IccExtend h f` is continuous if and only if `f` is continuous. -/ open Set Filter Topology variable {α β γ : Type*} [LinearOrder α] {a b c : α} {h : a ≤ b} protected theorem Filter.Tendsto.IccExtend (f : γ → Icc a b → β) {la : Filter α} {lb : Filter β} {lc : Filter γ} (hf : Tendsto ↿f (lc ×ˢ la.map (projIcc a b h)) lb) : Tendsto (↿(IccExtend h ∘ f)) (lc ×ˢ la) lb := hf.comp <\| tendsto_id.prodMap tendsto_map variable [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ] @[continuity, fun_prop] theorem continuous_projIcc : Continuous (projIcc a b h) := (continuous_const.max <\| continuous_const.min continuous_id).subtype_mk _ theorem isQuotientMap_projIcc : IsQuotientMap (projIcc a b h) := isQuotientMap_iff.2 ⟨projIcc_surjective h, fun s => ⟨fun hs => hs.preimage continuous_projIcc, fun hs => ⟨_, hs, by ext; simp⟩⟩⟩ @[simp] theorem continuous_IccExtend_iff {f : Icc a b → β} : Continuous (IccExtend h f) ↔ Continuous f := isQuotientMap_projIcc.continuous_iff.symm /-- See Note [continuity lemma statement]. -/ @[fun_prop] protected theorem Continuous.IccExtend {f : γ → Icc a b → β} {g : γ → α} (hf : Continuous ↿f) (hg : Continuous g) : Continuous fun a => IccExtend h (f a) (g a) := show Continuous (↿f ∘ fun x => (x, projIcc a b h (g x))) from hf.comp <\| continuous_id.prodMk <\| continuous_projIcc.comp hg /-- A useful special case of `Continuous.IccExtend`. -/ @[continuity, fun_prop] protected theorem Continuous.Icc_extend' {f : Icc a b → β} (hf : Continuous f) : Continuous (IccExtend h f) := hf.comp continuous_projIcc @[fun_prop] theorem ContinuousAt.IccExtend {x : γ} (f : γ → Icc a b → β) {g : γ → α} (hf : ContinuousAt ↿f (x, projIcc a b h (g x))) (hg : ContinuousAt g x) : ContinuousAt (fun a => IccExtend h (f a) (g a)) x := show ContinuousAt (↿f ∘ fun x => (x, projIcc a b h (g x))) x from ContinuousAt.comp hf <\| continuousAt_id.prodMk <\| continuous_projIcc.continuousAt.comp hg
.lake/packages/mathlib/Mathlib/Topology/Order/PartialSups.lean	import Mathlib.Topology.Order.Lattice import Mathlib.Order.PartialSups /-! # Continuity of `partialSups` In this file we prove that `partialSups` of a sequence of continuous functions is continuous as well as versions for `Filter.Tendsto`, `ContinuousAt`, `ContinuousWithinAt`, and `ContinuousOn`. -/ open Filter open scoped Topology variable {L : Type} [SemilatticeSup L] [TopologicalSpace L] [ContinuousSup L] namespace Filter.Tendsto variable {α : Type} {l : Filter α} {f : ℕ → α → L} {g : ℕ → L} {n : ℕ} protected lemma partialSups (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) : Tendsto (partialSups f n) l (𝓝 (partialSups g n)) := by simp only [partialSups_eq_sup'_range] refine finset_sup'_nhds _ ?_ simpa [Nat.lt_succ_iff] protected lemma partialSups_apply (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) : Tendsto (fun a ↦ partialSups (f · a) n) l (𝓝 (partialSups g n)) := by simpa only [← Pi.partialSups_apply] using Tendsto.partialSups hf end Filter.Tendsto variable {X : Type*} [TopologicalSpace X] {f : ℕ → X → L} {n : ℕ} {s : Set X} {x : X} protected lemma ContinuousAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousAt (f k) x) : ContinuousAt (fun a ↦ partialSups (f · a) n) x := Tendsto.partialSups_apply hf protected lemma ContinuousAt.partialSups (hf : ∀ k ≤ n, ContinuousAt (f k) x) : ContinuousAt (partialSups f n) x := by simpa only [← Pi.partialSups_apply] using ContinuousAt.partialSups_apply hf protected lemma ContinuousWithinAt.partialSups_apply (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) : ContinuousWithinAt (fun a ↦ partialSups (f · a) n) s x := Tendsto.partialSups_apply hf protected lemma ContinuousWithinAt.partialSups (hf : ∀ k ≤ n, ContinuousWithinAt (f k) s x) : ContinuousWithinAt (partialSups f n) s x := by simpa only [← Pi.partialSups_apply] using ContinuousWithinAt.partialSups_apply hf protected lemma ContinuousOn.partialSups_apply (hf : ∀ k ≤ n, ContinuousOn (f k) s) : ContinuousOn (fun a ↦ partialSups (f · a) n) s := fun x hx ↦ ContinuousWithinAt.partialSups_apply fun k hk ↦ hf k hk x hx protected lemma ContinuousOn.partialSups (hf : ∀ k ≤ n, ContinuousOn (f k) s) : ContinuousOn (partialSups f n) s := fun x hx ↦ ContinuousWithinAt.partialSups fun k hk ↦ hf k hk x hx protected lemma Continuous.partialSups_apply (hf : ∀ k ≤ n, Continuous (f k)) : Continuous (fun a ↦ partialSups (f · a) n) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.partialSups_apply fun k hk ↦ (hf k hk).continuousAt protected lemma Continuous.partialSups (hf : ∀ k ≤ n, Continuous (f k)) : Continuous (partialSups f n) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.partialSups fun k hk ↦ (hf k hk).continuousAt
.lake/packages/mathlib/Mathlib/Topology/Order/IsLUB.lean	import Mathlib.Topology.Order.LeftRightNhds /-! # Properties of LUB and GLB in an order topology -/ open Set Filter TopologicalSpace Topology Function open OrderDual (toDual ofDual) variable {α γ : Type} section OrderTopology variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] theorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : ∃ᶠ x in 𝓝[≤] a, x ∈ s := by rcases hs with ⟨a', ha'⟩ intro h rcases (ha.1 ha').eq_or_lt with (rfl \| ha'a) · exact h.self_of_nhdsWithin le_rfl ha' · rcases (mem_nhdsLE_iff_exists_Ioc_subset' ha'a).1 h with ⟨b, hba, hb⟩ rcases ha.exists_between hba with ⟨b', hb's, hb'⟩ exact hb hb' hb's theorem IsLUB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : ∃ᶠ x in 𝓝 a, x ∈ s := (ha.frequently_mem hs).filter_mono inf_le_left theorem IsGLB.frequently_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : ∃ᶠ x in 𝓝[≥] a, x ∈ s := IsLUB.frequently_mem (α := αᵒᵈ) ha hs theorem IsGLB.frequently_nhds_mem {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : ∃ᶠ x in 𝓝 a, x ∈ s := (ha.frequently_mem hs).filter_mono inf_le_left theorem IsLUB.mem_closure {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : a ∈ closure s := (ha.frequently_nhds_mem hs).mem_closure theorem IsGLB.mem_closure {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : a ∈ closure s := (ha.frequently_nhds_mem hs).mem_closure theorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) : NeBot (𝓝[s] a) := mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs) theorem IsGLB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) : NeBot (𝓝[s] a) := IsLUB.nhdsWithin_neBot (α := αᵒᵈ) ha hs theorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f) [NeBot (f ⊓ 𝓝 a)] : IsLUB s a := ⟨hsa, fun b hb => not_lt.1 fun hba => have : s ∩ { a \| b < a } ∈ f ⊓ 𝓝 a := inter_mem_inf hsf (IsOpen.mem_nhds (isOpen_lt' _) hba) let ⟨_x, ⟨hxs, hxb⟩⟩ := Filter.nonempty_of_mem this have : b < b := lt_of_lt_of_le hxb <\| hb hxs lt_irrefl b this⟩ theorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) : IsLUB s a := by rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf exact isLUB_of_mem_nhds hsa (mem_principal_self s) theorem isGLB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ lowerBounds s) (hsf : s ∈ f) [NeBot (f ⊓ 𝓝 a)] : IsGLB s a := isLUB_of_mem_nhds (α := αᵒᵈ) hsa hsf theorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) : IsGLB s a := isLUB_of_mem_closure (α := αᵒᵈ) hsa hsf theorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by rintro _ ⟨x, hx, rfl⟩ replace ha := ha.inter_Ici_of_mem hx haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩ refine ge_of_tendsto (hb.mono_left (nhdsWithin_mono a (inter_subset_left (t := Ici x)))) ?_ exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2 -- For a version of this theorem in which the convergence considered on the domain `α` is as `x : α` -- tends to infinity, rather than tending to a point `x` in `α`, see `isLUB_of_tendsto_atTop` theorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b := haveI := ha.nhdsWithin_neBot hs ⟨ha.mem_upperBounds_of_tendsto hf hb, fun _b' hb' => le_of_tendsto hb (mem_of_superset self_mem_nhdsWithin fun _ hx => hb' <\| mem_image_of_mem _ hx)⟩ theorem IsGLB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsGLB s a) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) := IsLUB.mem_upperBounds_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual ha hb -- For a version of this theorem in which the convergence considered on the domain `α` is as -- `x : α` tends to negative infinity, rather than tending to a point `x` in `α`, see -- `isGLB_of_tendsto_atBot` theorem IsGLB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) : IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b := IsLUB.isLUB_of_tendsto (α := αᵒᵈ) (γ := γᵒᵈ) hf.dual theorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) := IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb theorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsGLB (f '' s) b := IsLUB.isLUB_of_tendsto (γ := γᵒᵈ) hf ha hs hb theorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb theorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a) (hs : s.Nonempty) (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b := IsGLB.isGLB_of_tendsto (γ := γᵒᵈ) hf ha hs hb theorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) (sc : IsClosed s) : a ∈ s := sc.closure_subset <\| ha.mem_closure hs alias IsClosed.isLUB_mem := IsLUB.mem_of_isClosed theorem IsGLB.mem_of_isClosed {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) (sc : IsClosed s) : a ∈ s := sc.closure_subset <\| ha.mem_closure hs alias IsClosed.isGLB_mem := IsGLB.mem_of_isClosed theorem isLUB_iff_of_subset_of_subset_closure {α : Type} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) {x : α} : IsLUB s x ↔ IsLUB t x := isLUB_congr <\| (upperBounds_closure (s := s) ▸ upperBounds_mono_set hts).antisymm <\| upperBounds_mono_set hst theorem isGLB_iff_of_subset_of_subset_closure {α : Type} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {s t : Set α} (hst : s ⊆ t) (hts : t ⊆ closure s) {x : α} : IsGLB s x ↔ IsGLB t x := isLUB_iff_of_subset_of_subset_closure (α := αᵒᵈ) hst hts theorem Dense.isLUB_inter_iff {α : Type} [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {s t : Set α} (hs : Dense s) (ht : IsOpen t) {x : α} : IsLUB (t ∩ s) x ↔ IsLUB t x := isLUB_iff_of_subset_of_subset_closure (by simp) <\| hs.open_subset_closure_inter ht theorem Dense.isGLB_inter_iff {α : Type} [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {s t : Set α} (hs : Dense s) (ht : IsOpen t) {x : α} : IsGLB (t ∩ s) x ↔ IsGLB t x := hs.isLUB_inter_iff (α := αᵒᵈ) ht /-! ### Existence of sequences tending to `sInf` or `sSup` of a given set -/ theorem IsLUB.exists_seq_strictMono_tendsto_of_notMem {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (notMem : x ∉ t) (ht : t.Nonempty) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by obtain ⟨v, hvx, hvt⟩ := exists_seq_forall_of_frequently (htx.frequently_mem ht) replace hvx := hvx.mono_right nhdsWithin_le_nhds have hvx' : ∀ {n}, v n < x := (htx.1 (hvt _)).lt_of_ne (ne_of_mem_of_not_mem (hvt _) notMem) have : ∀ k, ∀ᶠ l in atTop, v k < v l := fun k => hvx.eventually (lt_mem_nhds hvx') choose N hN hvN using fun k => ((eventually_gt_atTop k).and (this k)).exists refine ⟨fun k => v (N^[k] 0), strictMono_nat_of_lt_succ fun _ => ?_, fun _ => hvx', hvx.comp (strictMono_nat_of_lt_succ fun _ => ?_).tendsto_atTop, fun _ => hvt _⟩ · rw [iterate_succ_apply']; exact hvN _ · rw [iterate_succ_apply']; exact hN _ @[deprecated (since := "2025-05-23")] alias IsLUB.exists_seq_strictMono_tendsto_of_not_mem := IsLUB.exists_seq_strictMono_tendsto_of_notMem theorem IsLUB.exists_seq_monotone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)] (htx : IsLUB t x) (ht : t.Nonempty) : ∃ u : ℕ → α, Monotone u ∧ (∀ n, u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := by by_cases h : x ∈ t · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩ · rcases htx.exists_seq_strictMono_tendsto_of_notMem h ht with ⟨u, hu⟩ exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩ theorem exists_seq_strictMono_tendsto' {α : Type} [LinearOrder α] [TopologicalSpace α] [DenselyOrdered α] [OrderTopology α] [FirstCountableTopology α] {x y : α} (hy : y < x) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x) := by have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim have ht : Set.Nonempty (Ioo y x) := nonempty_Ioo.2 hy rcases (isLUB_Ioo hy).exists_seq_strictMono_tendsto_of_notMem hx ht with ⟨u, hu⟩ exact ⟨u, hu.1, hu.2.2.symm⟩ theorem exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝 x) := by obtain ⟨y, hy⟩ : ∃ y, y < x := exists_lt x rcases exists_seq_strictMono_tendsto' hy with ⟨u, hu_mono, hu_mem, hux⟩ exact ⟨u, hu_mono, fun n => (hu_mem n).2, hux⟩ theorem exists_seq_strictMono_tendsto_nhdsWithin [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n < x) ∧ Tendsto u atTop (𝓝[<] x) := let ⟨u, hu, hx, h⟩ := exists_seq_strictMono_tendsto x ⟨u, hu, hx, tendsto_nhdsWithin_mono_right (range_subset_iff.2 hx) <\| tendsto_nhdsWithin_range.2 h⟩ theorem exists_seq_tendsto_sSup {α : Type} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty) (hS' : BddAbove S) : ∃ u : ℕ → α, Monotone u ∧ Tendsto u atTop (𝓝 (sSup S)) ∧ ∀ n, u n ∈ S := by rcases (isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with ⟨u, hu⟩ exact ⟨u, hu.1, hu.2.2⟩ theorem Dense.exists_seq_strictMono_tendsto_of_lt [DenselyOrdered α] [FirstCountableTopology α] {s : Set α} (hs : Dense s) {x y : α} (hy : y < x) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ (Ioo y x ∩ s)) ∧ Tendsto u atTop (𝓝 x) := by have hnonempty : (Ioo y x ∩ s).Nonempty := by obtain ⟨z, hyz, hzx⟩ := hs.exists_between hy exact ⟨z, mem_inter hzx hyz⟩ have hx : IsLUB (Ioo y x ∩ s) x := hs.isLUB_inter_iff isOpen_Ioo \|>.mpr <\| isLUB_Ioo hy apply hx.exists_seq_strictMono_tendsto_of_notMem (by simp) hnonempty \|>.imp simp_all theorem Dense.exists_seq_strictMono_tendsto [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] {s : Set α} (hs : Dense s) (x : α) : ∃ u : ℕ → α, StrictMono u ∧ (∀ n, u n ∈ (Iio x ∩ s)) ∧ Tendsto u atTop (𝓝 x) := by obtain ⟨y, hy⟩ := exists_lt x apply hs.exists_seq_strictMono_tendsto_of_lt (exists_lt x).choose_spec \|>.imp simp_all theorem DenseRange.exists_seq_strictMono_tendsto_of_lt {β : Type} [LinearOrder β] [DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f) (hmono : Monotone f) (hlt : y < x) : ∃ u : ℕ → β, StrictMono u ∧ (∀ n, f (u n) ∈ Ioo y x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := by rcases Dense.exists_seq_strictMono_tendsto_of_lt hf hlt with ⟨u, hu, huyxf, hlim⟩ have huyx (n : ℕ) : u n ∈ Ioo y x := (huyxf n).1 have huf (n : ℕ) : u n ∈ range f := (huyxf n).2 choose v hv using huf obtain rfl : f ∘ v = u := funext hv exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <\| hu hlt, huyx, hlim⟩ theorem DenseRange.exists_seq_strictMono_tendsto {β : Type} [LinearOrder β] [DenselyOrdered α] [NoMinOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f) (x : α) : ∃ u : ℕ → β, StrictMono u ∧ (∀ n, f (u n) ∈ Iio x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := by rcases Dense.exists_seq_strictMono_tendsto hf x with ⟨u, hu, huxf, hlim⟩ have hux (n : ℕ) : u n ∈ Iio x := (huxf n).1 have huf (n : ℕ) : u n ∈ range f := (huxf n).2 choose v hv using huf obtain rfl : f ∘ v = u := funext hv exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <\| hu hlt, hux, hlim⟩ theorem IsGLB.exists_seq_strictAnti_tendsto_of_notMem {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (notMem : x ∉ t) (ht : t.Nonempty) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := IsLUB.exists_seq_strictMono_tendsto_of_notMem (α := αᵒᵈ) htx notMem ht @[deprecated (since := "2025-05-23")] alias IsGLB.exists_seq_strictAnti_tendsto_of_not_mem := IsGLB.exists_seq_strictAnti_tendsto_of_notMem theorem IsGLB.exists_seq_antitone_tendsto {t : Set α} {x : α} [IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (ht : t.Nonempty) : ∃ u : ℕ → α, Antitone u ∧ (∀ n, x ≤ u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t := IsLUB.exists_seq_monotone_tendsto (α := αᵒᵈ) htx ht theorem exists_seq_strictAnti_tendsto' [DenselyOrdered α] [FirstCountableTopology α] {x y : α} (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x) := by simpa using exists_seq_strictMono_tendsto' (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) theorem exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) := exists_seq_strictMono_tendsto (α := αᵒᵈ) x theorem exists_seq_strictAnti_tendsto_nhdsWithin [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝[>] x) := exists_seq_strictMono_tendsto_nhdsWithin (α := αᵒᵈ) _ theorem exists_seq_strictAnti_strictMono_tendsto [DenselyOrdered α] [FirstCountableTopology α] {x y : α} (h : x < y) : ∃ u v : ℕ → α, StrictAnti u ∧ StrictMono v ∧ (∀ k, u k ∈ Ioo x y) ∧ (∀ l, v l ∈ Ioo x y) ∧ (∀ k l, u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y) := by rcases exists_seq_strictAnti_tendsto' h with ⟨u, hu_anti, hu_mem, hux⟩ rcases exists_seq_strictMono_tendsto' (hu_mem 0).2 with ⟨v, hv_mono, hv_mem, hvy⟩ exact ⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩, fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩ theorem exists_seq_tendsto_sInf {α : Type} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [FirstCountableTopology α] {S : Set α} (hS : S.Nonempty) (hS' : BddBelow S) : ∃ u : ℕ → α, Antitone u ∧ Tendsto u atTop (𝓝 (sInf S)) ∧ ∀ n, u n ∈ S := exists_seq_tendsto_sSup (α := αᵒᵈ) hS hS' theorem Dense.exists_seq_strictAnti_tendsto_of_lt [DenselyOrdered α] [FirstCountableTopology α] {s : Set α} (hs : Dense s) {x y : α} (hy : x < y) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ (Ioo x y ∩ s)) ∧ Tendsto u atTop (𝓝 x) := by simpa using hs.exists_seq_strictMono_tendsto_of_lt (α := αᵒᵈ) (OrderDual.toDual_lt_toDual.2 hy) theorem Dense.exists_seq_strictAnti_tendsto [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] {s : Set α} (hs : Dense s) (x : α) : ∃ u : ℕ → α, StrictAnti u ∧ (∀ n, u n ∈ (Ioi x ∩ s)) ∧ Tendsto u atTop (𝓝 x) := hs.exists_seq_strictMono_tendsto (α := αᵒᵈ) x theorem DenseRange.exists_seq_strictAnti_tendsto_of_lt {β : Type} [LinearOrder β] [DenselyOrdered α] [FirstCountableTopology α] {f : β → α} {x y : α} (hf : DenseRange f) (hmono : Monotone f) (hlt : x < y) : ∃ u : ℕ → β, StrictAnti u ∧ (∀ n, f (u n) ∈ Ioo x y) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := by simpa using hf.exists_seq_strictMono_tendsto_of_lt (α := αᵒᵈ) (β := βᵒᵈ) hmono.dual (OrderDual.toDual_lt_toDual.2 hlt) theorem DenseRange.exists_seq_strictAnti_tendsto {β : Type} [LinearOrder β] [DenselyOrdered α] [NoMaxOrder α] [FirstCountableTopology α] {f : β → α} (hf : DenseRange f) (hmono : Monotone f) (x : α) : ∃ u : ℕ → β, StrictAnti u ∧ (∀ n, f (u n) ∈ Ioi x) ∧ Tendsto (f ∘ u) atTop (𝓝 x) := hf.exists_seq_strictMono_tendsto (α := αᵒᵈ) (β := βᵒᵈ) hmono.dual x end OrderTopology
.lake/packages/mathlib/Mathlib/Topology/Order/IsNormal.lean	import Mathlib.Order.IsNormal import Mathlib.Topology.Order.IsLUB /-! # A normal function is strictly monotone and continuous We defined the predicate `Order.IsNormal` in terms of `IsLUB`, which avoids having to import topology in order theory files. This file shows that the predicate is equivalent to the definition in the literature, being that of a strictly monotonic function, continuous in the order topology. -/ open Set namespace Order variable {α β : Type*} [LinearOrder α] [WellFoundedLT α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] [WellFoundedLT β] [TopologicalSpace β] [OrderTopology β] attribute [local instance] WellFoundedLT.toOrderBot WellFoundedLT.conditionallyCompleteLinearOrderBot in theorem IsNormal.continuous {f : α → β} (hf : IsNormal f) : Continuous f := by rw [OrderTopology.continuous_iff] refine fun b ↦ ⟨?_, ((isLowerSet_Iio b).preimage hf.strictMono.monotone).isOpen⟩ rw [← isClosed_compl_iff, ← Set.preimage_compl, Set.compl_Ioi] obtain ha \| ⟨a, ha⟩ := ((isLowerSet_Iic b).preimage hf.strictMono.monotone).eq_univ_or_Iio · exact ha ▸ isClosed_univ · obtain h \| h := (f ⁻¹' Iic b).eq_empty_or_nonempty · exact h ▸ isClosed_empty · have : Nonempty α := ⟨a⟩ have : Nonempty β := ⟨b⟩ rw [hf.preimage_Iic h (ha ▸ bddAbove_Iio)] exact isClosed_Iic /-- A normal function between well-orders is equivalent to a strictly monotone, continuous function. -/ theorem isNormal_iff_strictMono_and_continuous {f : α → β} : IsNormal f ↔ StrictMono f ∧ Continuous f where mp hf := ⟨hf.strictMono, hf.continuous⟩ mpr := by rintro ⟨hs, hc⟩ refine ⟨hs, fun {a} ha ↦ (isLUB_of_mem_closure ?_ ?_).2⟩ · rintro _ ⟨b, hb, rfl⟩ exact (hs hb).le · apply image_closure_subset_closure_image hc (mem_image_of_mem ..) exact ha.isLUB_Iio.mem_closure (Iio_nonempty.2 ha.1) end Order
.lake/packages/mathlib/Mathlib/Topology/Order/Filter.lean	import Mathlib.Topology.Filter import Mathlib.Topology.Order.Basic /-! # Topology on filters of a space with order topology In this file we prove that `𝓝 (f x)` tends to `𝓝 Filter.atTop` provided that `f` tends to `Filter.atTop`, and similarly for `Filter.atBot`. -/ open Topology namespace Filter variable {α X : Type*} [TopologicalSpace X] [PartialOrder X] [OrderTopology X] protected theorem tendsto_nhds_atTop [NoMaxOrder X] : Tendsto 𝓝 (atTop : Filter X) (𝓝 atTop) := Filter.tendsto_nhds_atTop_iff.2 fun x => (eventually_gt_atTop x).mono fun _ => le_mem_nhds protected theorem tendsto_nhds_atBot [NoMinOrder X] : Tendsto 𝓝 (atBot : Filter X) (𝓝 atBot) := @Filter.tendsto_nhds_atTop Xᵒᵈ _ _ _ _ theorem Tendsto.nhds_atTop [NoMaxOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atTop) : Tendsto (𝓝 ∘ f) l (𝓝 atTop) := Filter.tendsto_nhds_atTop.comp h theorem Tendsto.nhds_atBot [NoMinOrder X] {f : α → X} {l : Filter α} (h : Tendsto f l atBot) : Tendsto (𝓝 ∘ f) l (𝓝 atBot) := @Tendsto.nhds_atTop α Xᵒᵈ _ _ _ _ _ _ h end Filter
.lake/packages/mathlib/Mathlib/Topology/Order/OrderClosed.lean	import Mathlib.Topology.Order.LeftRight import Mathlib.Topology.Separation.Hausdorff /-! # Order-closed topologies In this file we introduce 3 typeclass mixins that relate topology and order structures: - `ClosedIicTopology` says that all the intervals $(-∞, a]$ (formally, `Set.Iic a`) are closed sets; - `ClosedIciTopology` says that all the intervals $[a, +∞)$ (formally, `Set.Ici a`) are closed sets; - `OrderClosedTopology` says that the set of points `(x, y)` such that `x ≤ y` is closed in the product topology. The last predicate implies the first two. We prove many basic properties of such topologies. ## Main statements This file contains the proofs of the following facts. For exact requirements (`OrderClosedTopology` vs `ClosedIciTopology` vs `ClosedIicTopology, `Preorder` vs `PartialOrder` vs `LinearOrder` etc) see their statements. ### Open / closed sets * `isOpen_lt` : if `f` and `g` are continuous functions, then `{x \| f x < g x}` is open; * `isOpen_Iio`, `isOpen_Ioi`, `isOpen_Ioo` : open intervals are open; * `isClosed_le` : if `f` and `g` are continuous functions, then `{x \| f x ≤ g x}` is closed; * `isClosed_Iic`, `isClosed_Ici`, `isClosed_Icc` : closed intervals are closed; * `frontier_le_subset_eq`, `frontier_lt_subset_eq` : frontiers of both `{x \| f x ≤ g x}` and `{x \| f x < g x}` are included by `{x \| f x = g x}`; ### Convergence and inequalities * `le_of_tendsto_of_tendsto` : if `f` converges to `a`, `g` converges to `b`, and eventually `f x ≤ g x`, then `a ≤ b` * `le_of_tendsto`, `ge_of_tendsto` : if `f` converges to `a` and eventually `f x ≤ b` (resp., `b ≤ f x`), then `a ≤ b` (resp., `b ≤ a`); we also provide primed versions that assume the inequalities to hold for all `x`. ### Min, max, `sSup` and `sInf` * `Continuous.min`, `Continuous.max`: pointwise `min`/`max` of two continuous functions is continuous. * `Tendsto.min`, `Tendsto.max` : if `f` tends to `a` and `g` tends to `b`, then their pointwise `min`/`max` tend to `min a b` and `max a b`, respectively. -/ open Set Filter open OrderDual (toDual) open scoped Topology universe u v w variable {α : Type u} {β : Type v} {γ : Type w} /-- If `α` is a topological space and a preorder, `ClosedIicTopology α` means that `Iic a` is closed for all `a : α`. -/ class ClosedIicTopology (α : Type) [TopologicalSpace α] [Preorder α] : Prop where /-- For any `a`, the set `(-∞, a]` is closed. -/ isClosed_Iic (a : α) : IsClosed (Iic a) /-- If `α` is a topological space and a preorder, `ClosedIciTopology α` means that `Ici a` is closed for all `a : α`. -/ class ClosedIciTopology (α : Type) [TopologicalSpace α] [Preorder α] : Prop where /-- For any `a`, the set `[a, +∞)` is closed. -/ isClosed_Ici (a : α) : IsClosed (Ici a) /-- A topology on a set which is both a topological space and a preorder is _order-closed_ if the set of points `(x, y)` with `x ≤ y` is closed in the product space. We introduce this as a mixin. This property is satisfied for the order topology on a linear order, but it can be satisfied more generally, and suffices to derive many interesting properties relating order and topology. -/ class OrderClosedTopology (α : Type) [TopologicalSpace α] [Preorder α] : Prop where /-- The set `{ (x, y) \| x ≤ y }` is a closed set. -/ isClosed_le' : IsClosed { p : α × α \| p.1 ≤ p.2 } instance [TopologicalSpace α] [h : FirstCountableTopology α] : FirstCountableTopology αᵒᵈ := h instance [TopologicalSpace α] [h : SecondCountableTopology α] : SecondCountableTopology αᵒᵈ := h theorem Dense.orderDual [TopologicalSpace α] {s : Set α} (hs : Dense s) : Dense (OrderDual.ofDual ⁻¹' s) := hs section General variable [TopologicalSpace α] [Preorder α] {s : Set α} protected lemma BddAbove.of_closure : BddAbove (closure s) → BddAbove s := BddAbove.mono subset_closure protected lemma BddBelow.of_closure : BddBelow (closure s) → BddBelow s := BddBelow.mono subset_closure end General section ClosedIicTopology section Preorder variable [TopologicalSpace α] [Preorder α] [ClosedIicTopology α] {f : β → α} {a b : α} {s : Set α} theorem isClosed_Iic : IsClosed (Iic a) := ClosedIicTopology.isClosed_Iic a instance : ClosedIciTopology αᵒᵈ where isClosed_Ici _ := isClosed_Iic (α := α) @[simp] theorem closure_Iic (a : α) : closure (Iic a) = Iic a := isClosed_Iic.closure_eq theorem le_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a)) (h : ∃ᶠ c in x, f c ≤ b) : a ≤ b := isClosed_Iic.mem_of_frequently_of_tendsto h lim theorem le_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, f c ≤ b) : a ≤ b := isClosed_Iic.mem_of_tendsto lim h theorem le_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a)) (h : ∀ c, f c ≤ b) : a ≤ b := le_of_tendsto lim (Eventually.of_forall h) @[simp] lemma upperBounds_closure (s : Set α) : upperBounds (closure s : Set α) = upperBounds s := ext fun a ↦ by simp_rw [mem_upperBounds_iff_subset_Iic, isClosed_Iic.closure_subset_iff] @[simp] lemma bddAbove_closure : BddAbove (closure s) ↔ BddAbove s := by simp_rw [BddAbove, upperBounds_closure] protected alias ⟨_, BddAbove.closure⟩ := bddAbove_closure @[simp] theorem disjoint_nhds_atBot_iff : Disjoint (𝓝 a) atBot ↔ ¬IsBot a := by constructor · intro hd hbot rw [hbot.atBot_eq, disjoint_principal_right] at hd exact mem_of_mem_nhds hd le_rfl · simp only [IsBot, not_forall] rintro ⟨b, hb⟩ refine disjoint_of_disjoint_of_mem disjoint_compl_left ?_ (Iic_mem_atBot b) exact isClosed_Iic.isOpen_compl.mem_nhds hb theorem IsLUB.range_of_tendsto {F : Filter β} [F.NeBot] (hle : ∀ i, f i ≤ a) (hlim : Tendsto f F (𝓝 a)) : IsLUB (range f) a := ⟨forall_mem_range.mpr hle, fun _c hc ↦ le_of_tendsto' hlim fun i ↦ hc <\| mem_range_self i⟩ end Preorder section NoBotOrder variable [Preorder α] [NoBotOrder α] [TopologicalSpace α] [ClosedIicTopology α] {a : α} {l : Filter β} [NeBot l] {f : β → α} theorem disjoint_nhds_atBot (a : α) : Disjoint (𝓝 a) atBot := by simp @[simp] theorem inf_nhds_atBot (a : α) : 𝓝 a ⊓ atBot = ⊥ := (disjoint_nhds_atBot a).eq_bot theorem not_tendsto_nhds_of_tendsto_atBot (hf : Tendsto f l atBot) (a : α) : ¬Tendsto f l (𝓝 a) := hf.not_tendsto (disjoint_nhds_atBot a).symm theorem not_tendsto_atBot_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atBot := hf.not_tendsto (disjoint_nhds_atBot a) end NoBotOrder theorem iSup_eq_of_forall_le_of_tendsto {ι : Type} {F : Filter ι} [Filter.NeBot F] [ConditionallyCompleteLattice α] [TopologicalSpace α] [ClosedIicTopology α] {a : α} {f : ι → α} (hle : ∀ i, f i ≤ a) (hlim : Filter.Tendsto f F (𝓝 a)) : ⨆ i, f i = a := have := F.nonempty_of_neBot (IsLUB.range_of_tendsto hle hlim).ciSup_eq theorem iUnion_Iic_eq_Iio_of_lt_of_tendsto {ι : Type} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] {a : α} {f : ι → α} (hlt : ∀ i, f i < a) (hlim : Tendsto f F (𝓝 a)) : ⋃ i : ι, Iic (f i) = Iio a := by have obs : a ∉ range f := by rw [mem_range] rintro ⟨i, rfl⟩ exact (hlt i).false rw [← biUnion_range, (IsLUB.range_of_tendsto (le_of_lt <\| hlt ·) hlim).biUnion_Iic_eq_Iio obs] section LinearOrder variable [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] [TopologicalSpace β] {a b c : α} {f : α → β} theorem isOpen_Ioi : IsOpen (Ioi a) := by rw [← compl_Iic] exact isClosed_Iic.isOpen_compl @[simp] theorem interior_Ioi : interior (Ioi a) = Ioi a := isOpen_Ioi.interior_eq theorem Ioi_mem_nhds (h : a < b) : Ioi a ∈ 𝓝 b := IsOpen.mem_nhds isOpen_Ioi h theorem eventually_gt_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b < x := Ioi_mem_nhds hab theorem Ici_mem_nhds (h : a < b) : Ici a ∈ 𝓝 b := mem_of_superset (Ioi_mem_nhds h) Ioi_subset_Ici_self theorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab theorem Filter.Tendsto.eventually_const_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a := h.eventually <\| eventually_gt_nhds hv theorem Filter.Tendsto.eventually_const_le {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v) (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a := h.eventually <\| eventually_ge_nhds hv protected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) : ∃ y ∈ s, x < y := hs.exists_mem_open isOpen_Ioi (exists_gt x) protected theorem Dense.exists_ge [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) : ∃ y ∈ s, x ≤ y := (hs.exists_gt x).imp fun _ h ↦ ⟨h.1, h.2.le⟩ theorem Dense.exists_ge' {s : Set α} (hs : Dense s) (htop : ∀ x, IsTop x → x ∈ s) (x : α) : ∃ y ∈ s, x ≤ y := by by_cases hx : IsTop x · exact ⟨x, htop x hx, le_rfl⟩ · simp only [IsTop, not_forall, not_le] at hx rcases hs.exists_mem_open isOpen_Ioi hx with ⟨y, hys, hy : x < y⟩ exact ⟨y, hys, hy.le⟩ /-! ### Left neighborhoods on a `ClosedIicTopology` Limits to the left of real functions are defined in terms of neighborhoods to the left, either open or closed, i.e., members of `𝓝[<] a` and `𝓝[≤] a`. Here we prove that all left-neighborhoods of a point are equal, and we prove other useful characterizations which require the stronger hypothesis `OrderTopology α` in another file. -/ /-! #### Point excluded -/ theorem Ioo_mem_nhdsLT (H : a < b) : Ioo a b ∈ 𝓝[<] b := by simpa only [← Iio_inter_Ioi] using inter_mem_nhdsWithin _ (Ioi_mem_nhds H) theorem Ioo_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ioo a c ∈ 𝓝[<] b := mem_of_superset (Ioo_mem_nhdsLT H.1) <\| Ioo_subset_Ioo_right H.2 protected theorem CovBy.nhdsLT (h : a ⋖ b) : 𝓝[<] b = ⊥ := empty_mem_iff_bot.mp <\| h.Ioo_eq ▸ Ioo_mem_nhdsLT h.1 protected theorem PredOrder.nhdsLT [PredOrder α] : 𝓝[<] a = ⊥ := by if h : IsMin a then simp [h.Iio_eq] else exact (Order.pred_covBy_of_not_isMin h).nhdsLT theorem PredOrder.nhdsGT_eq_nhdsNE [PredOrder α] (a : α) : 𝓝[>] a = 𝓝[≠] a := by rw [← nhdsLT_sup_nhdsGT, PredOrder.nhdsLT, bot_sup_eq] theorem PredOrder.nhdsGE_eq_nhds [PredOrder α] (a : α) : 𝓝[≥] a = 𝓝 a := by rw [← nhdsLT_sup_nhdsGE, PredOrder.nhdsLT, bot_sup_eq] theorem Ico_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ico a c ∈ 𝓝[<] b := mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Ico_self theorem Ico_mem_nhdsLT (H : a < b) : Ico a b ∈ 𝓝[<] b := Ico_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ theorem Ioc_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[<] b := mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Ioc_self theorem Ioc_mem_nhdsLT (H : a < b) : Ioc a b ∈ 𝓝[<] b := Ioc_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ theorem Icc_mem_nhdsLT_of_mem (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[<] b := mem_of_superset (Ioo_mem_nhdsLT_of_mem H) Ioo_subset_Icc_self theorem Icc_mem_nhdsLT (H : a < b) : Icc a b ∈ 𝓝[<] b := Icc_mem_nhdsLT_of_mem ⟨H, le_rfl⟩ @[simp] theorem nhdsWithin_Ico_eq_nhdsLT (h : a < b) : 𝓝[Ico a b] b = 𝓝[<] b := nhdsWithin_inter_of_mem <\| nhdsWithin_le_nhds <\| Ici_mem_nhds h @[simp] theorem nhdsWithin_Ioo_eq_nhdsLT (h : a < b) : 𝓝[Ioo a b] b = 𝓝[<] b := nhdsWithin_inter_of_mem <\| nhdsWithin_le_nhds <\| Ioi_mem_nhds h @[simp] theorem continuousWithinAt_Ico_iff_Iio (h : a < b) : ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b := by simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsLT h] @[simp] theorem continuousWithinAt_Ioo_iff_Iio (h : a < b) : ContinuousWithinAt f (Ioo a b) b ↔ ContinuousWithinAt f (Iio b) b := by simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsLT h] /-! #### Point included -/ protected theorem CovBy.nhdsLE (H : a ⋖ b) : 𝓝[≤] b = pure b := by rw [← Iio_insert, nhdsWithin_insert, H.nhdsLT, sup_bot_eq] protected theorem PredOrder.nhdsLE [PredOrder α] : 𝓝[≤] b = pure b := by rw [← Iio_insert, nhdsWithin_insert, PredOrder.nhdsLT, sup_bot_eq] theorem Ioc_mem_nhdsLE (H : a < b) : Ioc a b ∈ 𝓝[≤] b := inter_mem (nhdsWithin_le_nhds <\| Ioi_mem_nhds H) self_mem_nhdsWithin theorem Ioo_mem_nhdsLE_of_mem (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≤] b := mem_of_superset (Ioc_mem_nhdsLE H.1) <\| Ioc_subset_Ioo_right H.2 theorem Ico_mem_nhdsLE_of_mem (H : b ∈ Ioo a c) : Ico a c ∈ 𝓝[≤] b := mem_of_superset (Ioo_mem_nhdsLE_of_mem H) Ioo_subset_Ico_self theorem Ioc_mem_nhdsLE_of_mem (H : b ∈ Ioc a c) : Ioc a c ∈ 𝓝[≤] b := mem_of_superset (Ioc_mem_nhdsLE H.1) <\| Ioc_subset_Ioc_right H.2 theorem Icc_mem_nhdsLE_of_mem (H : b ∈ Ioc a c) : Icc a c ∈ 𝓝[≤] b := mem_of_superset (Ioc_mem_nhdsLE_of_mem H) Ioc_subset_Icc_self theorem Icc_mem_nhdsLE (H : a < b) : Icc a b ∈ 𝓝[≤] b := Icc_mem_nhdsLE_of_mem ⟨H, le_rfl⟩ @[simp] theorem nhdsWithin_Icc_eq_nhdsLE (h : a < b) : 𝓝[Icc a b] b = 𝓝[≤] b := nhdsWithin_inter_of_mem <\| nhdsWithin_le_nhds <\| Ici_mem_nhds h @[simp] theorem nhdsWithin_Ioc_eq_nhdsLE (h : a < b) : 𝓝[Ioc a b] b = 𝓝[≤] b := nhdsWithin_inter_of_mem <\| nhdsWithin_le_nhds <\| Ioi_mem_nhds h @[simp] theorem continuousWithinAt_Icc_iff_Iic (h : a < b) : ContinuousWithinAt f (Icc a b) b ↔ ContinuousWithinAt f (Iic b) b := by simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsLE h] @[simp] theorem continuousWithinAt_Ioc_iff_Iic (h : a < b) : ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b := by simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] end LinearOrder end ClosedIicTopology section ClosedIciTopology section Preorder variable [TopologicalSpace α] [Preorder α] [ClosedIciTopology α] {f : β → α} {a b : α} {s : Set α} theorem isClosed_Ici {a : α} : IsClosed (Ici a) := ClosedIciTopology.isClosed_Ici a instance : ClosedIicTopology αᵒᵈ where isClosed_Iic _ := isClosed_Ici (α := α) @[simp] theorem closure_Ici (a : α) : closure (Ici a) = Ici a := isClosed_Ici.closure_eq lemma ge_of_tendsto_of_frequently {x : Filter β} (lim : Tendsto f x (𝓝 a)) (h : ∃ᶠ c in x, b ≤ f c) : b ≤ a := isClosed_Ici.mem_of_frequently_of_tendsto h lim theorem ge_of_tendsto {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a)) (h : ∀ᶠ c in x, b ≤ f c) : b ≤ a := isClosed_Ici.mem_of_tendsto lim h theorem ge_of_tendsto' {x : Filter β} [NeBot x] (lim : Tendsto f x (𝓝 a)) (h : ∀ c, b ≤ f c) : b ≤ a := ge_of_tendsto lim (Eventually.of_forall h) @[simp] lemma lowerBounds_closure (s : Set α) : lowerBounds (closure s : Set α) = lowerBounds s := ext fun a ↦ by simp_rw [mem_lowerBounds_iff_subset_Ici, isClosed_Ici.closure_subset_iff] @[simp] lemma bddBelow_closure : BddBelow (closure s) ↔ BddBelow s := by simp_rw [BddBelow, lowerBounds_closure] protected alias ⟨_, BddBelow.closure⟩ := bddBelow_closure @[simp] theorem disjoint_nhds_atTop_iff : Disjoint (𝓝 a) atTop ↔ ¬IsTop a := disjoint_nhds_atBot_iff (α := αᵒᵈ) theorem IsGLB.range_of_tendsto {F : Filter β} [F.NeBot] (hle : ∀ i, a ≤ f i) (hlim : Tendsto f F (𝓝 a)) : IsGLB (range f) a := IsLUB.range_of_tendsto (α := αᵒᵈ) hle hlim end Preorder section NoTopOrder variable [Preorder α] [NoTopOrder α] [TopologicalSpace α] [ClosedIciTopology α] {a : α} {l : Filter β} [NeBot l] {f : β → α} theorem disjoint_nhds_atTop (a : α) : Disjoint (𝓝 a) atTop := disjoint_nhds_atBot (toDual a) @[simp] theorem inf_nhds_atTop (a : α) : 𝓝 a ⊓ atTop = ⊥ := (disjoint_nhds_atTop a).eq_bot theorem not_tendsto_nhds_of_tendsto_atTop (hf : Tendsto f l atTop) (a : α) : ¬Tendsto f l (𝓝 a) := hf.not_tendsto (disjoint_nhds_atTop a).symm theorem not_tendsto_atTop_of_tendsto_nhds (hf : Tendsto f l (𝓝 a)) : ¬Tendsto f l atTop := hf.not_tendsto (disjoint_nhds_atTop a) end NoTopOrder theorem iInf_eq_of_forall_le_of_tendsto {ι : Type} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLattice α] [TopologicalSpace α] [ClosedIciTopology α] {a : α} {f : ι → α} (hle : ∀ i, a ≤ f i) (hlim : Tendsto f F (𝓝 a)) : ⨅ i, f i = a := iSup_eq_of_forall_le_of_tendsto (α := αᵒᵈ) hle hlim theorem iUnion_Ici_eq_Ioi_of_lt_of_tendsto {ι : Type} {F : Filter ι} [F.NeBot] [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] {a : α} {f : ι → α} (hlt : ∀ i, a < f i) (hlim : Tendsto f F (𝓝 a)) : ⋃ i : ι, Ici (f i) = Ioi a := iUnion_Iic_eq_Iio_of_lt_of_tendsto (α := αᵒᵈ) hlt hlim section LinearOrder variable [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] [TopologicalSpace β] {a b c : α} {f : α → β} theorem isOpen_Iio : IsOpen (Iio a) := isOpen_Ioi (α := αᵒᵈ) @[simp] theorem interior_Iio : interior (Iio a) = Iio a := isOpen_Iio.interior_eq theorem Iio_mem_nhds (h : a < b) : Iio b ∈ 𝓝 a := isOpen_Iio.mem_nhds h theorem eventually_lt_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x < b := Iio_mem_nhds hab theorem Iic_mem_nhds (h : a < b) : Iic b ∈ 𝓝 a := mem_of_superset (Iio_mem_nhds h) Iio_subset_Iic_self theorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab theorem Filter.Tendsto.eventually_lt_const {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u := h.eventually <\| eventually_lt_nhds hv theorem Filter.Tendsto.eventually_le_const {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u) (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u := h.eventually <\| eventually_le_nhds hv protected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) : ∃ y ∈ s, y < x := hs.orderDual.exists_gt x protected theorem Dense.exists_le [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) : ∃ y ∈ s, y ≤ x := hs.orderDual.exists_ge x theorem Dense.exists_le' {s : Set α} (hs : Dense s) (hbot : ∀ x, IsBot x → x ∈ s) (x : α) : ∃ y ∈ s, y ≤ x := hs.orderDual.exists_ge' hbot x /-! ### Right neighborhoods on a `ClosedIciTopology` Limits to the right of real functions are defined in terms of neighborhoods to the right, either open or closed, i.e., members of `𝓝[>] a` and `𝓝[≥] a`. Here we prove that all right-neighborhoods of a point are equal, and we prove other useful characterizations which require the stronger hypothesis `OrderTopology α` in another file. -/ /-! #### Point excluded -/ theorem Ioo_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Ioo a c ∈ 𝓝[>] b := mem_nhdsWithin.2 ⟨Iio c, isOpen_Iio, H.2, by rw [inter_comm, Ioi_inter_Iio]; exact Ioo_subset_Ioo_left H.1⟩ theorem Ioo_mem_nhdsGT (H : a < b) : Ioo a b ∈ 𝓝[>] a := Ioo_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ protected theorem CovBy.nhdsGT (h : a ⋖ b) : 𝓝[>] a = ⊥ := h.toDual.nhdsLT protected theorem SuccOrder.nhdsGT [SuccOrder α] : 𝓝[>] a = ⊥ := PredOrder.nhdsLT (α := αᵒᵈ) theorem SuccOrder.nhdsLT_eq_nhdsNE [SuccOrder α] (a : α) : 𝓝[<] a = 𝓝[≠] a := PredOrder.nhdsGT_eq_nhdsNE (α := αᵒᵈ) a theorem SuccOrder.nhdsLE_eq_nhds [SuccOrder α] (a : α) : 𝓝[≤] a = 𝓝 a := PredOrder.nhdsGE_eq_nhds (α := αᵒᵈ) a theorem Ioc_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Ioc a c ∈ 𝓝[>] b := mem_of_superset (Ioo_mem_nhdsGT_of_mem H) Ioo_subset_Ioc_self theorem Ioc_mem_nhdsGT (H : a < b) : Ioc a b ∈ 𝓝[>] a := Ioc_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ theorem Ico_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[>] b := mem_of_superset (Ioo_mem_nhdsGT_of_mem H) Ioo_subset_Ico_self theorem Ico_mem_nhdsGT (H : a < b) : Ico a b ∈ 𝓝[>] a := Ico_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ theorem Icc_mem_nhdsGT_of_mem (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[>] b := mem_of_superset (Ioo_mem_nhdsGT_of_mem H) Ioo_subset_Icc_self theorem Icc_mem_nhdsGT (H : a < b) : Icc a b ∈ 𝓝[>] a := Icc_mem_nhdsGT_of_mem ⟨le_rfl, H⟩ @[simp] theorem nhdsWithin_Ioc_eq_nhdsGT (h : a < b) : 𝓝[Ioc a b] a = 𝓝[>] a := nhdsWithin_inter_of_mem' <\| nhdsWithin_le_nhds <\| Iic_mem_nhds h @[simp] theorem nhdsWithin_Ioo_eq_nhdsGT (h : a < b) : 𝓝[Ioo a b] a = 𝓝[>] a := nhdsWithin_inter_of_mem' <\| nhdsWithin_le_nhds <\| Iio_mem_nhds h @[simp] theorem continuousWithinAt_Ioc_iff_Ioi (h : a < b) : ContinuousWithinAt f (Ioc a b) a ↔ ContinuousWithinAt f (Ioi a) a := by simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsGT h] @[simp] theorem continuousWithinAt_Ioo_iff_Ioi (h : a < b) : ContinuousWithinAt f (Ioo a b) a ↔ ContinuousWithinAt f (Ioi a) a := by simp only [ContinuousWithinAt, nhdsWithin_Ioo_eq_nhdsGT h] /-! ### Point included -/ protected theorem CovBy.nhdsGE (H : a ⋖ b) : 𝓝[≥] a = pure a := H.toDual.nhdsLE protected theorem SuccOrder.nhdsGE [SuccOrder α] : 𝓝[≥] a = pure a := PredOrder.nhdsLE (α := αᵒᵈ) theorem Ico_mem_nhdsGE (H : a < b) : Ico a b ∈ 𝓝[≥] a := inter_mem_nhdsWithin _ <\| Iio_mem_nhds H theorem Ico_mem_nhdsGE_of_mem (H : b ∈ Ico a c) : Ico a c ∈ 𝓝[≥] b := mem_of_superset (Ico_mem_nhdsGE H.2) <\| Ico_subset_Ico_left H.1 theorem Ioo_mem_nhdsGE_of_mem (H : b ∈ Ioo a c) : Ioo a c ∈ 𝓝[≥] b := mem_of_superset (Ico_mem_nhdsGE H.2) <\| Ico_subset_Ioo_left H.1 theorem Ioc_mem_nhdsGE_of_mem (H : b ∈ Ioo a c) : Ioc a c ∈ 𝓝[≥] b := mem_of_superset (Ioo_mem_nhdsGE_of_mem H) Ioo_subset_Ioc_self theorem Icc_mem_nhdsGE_of_mem (H : b ∈ Ico a c) : Icc a c ∈ 𝓝[≥] b := mem_of_superset (Ico_mem_nhdsGE_of_mem H) Ico_subset_Icc_self theorem Icc_mem_nhdsGE (H : a < b) : Icc a b ∈ 𝓝[≥] a := Icc_mem_nhdsGE_of_mem ⟨le_rfl, H⟩ @[simp] theorem nhdsWithin_Icc_eq_nhdsGE (h : a < b) : 𝓝[Icc a b] a = 𝓝[≥] a := nhdsWithin_inter_of_mem' <\| nhdsWithin_le_nhds <\| Iic_mem_nhds h @[simp] theorem nhdsWithin_Ico_eq_nhdsGE (h : a < b) : 𝓝[Ico a b] a = 𝓝[≥] a := nhdsWithin_inter_of_mem' <\| nhdsWithin_le_nhds <\| Iio_mem_nhds h @[simp] theorem continuousWithinAt_Icc_iff_Ici (h : a < b) : ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a := by simp only [ContinuousWithinAt, nhdsWithin_Icc_eq_nhdsGE h] @[simp] theorem continuousWithinAt_Ico_iff_Ici (h : a < b) : ContinuousWithinAt f (Ico a b) a ↔ ContinuousWithinAt f (Ici a) a := by simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsGE h] end LinearOrder end ClosedIciTopology section OrderClosedTopology section Preorder variable [TopologicalSpace α] [Preorder α] [t : OrderClosedTopology α] namespace Subtype -- todo: add `OrderEmbedding.orderClosedTopology` instance {p : α → Prop} : OrderClosedTopology (Subtype p) := have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) := continuous_subtype_val.prodMap continuous_subtype_val OrderClosedTopology.mk (t.isClosed_le'.preimage this) end Subtype theorem isClosed_le_prod : IsClosed { p : α × α \| p.1 ≤ p.2 } := t.isClosed_le' theorem isClosed_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : IsClosed { b \| f b ≤ g b } := continuous_iff_isClosed.mp (hf.prodMk hg) _ isClosed_le_prod instance : ClosedIicTopology α where isClosed_Iic _ := isClosed_le continuous_id continuous_const instance : ClosedIciTopology α where isClosed_Ici _ := isClosed_le continuous_const continuous_id instance : OrderClosedTopology αᵒᵈ := ⟨(OrderClosedTopology.isClosed_le' (α := α)).preimage continuous_swap⟩ theorem isClosed_Icc {a b : α} : IsClosed (Icc a b) := IsClosed.inter isClosed_Ici isClosed_Iic @[simp] theorem closure_Icc (a b : α) : closure (Icc a b) = Icc a b := isClosed_Icc.closure_eq theorem le_of_tendsto_of_tendsto_of_frequently {f g : β → α} {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∃ᶠ x in b, f x ≤ g x) : a₁ ≤ a₂ := t.isClosed_le'.mem_of_frequently_of_tendsto h (hf.prodMk_nhds hg) theorem le_of_tendsto_of_tendsto {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b] (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : f ≤ᶠ[b] g) : a₁ ≤ a₂ := le_of_tendsto_of_tendsto_of_frequently hf hg <\| Eventually.frequently h alias tendsto_le_of_eventuallyLE := le_of_tendsto_of_tendsto theorem le_of_tendsto_of_tendsto' {f g : β → α} {b : Filter β} {a₁ a₂ : α} [NeBot b] (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) (h : ∀ x, f x ≤ g x) : a₁ ≤ a₂ := le_of_tendsto_of_tendsto hf hg (Eventually.of_forall h) @[simp] theorem closure_le_eq [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : closure { b \| f b ≤ g b } = { b \| f b ≤ g b } := (isClosed_le hf hg).closure_eq theorem closure_lt_subset_le [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : closure { b \| f b < g b } ⊆ { b \| f b ≤ g b } := (closure_minimal fun _ => le_of_lt) <\| isClosed_le hf hg theorem ContinuousWithinAt.closure_le [TopologicalSpace β] {f g : β → α} {s : Set β} {x : β} (hx : x ∈ closure s) (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) (h : ∀ y ∈ s, f y ≤ g y) : f x ≤ g x := show (f x, g x) ∈ { p : α × α \| p.1 ≤ p.2 } from OrderClosedTopology.isClosed_le'.closure_subset ((hf.prodMk hg).mem_closure hx h) /-- If `s` is a closed set and two functions `f` and `g` are continuous on `s`, then the set `{x ∈ s \| f x ≤ g x}` is a closed set. -/ theorem IsClosed.isClosed_le [TopologicalSpace β] {f g : β → α} {s : Set β} (hs : IsClosed s) (hf : ContinuousOn f s) (hg : ContinuousOn g s) : IsClosed ({ x ∈ s \| f x ≤ g x }) := (hf.prodMk hg).preimage_isClosed_of_isClosed hs OrderClosedTopology.isClosed_le' theorem le_on_closure [TopologicalSpace β] {f g : β → α} {s : Set β} (h : ∀ x ∈ s, f x ≤ g x) (hf : ContinuousOn f (closure s)) (hg : ContinuousOn g (closure s)) ⦃x⦄ (hx : x ∈ closure s) : f x ≤ g x := have : s ⊆ { y ∈ closure s \| f y ≤ g y } := fun y hy => ⟨subset_closure hy, h y hy⟩ (closure_minimal this (isClosed_closure.isClosed_le hf hg) hx).2 theorem IsClosed.epigraph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s) (hf : ContinuousOn f s) : IsClosed { p : β × α \| p.1 ∈ s ∧ f p.1 ≤ p.2 } := (hs.preimage continuous_fst).isClosed_le (hf.comp continuousOn_fst Subset.rfl) continuousOn_snd theorem IsClosed.hypograph [TopologicalSpace β] {f : β → α} {s : Set β} (hs : IsClosed s) (hf : ContinuousOn f s) : IsClosed { p : β × α \| p.1 ∈ s ∧ p.2 ≤ f p.1 } := (hs.preimage continuous_fst).isClosed_le continuousOn_snd (hf.comp continuousOn_fst Subset.rfl) /-- The set of monotone functions on a set is closed. -/ theorem isClosed_monotoneOn [Preorder β] {s : Set β} : IsClosed {f : β → α \| MonotoneOn f s} := by simp only [isClosed_iff_clusterPt, clusterPt_principal_iff_frequently] intro g hg a ha b hb hab have hmain (x) : Tendsto (fun f' ↦ f' x) (𝓝 g) (𝓝 (g x)) := continuousAt_apply x _ exact le_of_tendsto_of_tendsto_of_frequently (hmain a) (hmain b) (hg.mono fun g h ↦ h ha hb hab) /-- The set of monotone functions is closed. -/ theorem isClosed_monotone [Preorder β] : IsClosed {f : β → α \| Monotone f} := by simp_rw [← monotoneOn_univ] exact isClosed_monotoneOn /-- The set of antitone functions on a set is closed. -/ theorem isClosed_antitoneOn [Preorder β] {s : Set β} : IsClosed {f : β → α \| AntitoneOn f s} := isClosed_monotoneOn (α := αᵒᵈ) (β := β) /-- The set of antitone functions is closed. -/ theorem isClosed_antitone [Preorder β] : IsClosed {f : β → α \| Antitone f} := isClosed_monotone (α := αᵒᵈ) (β := β) end Preorder section PartialOrder variable [TopologicalSpace α] [PartialOrder α] [t : OrderClosedTopology α] -- see Note [lower instance priority] instance (priority := 90) OrderClosedTopology.to_t2Space : T2Space α := t2_iff_isClosed_diagonal.2 <\| by simpa only [diagonal, le_antisymm_iff] using t.isClosed_le'.inter (isClosed_le continuous_snd continuous_fst) end PartialOrder section LinearOrder variable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] theorem isOpen_lt [TopologicalSpace β] {f g : β → α} (hf : Continuous f) (hg : Continuous g) : IsOpen { b \| f b < g b } := by simpa only [lt_iff_not_ge] using (isClosed_le hg hf).isOpen_compl theorem isOpen_lt_prod : IsOpen { p : α × α \| p.1 < p.2 } := isOpen_lt continuous_fst continuous_snd variable {a b : α} theorem isOpen_Ioo : IsOpen (Ioo a b) := IsOpen.inter isOpen_Ioi isOpen_Iio @[simp] theorem interior_Ioo : interior (Ioo a b) = Ioo a b := isOpen_Ioo.interior_eq theorem Ioo_subset_closure_interior : Ioo a b ⊆ closure (interior (Ioo a b)) := by simp only [interior_Ioo, subset_closure] theorem Ioo_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioo a b ∈ 𝓝 x := IsOpen.mem_nhds isOpen_Ioo ⟨ha, hb⟩ theorem Ioc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ioc a b ∈ 𝓝 x := mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ioc_self theorem Ico_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x := mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Ico_self theorem Icc_mem_nhds {a b x : α} (ha : a < x) (hb : x < b) : Icc a b ∈ 𝓝 x := mem_of_superset (Ioo_mem_nhds ha hb) Ioo_subset_Icc_self /-- The only order closed topology on a linear order which is a `PredOrder` and a `SuccOrder` is the discrete topology. This theorem is not an instance, because it causes searches for `PredOrder` and `SuccOrder` with their `Preorder` arguments and very rarely matches. -/ theorem DiscreteTopology.of_predOrder_succOrder [PredOrder α] [SuccOrder α] : DiscreteTopology α := by refine discreteTopology_iff_nhds.mpr fun a ↦ ?_ rw [← nhdsWithin_univ, ← Iic_union_Ioi, nhdsWithin_union, PredOrder.nhdsLE, SuccOrder.nhdsGT, sup_bot_eq] end LinearOrder section LinearOrder variable [TopologicalSpace α] [LinearOrder α] [OrderClosedTopology α] {f g : β → α} section variable [TopologicalSpace β] theorem lt_subset_interior_le (hf : Continuous f) (hg : Continuous g) : { b \| f b < g b } ⊆ interior { b \| f b ≤ g b } := (interior_maximal fun _ => le_of_lt) <\| isOpen_lt hf hg theorem frontier_le_subset_eq (hf : Continuous f) (hg : Continuous g) : frontier { b \| f b ≤ g b } ⊆ { b \| f b = g b } := by rw [frontier_eq_closure_inter_closure, closure_le_eq hf hg] rintro b ⟨hb₁, hb₂⟩ refine le_antisymm hb₁ (closure_lt_subset_le hg hf ?_) convert hb₂ using 2; simp only [not_le.symm]; rfl theorem frontier_Iic_subset (a : α) : frontier (Iic a) ⊆ {a} := frontier_le_subset_eq (@continuous_id α _) continuous_const theorem frontier_Ici_subset (a : α) : frontier (Ici a) ⊆ {a} := frontier_Iic_subset (α := αᵒᵈ) _ theorem frontier_lt_subset_eq (hf : Continuous f) (hg : Continuous g) : frontier { b \| f b < g b } ⊆ { b \| f b = g b } := by simpa only [← not_lt, ← compl_setOf, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf theorem continuous_if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ} (hf : Continuous f) (hg : Continuous g) (hf' : ContinuousOn f' { x \| f x ≤ g x }) (hg' : ContinuousOn g' { x \| g x ≤ f x }) (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x := by refine continuous_if (fun a ha => hfg _ (frontier_le_subset_eq hf hg ha)) ?_ (hg'.mono ?_) · rwa [(isClosed_le hf hg).closure_eq] · simp only [not_le] exact closure_lt_subset_le hg hf theorem Continuous.if_le [TopologicalSpace γ] [∀ x, Decidable (f x ≤ g x)] {f' g' : β → γ} (hf' : Continuous f') (hg' : Continuous g') (hf : Continuous f) (hg : Continuous g) (hfg : ∀ x, f x = g x → f' x = g' x) : Continuous fun x => if f x ≤ g x then f' x else g' x := continuous_if_le hf hg hf'.continuousOn hg'.continuousOn hfg theorem Filter.Tendsto.eventually_lt {l : Filter γ} {f g : γ → α} {y z : α} (hf : Tendsto f l (𝓝 y)) (hg : Tendsto g l (𝓝 z)) (hyz : y < z) : ∀ᶠ x in l, f x < g x := let ⟨_a, ha, _b, hb, h⟩ := hyz.exists_disjoint_Iio_Ioi (hg.eventually (Ioi_mem_nhds hb)).mp <\| (hf.eventually (Iio_mem_nhds ha)).mono fun _ h₁ h₂ => h _ h₁ _ h₂ nonrec theorem ContinuousAt.eventually_lt {x₀ : β} (hf : ContinuousAt f x₀) (hg : ContinuousAt g x₀) (hfg : f x₀ < g x₀) : ∀ᶠ x in 𝓝 x₀, f x < g x := hf.eventually_lt hg hfg @[continuity, fun_prop] protected theorem Continuous.min (hf : Continuous f) (hg : Continuous g) : Continuous fun b => min (f b) (g b) := by simp only [min_def] exact hf.if_le hg hf hg fun x => id @[continuity, fun_prop] protected theorem Continuous.max (hf : Continuous f) (hg : Continuous g) : Continuous fun b => max (f b) (g b) := Continuous.min (α := αᵒᵈ) hf hg end theorem continuous_min : Continuous fun p : α × α => min p.1 p.2 := continuous_fst.min continuous_snd theorem continuous_max : Continuous fun p : α × α => max p.1 p.2 := continuous_fst.max continuous_snd protected theorem Filter.Tendsto.max {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => max (f b) (g b)) b (𝓝 (max a₁ a₂)) := (continuous_max.tendsto (a₁, a₂)).comp (hf.prodMk_nhds hg) protected theorem Filter.Tendsto.min {b : Filter β} {a₁ a₂ : α} (hf : Tendsto f b (𝓝 a₁)) (hg : Tendsto g b (𝓝 a₂)) : Tendsto (fun b => min (f b) (g b)) b (𝓝 (min a₁ a₂)) := (continuous_min.tendsto (a₁, a₂)).comp (hf.prodMk_nhds hg) protected theorem Filter.Tendsto.max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) : Tendsto (fun i => max a (f i)) l (𝓝 a) := by simpa only [sup_idem] using (tendsto_const_nhds (x := a)).max h protected theorem Filter.Tendsto.max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) : Tendsto (fun i => max (f i) a) l (𝓝 a) := by simp_rw [max_comm _ a] exact h.max_right theorem Filter.tendsto_nhds_max_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) : Tendsto (fun i => max a (f i)) l (𝓝[>] a) := by obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩ theorem Filter.tendsto_nhds_max_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[>] a)) : Tendsto (fun i => max (f i) a) l (𝓝[>] a) := by simp_rw [max_comm _ a] exact Filter.tendsto_nhds_max_right h theorem Filter.Tendsto.min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) : Tendsto (fun i => min a (f i)) l (𝓝 a) := Filter.Tendsto.max_right (α := αᵒᵈ) h theorem Filter.Tendsto.min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝 a)) : Tendsto (fun i => min (f i) a) l (𝓝 a) := Filter.Tendsto.max_left (α := αᵒᵈ) h theorem Filter.tendsto_nhds_min_right {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) : Tendsto (fun i => min a (f i)) l (𝓝[<] a) := Filter.tendsto_nhds_max_right (α := αᵒᵈ) h theorem Filter.tendsto_nhds_min_left {l : Filter β} {a : α} (h : Tendsto f l (𝓝[<] a)) : Tendsto (fun i => min (f i) a) l (𝓝[<] a) := Filter.tendsto_nhds_max_left (α := αᵒᵈ) h theorem Dense.exists_between [DenselyOrdered α] {s : Set α} (hs : Dense s) {x y : α} (h : x < y) : ∃ z ∈ s, z ∈ Ioo x y := hs.exists_mem_open isOpen_Ioo (nonempty_Ioo.2 h) theorem Dense.Ioi_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) : Ioi x = ⋃ y ∈ s ∩ Ioi x, Ioi y := by refine Subset.antisymm (fun z hz ↦ ?_) (iUnion₂_subset fun y hy ↦ Ioi_subset_Ioi (le_of_lt hy.2)) rcases hs.exists_between hz with ⟨y, hys, hxy, hyz⟩ exact mem_iUnion₂.2 ⟨y, ⟨hys, hxy⟩, hyz⟩ theorem Dense.Iio_eq_biUnion [DenselyOrdered α] {s : Set α} (hs : Dense s) (x : α) : Iio x = ⋃ y ∈ s ∩ Iio x, Iio y := Dense.Ioi_eq_biUnion (α := αᵒᵈ) hs x end LinearOrder end OrderClosedTopology instance [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] : OrderClosedTopology (α × β) := ⟨(isClosed_le continuous_fst.fst continuous_snd.fst).inter (isClosed_le continuous_fst.snd continuous_snd.snd)⟩ instance {ι : Type} {α : ι → Type*} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)] [∀ i, OrderClosedTopology (α i)] : OrderClosedTopology (∀ i, α i) := by constructor simp only [Pi.le_def, setOf_forall] exact isClosed_iInter fun i => isClosed_le (continuous_apply i).fst' (continuous_apply i).snd' instance Pi.orderClosedTopology' [Preorder β] [TopologicalSpace β] [OrderClosedTopology β] : OrderClosedTopology (α → β) := inferInstance
.lake/packages/mathlib/Mathlib/Topology/Order/LeftRight.lean	import Mathlib.Order.Antichain import Mathlib.Topology.ContinuousOn /-! # Left and right continuity In this file we prove a few lemmas about left and right continuous functions: * `continuousWithinAt_Ioi_iff_Ici`: two definitions of right continuity (with `(a, ∞)` and with `[a, ∞)`) are equivalent; * `continuousWithinAt_Iio_iff_Iic`: two definitions of left continuity (with `(-∞, a)` and with `(-∞, a]`) are equivalent; * `continuousAt_iff_continuous_left_right`, `continuousAt_iff_continuous_left'_right'` : a function is continuous at `a` if and only if it is left and right continuous at `a`. ## Tags left continuous, right continuous -/ open Set Filter Topology section Preorder variable {α : Type} [TopologicalSpace α] [Preorder α] lemma frequently_lt_nhds (a : α) [NeBot (𝓝[<] a)] : ∃ᶠ x in 𝓝 a, x < a := frequently_iff_neBot.2 ‹_› lemma frequently_gt_nhds (a : α) [NeBot (𝓝[>] a)] : ∃ᶠ x in 𝓝 a, a < x := frequently_iff_neBot.2 ‹_› theorem Filter.Eventually.exists_lt {a : α} [NeBot (𝓝[<] a)] {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) : ∃ b < a, p b := ((frequently_lt_nhds a).and_eventually h).exists theorem Filter.Eventually.exists_gt {a : α} [NeBot (𝓝[>] a)] {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) : ∃ b > a, p b := ((frequently_gt_nhds a).and_eventually h).exists theorem nhdsWithin_Ici_neBot {a b : α} (H₂ : a ≤ b) : NeBot (𝓝[Ici a] b) := nhdsWithin_neBot_of_mem H₂ instance nhdsGE_neBot (a : α) : NeBot (𝓝[≥] a) := nhdsWithin_Ici_neBot (le_refl a) theorem nhdsWithin_Iic_neBot {a b : α} (H : a ≤ b) : NeBot (𝓝[Iic b] a) := nhdsWithin_neBot_of_mem H instance nhdsLE_neBot (a : α) : NeBot (𝓝[≤] a) := nhdsWithin_Iic_neBot (le_refl a) theorem nhdsLT_le_nhdsNE (a : α) : 𝓝[<] a ≤ 𝓝[≠] a := nhdsWithin_mono a fun _ => ne_of_lt theorem nhdsGT_le_nhdsNE (a : α) : 𝓝[>] a ≤ 𝓝[≠] a := nhdsWithin_mono a fun _ => ne_of_gt -- TODO: add instances for `NeBot (𝓝[<] x)` on (indexed) product types lemma IsAntichain.interior_eq_empty [∀ x : α, (𝓝[<] x).NeBot] {s : Set α} (hs : IsAntichain (· ≤ ·) s) : interior s = ∅ := by refine eq_empty_of_forall_notMem fun x hx ↦ ?_ have : ∀ᶠ y in 𝓝 x, y ∈ s := mem_interior_iff_mem_nhds.1 hx rcases this.exists_lt with ⟨y, hyx, hys⟩ exact hs hys (interior_subset hx) hyx.ne hyx.le lemma IsAntichain.interior_eq_empty' [∀ x : α, (𝓝[>] x).NeBot] {s : Set α} (hs : IsAntichain (· ≤ ·) s) : interior s = ∅ := have : ∀ x : αᵒᵈ, NeBot (𝓝[<] x) := ‹_› hs.to_dual.interior_eq_empty end Preorder section PartialOrder variable {α β : Type} [TopologicalSpace α] [PartialOrder α] [TopologicalSpace β] theorem continuousWithinAt_Ioi_iff_Ici {a : α} {f : α → β} : ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a := by simp only [← Ici_diff_left, continuousWithinAt_diff_self] theorem continuousWithinAt_Iio_iff_Iic {a : α} {f : α → β} : ContinuousWithinAt f (Iio a) a ↔ ContinuousWithinAt f (Iic a) a := continuousWithinAt_Ioi_iff_Ici (α := αᵒᵈ) theorem continuousWithinAt_inter_Ioi_iff_Ici {a : α} {f : α → β} {s : Set α} : ContinuousWithinAt f (s ∩ Ioi a) a ↔ ContinuousWithinAt f (s ∩ Ici a) a := by simp [← Ici_diff_left, ← inter_diff_assoc, continuousWithinAt_diff_self] theorem continuousWithinAt_inter_Iio_iff_Iic {a : α} {f : α → β} {s : Set α} : ContinuousWithinAt f (s ∩ Iio a) a ↔ ContinuousWithinAt f (s ∩ Iic a) a := continuousWithinAt_inter_Ioi_iff_Ici (α := αᵒᵈ) end PartialOrder section TopologicalSpace variable {α β : Type*} [TopologicalSpace α] [LinearOrder α] [TopologicalSpace β] {s : Set α} theorem nhdsLE_sup_nhdsGE (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a := by rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] theorem nhdsWithinLE_sup_nhdsWithinGE (a : α) : 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a := by rw [← nhdsWithin_union, ← inter_union_distrib_left, Iic_union_Ici, inter_univ] theorem nhdsLT_sup_nhdsGE (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a := by rw [← nhdsWithin_union, Iio_union_Ici, nhdsWithin_univ] theorem nhdsWithinLT_sup_nhdsWithinGE (a : α) : 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a := by rw [← nhdsWithin_union, ← inter_union_distrib_left, Iio_union_Ici, inter_univ] theorem nhdsLE_sup_nhdsGT (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a := by rw [← nhdsWithin_union, Iic_union_Ioi, nhdsWithin_univ] theorem nhdsWithinLE_sup_nhdsWithinGT (a : α) : 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s] a := by rw [← nhdsWithin_union, ← inter_union_distrib_left, Iic_union_Ioi, inter_univ] theorem nhdsLT_sup_nhdsGT (a : α) : 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a := by rw [← nhdsWithin_union, Iio_union_Ioi] theorem nhdsWithinLT_sup_nhdsWithinGT (a : α) : 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s \ {a}] a := by rw [← nhdsWithin_union, ← inter_union_distrib_left, Iio_union_Ioi, compl_eq_univ_diff, inter_sdiff_left_comm, univ_inter] lemma nhdsGT_sup_nhdsWithin_singleton (a : α) : 𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a := by simp only [union_singleton, Ioi_insert, ← nhdsWithin_union] @[deprecated (since := "2025-06-15")] alias nhdsWithin_right_sup_nhds_singleton := nhdsGT_sup_nhdsWithin_singleton theorem continuousAt_iff_continuous_left_right {a : α} {f : α → β} : ContinuousAt f a ↔ ContinuousWithinAt f (Iic a) a ∧ ContinuousWithinAt f (Ici a) a := by simp only [ContinuousWithinAt, ContinuousAt, ← tendsto_sup, nhdsLE_sup_nhdsGE] theorem continuousAt_iff_continuous_left'_right' {a : α} {f : α → β} : ContinuousAt f a ↔ ContinuousWithinAt f (Iio a) a ∧ ContinuousWithinAt f (Ioi a) a := by rw [continuousWithinAt_Ioi_iff_Ici, continuousWithinAt_Iio_iff_Iic, continuousAt_iff_continuous_left_right] theorem continuousWithinAt_iff_continuous_left_right {a : α} {f : α → β} : ContinuousWithinAt f s a ↔ ContinuousWithinAt f (s ∩ Iic a) a ∧ ContinuousWithinAt f (s ∩ Ici a) a := by simp only [ContinuousWithinAt, ← tendsto_sup, nhdsWithinLE_sup_nhdsWithinGE] theorem continuousWithinAt_iff_continuous_left'_right' {a : α} {f : α → β} : ContinuousWithinAt f s a ↔ ContinuousWithinAt f (s ∩ Iio a) a ∧ ContinuousWithinAt f (s ∩ Ioi a) a := by rw [continuousWithinAt_inter_Ioi_iff_Ici, continuousWithinAt_inter_Iio_iff_Iic, continuousWithinAt_iff_continuous_left_right] end TopologicalSpace
.lake/packages/mathlib/Mathlib/Topology/Order/LeftRightLim.lean	import Mathlib.Topology.Order.LeftRight import Mathlib.Topology.Order.Monotone /-! # Left and right limits We define the (strict) left and right limits of a function. * `leftLim f x` is the strict left limit of `f` at `x` (using `f x` as a garbage value if `x` is isolated to its left). * `rightLim f x` is the strict right limit of `f` at `x` (using `f x` as a garbage value if `x` is isolated to its right). We develop a comprehensive API for monotone functions. Notably, * `Monotone.continuousAt_iff_leftLim_eq_rightLim` states that a monotone function is continuous at a point if and only if its left and right limits coincide. * `Monotone.countable_not_continuousAt` asserts that a monotone function taking values in a second-countable space has at most countably many discontinuity points. We also port the API to antitone functions. ## TODO Prove corresponding stronger results for `StrictMono` and `StrictAnti` functions. -/ open Set Filter open Topology section variable {α β : Type} [LinearOrder α] [TopologicalSpace β] /-- Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and let `a : α`. The limit strictly to the left of `f` at `a`, denoted with `leftLim f a`, is defined by using the order topology on `α`. If `a` is isolated to its left or the function has no left limit, we use `f a` instead to guarantee a good behavior in most cases. -/ noncomputable def Function.leftLim (f : α → β) (a : α) : β := by classical haveI : Nonempty β := ⟨f a⟩ letI : TopologicalSpace α := Preorder.topology α exact if 𝓝[<] a = ⊥ ∨ ¬∃ y, Tendsto f (𝓝[<] a) (𝓝 y) then f a else limUnder (𝓝[<] a) f /-- Let `f : α → β` be a function from a linear order `α` to a topological space `β`, and let `a : α`. The limit strictly to the right of `f` at `a`, denoted with `rightLim f a`, is defined by using the order topology on `α`. If `a` is isolated to its right or the function has no right limit, we use `f a` instead to guarantee a good behavior in most cases. -/ noncomputable def Function.rightLim (f : α → β) (a : α) : β := @Function.leftLim αᵒᵈ β _ _ f a open Function theorem leftLim_eq_of_tendsto [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β] {f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Tendsto f (𝓝[<] a) (𝓝 y)) : leftLim f a = y := by have h'' : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y) := ⟨y, h'⟩ rw [h'α.topology_eq_generate_intervals] at h h' h'' simp only [leftLim, h, h'', not_true, or_self_iff, if_false] haveI := neBot_iff.2 h exact lim_eq h' theorem leftLim_eq_of_eq_bot [hα : TopologicalSpace α] [h'α : OrderTopology α] (f : α → β) {a : α} (h : 𝓝[<] a = ⊥) : leftLim f a = f a := by rw [h'α.topology_eq_generate_intervals] at h simp [leftLim, h] theorem rightLim_eq_of_tendsto [TopologicalSpace α] [OrderTopology α] [T2Space β] {f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Tendsto f (𝓝[>] a) (𝓝 y)) : Function.rightLim f a = y := @leftLim_eq_of_tendsto αᵒᵈ _ _ _ _ _ _ f a y h h' theorem rightLim_eq_of_eq_bot [TopologicalSpace α] [OrderTopology α] (f : α → β) {a : α} (h : 𝓝[>] a = ⊥) : rightLim f a = f a := @leftLim_eq_of_eq_bot αᵒᵈ _ _ _ _ _ f a h end open Function namespace Monotone variable {α β : Type} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Monotone f) {x y : α} include hf theorem leftLim_eq_sSup [TopologicalSpace α] [OrderTopology α] (h : 𝓝[<] x ≠ ⊥) : leftLim f x = sSup (f '' Iio x) := leftLim_eq_of_tendsto h (hf.tendsto_nhdsLT x) theorem rightLim_eq_sInf [TopologicalSpace α] [OrderTopology α] (h : 𝓝[>] x ≠ ⊥) : rightLim f x = sInf (f '' Ioi x) := rightLim_eq_of_tendsto h (hf.tendsto_nhdsGT x) theorem leftLim_le (h : x ≤ y) : leftLim f x ≤ f y := by letI : TopologicalSpace α := Preorder.topology α haveI : OrderTopology α := ⟨rfl⟩ rcases eq_or_ne (𝓝[<] x) ⊥ with (h' \| h') · simpa [leftLim, h'] using hf h haveI A : NeBot (𝓝[<] x) := neBot_iff.2 h' rw [leftLim_eq_sSup hf h'] refine csSup_le ?_ ?_ · simp only [image_nonempty] exact (forall_mem_nonempty_iff_neBot.2 A) _ self_mem_nhdsWithin · simp only [mem_image, mem_Iio, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] intro z hz exact hf (hz.le.trans h) theorem le_leftLim (h : x < y) : f x ≤ leftLim f y := by letI : TopologicalSpace α := Preorder.topology α haveI : OrderTopology α := ⟨rfl⟩ rcases eq_or_ne (𝓝[<] y) ⊥ with (h' \| h') · rw [leftLim_eq_of_eq_bot _ h'] exact hf h.le rw [leftLim_eq_sSup hf h'] refine le_csSup ⟨f y, ?_⟩ (mem_image_of_mem _ h) simp only [upperBounds, mem_image, mem_Iio, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_setOf_eq] intro z hz exact hf hz.le @[mono] protected theorem leftLim : Monotone (leftLim f) := by intro x y h rcases eq_or_lt_of_le h with (rfl \| hxy) · exact le_rfl · exact (hf.leftLim_le le_rfl).trans (hf.le_leftLim hxy) theorem le_rightLim (h : x ≤ y) : f x ≤ rightLim f y := hf.dual.leftLim_le h theorem rightLim_le (h : x < y) : rightLim f x ≤ f y := hf.dual.le_leftLim h @[mono] protected theorem rightLim : Monotone (rightLim f) := fun _ _ h => hf.dual.leftLim h theorem leftLim_le_rightLim (h : x ≤ y) : leftLim f x ≤ rightLim f y := (hf.leftLim_le le_rfl).trans (hf.le_rightLim h) theorem rightLim_le_leftLim (h : x < y) : rightLim f x ≤ leftLim f y := by letI : TopologicalSpace α := Preorder.topology α haveI : OrderTopology α := ⟨rfl⟩ rcases eq_or_neBot (𝓝[<] y) with (h' \| h') · simpa [leftLim, h'] using rightLim_le hf h obtain ⟨a, ⟨xa, ay⟩⟩ : (Ioo x y).Nonempty := nonempty_of_mem (Ioo_mem_nhdsLT h) calc rightLim f x ≤ f a := hf.rightLim_le xa _ ≤ leftLim f y := hf.le_leftLim ay variable [TopologicalSpace α] [OrderTopology α] theorem tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) := by rcases eq_or_ne (𝓝[<] x) ⊥ with (h' \| h') · simp [h'] rw [leftLim_eq_sSup hf h'] exact hf.tendsto_nhdsLT x theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x) := by apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within f (hf.tendsto_leftLim x) filter_upwards [@self_mem_nhdsWithin _ _ x (Iio x)] with y hy using hf.le_leftLim hy theorem tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) := hf.dual.tendsto_leftLim x theorem tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≥] rightLim f x) := hf.dual.tendsto_leftLim_within x /-- A monotone function is continuous to the left at a point if and only if its left limit coincides with the value of the function. -/ theorem continuousWithinAt_Iio_iff_leftLim_eq : ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x := by rcases eq_or_ne (𝓝[<] x) ⊥ with (h' \| h') · simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h'] haveI : (𝓝[Iio x] x).NeBot := neBot_iff.2 h' refine ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => ?_⟩ have := hf.tendsto_leftLim x rwa [h] at this /-- A monotone function is continuous to the right at a point if and only if its right limit coincides with the value of the function. -/ theorem continuousWithinAt_Ioi_iff_rightLim_eq : ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x := hf.dual.continuousWithinAt_Iio_iff_leftLim_eq /-- A monotone function is continuous at a point if and only if its left and right limits coincide. -/ theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x := by refine ⟨fun h => ?_, fun h => ?_⟩ · have A : leftLim f x = f x := hf.continuousWithinAt_Iio_iff_leftLim_eq.1 h.continuousWithinAt have B : rightLim f x = f x := hf.continuousWithinAt_Ioi_iff_rightLim_eq.1 h.continuousWithinAt exact A.trans B.symm · have h' : leftLim f x = f x := by apply le_antisymm (leftLim_le hf (le_refl _)) rw [h] exact le_rightLim hf (le_refl _) refine continuousAt_iff_continuous_left'_right'.2 ⟨?_, ?_⟩ · exact hf.continuousWithinAt_Iio_iff_leftLim_eq.2 h' · rw [h] at h' exact hf.continuousWithinAt_Ioi_iff_rightLim_eq.2 h' end Monotone namespace Antitone variable {α β : Type*} [LinearOrder α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (hf : Antitone f) {x y : α} include hf theorem le_leftLim (h : x ≤ y) : f y ≤ leftLim f x := hf.dual_right.leftLim_le h theorem leftLim_le (h : x < y) : leftLim f y ≤ f x := hf.dual_right.le_leftLim h @[mono] protected theorem leftLim : Antitone (leftLim f) := hf.dual_right.leftLim theorem rightLim_le (h : x ≤ y) : rightLim f y ≤ f x := hf.dual_right.le_rightLim h theorem le_rightLim (h : x < y) : f y ≤ rightLim f x := hf.dual_right.rightLim_le h @[mono] protected theorem rightLim : Antitone (rightLim f) := hf.dual_right.rightLim theorem rightLim_le_leftLim (h : x ≤ y) : rightLim f y ≤ leftLim f x := hf.dual_right.leftLim_le_rightLim h theorem leftLim_le_rightLim (h : x < y) : leftLim f y ≤ rightLim f x := hf.dual_right.rightLim_le_leftLim h variable [TopologicalSpace α] [OrderTopology α] theorem tendsto_leftLim (x : α) : Tendsto f (𝓝[<] x) (𝓝 (leftLim f x)) := hf.dual_right.tendsto_leftLim x theorem tendsto_leftLim_within (x : α) : Tendsto f (𝓝[<] x) (𝓝[≥] leftLim f x) := hf.dual_right.tendsto_leftLim_within x theorem tendsto_rightLim (x : α) : Tendsto f (𝓝[>] x) (𝓝 (rightLim f x)) := hf.dual_right.tendsto_rightLim x theorem tendsto_rightLim_within (x : α) : Tendsto f (𝓝[>] x) (𝓝[≤] rightLim f x) := hf.dual_right.tendsto_rightLim_within x /-- An antitone function is continuous to the left at a point if and only if its left limit coincides with the value of the function. -/ theorem continuousWithinAt_Iio_iff_leftLim_eq : ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x := hf.dual_right.continuousWithinAt_Iio_iff_leftLim_eq /-- An antitone function is continuous to the right at a point if and only if its right limit coincides with the value of the function. -/ theorem continuousWithinAt_Ioi_iff_rightLim_eq : ContinuousWithinAt f (Ioi x) x ↔ rightLim f x = f x := hf.dual_right.continuousWithinAt_Ioi_iff_rightLim_eq /-- An antitone function is continuous at a point if and only if its left and right limits coincide. -/ theorem continuousAt_iff_leftLim_eq_rightLim : ContinuousAt f x ↔ leftLim f x = rightLim f x := hf.dual_right.continuousAt_iff_leftLim_eq_rightLim end Antitone
.lake/packages/mathlib/Mathlib/Topology/Order/LawsonTopology.lean	import Mathlib.Topology.Order.LowerUpperTopology import Mathlib.Topology.Order.ScottTopology /-! # Lawson topology This file introduces the Lawson topology on a preorder. ## Main definitions - `Topology.lawson` - the Lawson topology is defined as the meet of the lower topology and the Scott topology. - `Topology.IsLawson.lawsonBasis` - The complements of the upper closures of finite sets intersected with Scott open sets. ## Main statements - `Topology.IsLawson.isTopologicalBasis` - `Topology.IsLawson.lawsonBasis` is a basis for `Topology.IsLawson` - `Topology.lawsonOpen_iff_scottOpen_of_isUpperSet'` - An upper set is Lawson open if and only if it is Scott open - `Topology.lawsonClosed_iff_dirSupClosed_of_isLowerSet` - A lower set is Lawson closed if and only if it is closed under sups of directed sets - `Topology.IsLawson.t0Space` - The Lawson topology is T₀ ## Implementation notes A type synonym `Topology.WithLawson` is introduced and for a preorder `α`, `Topology.WithLawson α` is made an instance of `TopologicalSpace` by `Topology.lawson`. We define a mixin class `Topology.IsLawson` for the class of types which are both a preorder and a topology and where the topology is `Topology.lawson`. It is shown that `Topology.WithLawson α` is an instance of `Topology.IsLawson`. ## References * [Gierz et al, A Compendium of Continuous Lattices][GierzEtAl1980] ## Tags Lawson topology, preorder -/ open Set TopologicalSpace variable {α : Type} namespace Topology /-! ### Lawson topology -/ section Lawson section Preorder /-- The Lawson topology is defined as the meet of `Topology.lower` and the `Topology.scott`. -/ def lawson (α : Type) [Preorder α] : TopologicalSpace α := lower α ⊓ scott α univ variable (α) [Preorder α] [TopologicalSpace α] /-- Predicate for an ordered topological space to be equipped with its Lawson topology. The Lawson topology is defined as the meet of `Topology.lower` and the `Topology.scott`. -/ class IsLawson : Prop where topology_eq_lawson : ‹TopologicalSpace α› = lawson α end Preorder namespace IsLawson section Preorder variable (α) [Preorder α] [TopologicalSpace α] [IsLawson α] /-- The complements of the upper closures of finite sets intersected with Scott open sets form a basis for the lawson topology. -/ def lawsonBasis := { s : Set α \| ∃ t : Set α, t.Finite ∧ ∃ u : Set α, IsOpen[scott α univ] u ∧ u \ upperClosure t = s } protected theorem isTopologicalBasis : TopologicalSpace.IsTopologicalBasis (lawsonBasis α) := by have lawsonBasis_image2 : lawsonBasis α = (image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U \| IsOpen[scott α univ] U}) := by rw [lawsonBasis, image2, IsLower.lowerBasis] simp_rw [diff_eq_compl_inter] aesop rw [lawsonBasis_image2] convert IsTopologicalBasis.inf_induced IsLower.isTopologicalBasis (isTopologicalBasis_opens (α := WithScott α)) WithLower.toLower WithScott.toScott rw [@topology_eq_lawson α _ _ _, lawson] apply (congrArg₂ min _) _ · letI _ := lower α exact (@IsLower.withLowerHomeomorph α ‹_› (lower α) ⟨rfl⟩).isInducing.eq_induced · letI _ := scott α univ exact (@IsScott.withScottHomeomorph α _ (scott α univ) ⟨rfl⟩).isInducing.eq_induced end Preorder end IsLawson /-- Type synonym for a preorder equipped with the Lawson topology. -/ def WithLawson (α : Type) := α namespace WithLawson /-- `toLawson` is the identity function to the `WithLawson` of a type. -/ @[match_pattern] def toLawson : α ≃ WithLawson α := Equiv.refl _ /-- `ofLawson` is the identity function from the `WithLawson` of a type. -/ @[match_pattern] def ofLawson : WithLawson α ≃ α := Equiv.refl _ @[simp] lemma to_Lawson_symm_eq : (@toLawson α).symm = ofLawson := rfl @[simp] lemma of_Lawson_symm_eq : (@ofLawson α).symm = toLawson := rfl @[simp] lemma toLawson_ofLawson (a : WithLawson α) : toLawson (ofLawson a) = a := rfl @[simp] lemma ofLawson_toLawson (a : α) : ofLawson (toLawson a) = a := rfl lemma toLawson_inj {a b : α} : toLawson a = toLawson b ↔ a = b := Iff.rfl lemma ofLawson_inj {a b : WithLawson α} : ofLawson a = ofLawson b ↔ a = b := Iff.rfl /-- A recursor for `WithLawson`. Use as `induction x`. -/ @[elab_as_elim, cases_eliminator, induction_eliminator] protected def rec {β : WithLawson α → Sort} (h : ∀ a, β (toLawson a)) : ∀ a, β a := fun a => h (ofLawson a) instance [Nonempty α] : Nonempty (WithLawson α) := ‹Nonempty α› instance [Inhabited α] : Inhabited (WithLawson α) := ‹Inhabited α› variable [Preorder α] instance instPreorder : Preorder (WithLawson α) := ‹Preorder α› instance instTopologicalSpace : TopologicalSpace (WithLawson α) := lawson α instance instIsLawson : IsLawson (WithLawson α) := ⟨rfl⟩ /-- If `α` is equipped with the Lawson topology, then it is homeomorphic to `WithLawson α`. -/ def homeomorph [TopologicalSpace α] [IsLawson α] : WithLawson α ≃ₜ α := ofLawson.toHomeomorphOfIsInducing ⟨IsLawson.topology_eq_lawson (α := α) ▸ induced_id.symm⟩ theorem isOpen_preimage_ofLawson {S : Set α} : IsOpen (ofLawson ⁻¹' S) ↔ (lawson α).IsOpen S := Iff.rfl theorem isClosed_preimage_ofLawson {S : Set α} : IsClosed (ofLawson ⁻¹' S) ↔ IsClosed[lawson α] S := Iff.rfl theorem isOpen_def {T : Set (WithLawson α)} : IsOpen T ↔ (lawson α).IsOpen (toLawson ⁻¹' T) := Iff.rfl end WithLawson end Lawson section Preorder variable [Preorder α] lemma lawson_le_scott : lawson α ≤ scott α univ := inf_le_right lemma lawson_le_lower : lawson α ≤ lower α := inf_le_left lemma scottHausdorff_le_lawson : scottHausdorff α univ ≤ lawson α := le_inf scottHausdorff_le_lower scottHausdorff_le_scott lemma lawsonClosed_of_scottClosed (s : Set α) (h : IsClosed (WithScott.ofScott ⁻¹' s)) : IsClosed (WithLawson.ofLawson ⁻¹' s) := h.mono lawson_le_scott lemma lawsonClosed_of_lowerClosed (s : Set α) (h : IsClosed (WithLower.ofLower ⁻¹' s)) : IsClosed (WithLawson.ofLawson ⁻¹' s) := h.mono lawson_le_lower /-- An upper set is Lawson open if and only if it is Scott open -/ lemma lawsonOpen_iff_scottOpen_of_isUpperSet {s : Set α} (h : IsUpperSet s) : IsOpen (WithLawson.ofLawson ⁻¹' s) ↔ IsOpen (WithScott.ofScott ⁻¹' s) := ⟨fun hs => IsScott.isOpen_iff_isUpperSet_and_scottHausdorff_open (D := univ).mpr ⟨h, (scottHausdorff_le_lawson s) hs⟩, lawson_le_scott _⟩ variable (L : TopologicalSpace α) (S : TopologicalSpace α) variable [@IsLawson α _ L] [@IsScott α univ _ S] lemma isLawson_le_isScott : L ≤ S := by rw [@IsScott.topology_eq α univ _ S _, @IsLawson.topology_eq_lawson α _ L _] exact inf_le_right lemma scottHausdorff_le_isLawson : scottHausdorff α univ ≤ L := by rw [@IsLawson.topology_eq_lawson α _ L _] exact scottHausdorff_le_lawson /-- An upper set is Lawson open if and only if it is Scott open -/ lemma lawsonOpen_iff_scottOpen_of_isUpperSet' (s : Set α) (h : IsUpperSet s) : IsOpen[L] s ↔ IsOpen[S] s := by rw [@IsLawson.topology_eq_lawson α _ L _, @IsScott.topology_eq α univ _ S _] exact lawsonOpen_iff_scottOpen_of_isUpperSet h lemma lawsonClosed_iff_scottClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) : IsClosed[L] s ↔ IsClosed[S] s := by rw [← @isOpen_compl_iff, ← isOpen_compl_iff, (lawsonOpen_iff_scottOpen_of_isUpperSet' L S _ (isUpperSet_compl.mpr h))] include S in /-- A lower set is Lawson closed if and only if it is closed under sups of directed sets -/ lemma lawsonClosed_iff_dirSupClosed_of_isLowerSet (s : Set α) (h : IsLowerSet s) : IsClosed[L] s ↔ DirSupClosed s := by rw [lawsonClosed_iff_scottClosed_of_isLowerSet L S _ h, @IsScott.isClosed_iff_isLowerSet_and_dirSupClosed] simp_all end Preorder namespace IsLawson variable [PartialOrder α] [TopologicalSpace α] [IsLawson α] /-- The Lawson topology on a partial order is T₁. -/ -- see Note [lower instance priority] instance (priority := 90) toT1Space : T1Space α where t1 a := by simp only [IsLawson.topology_eq_lawson] rw [← (Set.OrdConnected.upperClosure_inter_lowerClosure ordConnected_singleton), ← WithLawson.isClosed_preimage_ofLawson] apply IsClosed.inter (lawsonClosed_of_lowerClosed _ (IsLower.isClosed_upperClosure (finite_singleton a))) rw [lowerClosure_singleton, LowerSet.coe_Iic, ← WithLawson.isClosed_preimage_ofLawson] exact lawsonClosed_of_scottClosed _ isClosed_Iic @[deprecated (since := "2025-07-02")] protected alias singleton_isClosed := isClosed_singleton end IsLawson end Topology
.lake/packages/mathlib/Mathlib/Topology/Order/MonotoneConvergence.lean	import Mathlib.Topology.Order.Basic /-! # Bounded monotone sequences converge In this file we prove a few theorems of the form “if the range of a monotone function `f : ι → α` admits a least upper bound `a`, then `f x` tends to `a` as `x → ∞`”, as well as version of this statement for (conditionally) complete lattices that use `⨆ x, f x` instead of `IsLUB`. These theorems work for linear orders with order topologies as well as their products (both in terms of `Prod` and in terms of function types). In order to reduce code duplication, we introduce two typeclasses (one for the property formulated above and one for the dual property), prove theorems assuming one of these typeclasses, and provide instances for linear orders and their products. We also prove some "inverse" results: if `f n` is a monotone sequence and `a` is its limit, then `f n ≤ a` for all `n`. ## Tags monotone convergence -/ open Filter Set Function open scoped Topology variable {α β : Type} /-- We say that `α` is a `SupConvergenceClass` if the following holds. Let `f : ι → α` be a monotone function, let `a : α` be a least upper bound of `Set.range f`. Then `f x` tends to `𝓝 a` as `x → ∞` (formally, at the filter `Filter.atTop`). We require this for `ι = (s : Set α)`, `f = (↑)` in the definition, then prove it for any `f` in `tendsto_atTop_isLUB`. This property holds for linear orders with order topology as well as their products. -/ class SupConvergenceClass (α : Type) [Preorder α] [TopologicalSpace α] : Prop where /-- proof that a monotone function tends to `𝓝 a` as `x → ∞` -/ tendsto_coe_atTop_isLUB : ∀ (a : α) (s : Set α), IsLUB s a → Tendsto ((↑) : s → α) atTop (𝓝 a) /-- We say that `α` is an `InfConvergenceClass` if the following holds. Let `f : ι → α` be a monotone function, let `a : α` be a greatest lower bound of `Set.range f`. Then `f x` tends to `𝓝 a` as `x → -∞` (formally, at the filter `Filter.atBot`). We require this for `ι = (s : Set α)`, `f = (↑)` in the definition, then prove it for any `f` in `tendsto_atBot_isGLB`. This property holds for linear orders with order topology as well as their products. -/ class InfConvergenceClass (α : Type) [Preorder α] [TopologicalSpace α] : Prop where /-- proof that a monotone function tends to `𝓝 a` as `x → -∞` -/ tendsto_coe_atBot_isGLB : ∀ (a : α) (s : Set α), IsGLB s a → Tendsto ((↑) : s → α) atBot (𝓝 a) instance OrderDual.supConvergenceClass [Preorder α] [TopologicalSpace α] [InfConvergenceClass α] : SupConvergenceClass αᵒᵈ := ⟨‹InfConvergenceClass α›.1⟩ instance OrderDual.infConvergenceClass [Preorder α] [TopologicalSpace α] [SupConvergenceClass α] : InfConvergenceClass αᵒᵈ := ⟨‹SupConvergenceClass α›.1⟩ -- see Note [lower instance priority] instance (priority := 100) LinearOrder.supConvergenceClass [TopologicalSpace α] [LinearOrder α] [OrderTopology α] : SupConvergenceClass α := by refine ⟨fun a s ha => tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩⟩ · rcases ha.exists_between hb with ⟨c, hcs, bc, bca⟩ lift c to s using hcs exact (eventually_ge_atTop c).mono fun x hx => bc.trans_le hx · exact Eventually.of_forall fun x => (ha.1 x.2).trans_lt hb -- see Note [lower instance priority] instance (priority := 100) LinearOrder.infConvergenceClass [TopologicalSpace α] [LinearOrder α] [OrderTopology α] : InfConvergenceClass α := show InfConvergenceClass αᵒᵈᵒᵈ from OrderDual.infConvergenceClass section variable {ι : Type} [Preorder ι] [TopologicalSpace α] section IsLUB variable [Preorder α] [SupConvergenceClass α] {f : ι → α} {a : α} theorem tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) : Tendsto f atTop (𝓝 a) := by suffices Tendsto (rangeFactorization f) atTop atTop from (SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge theorem tendsto_atBot_isLUB (h_anti : Antitone f) (ha : IsLUB (Set.range f) a) : Tendsto f atBot (𝓝 a) := by convert tendsto_atTop_isLUB h_anti.dual_left ha using 1 end IsLUB section IsGLB variable [Preorder α] [InfConvergenceClass α] {f : ι → α} {a : α} theorem tendsto_atBot_isGLB (h_mono : Monotone f) (ha : IsGLB (Set.range f) a) : Tendsto f atBot (𝓝 a) := by convert tendsto_atTop_isLUB h_mono.dual ha.dual using 1 theorem tendsto_atTop_isGLB (h_anti : Antitone f) (ha : IsGLB (Set.range f) a) : Tendsto f atTop (𝓝 a) := by convert tendsto_atBot_isLUB h_anti.dual ha.dual using 1 end IsGLB section CiSup variable [ConditionallyCompleteLattice α] [SupConvergenceClass α] {f : ι → α} theorem tendsto_atTop_ciSup (h_mono : Monotone f) (hbdd : BddAbove <\| range f) : Tendsto f atTop (𝓝 (⨆ i, f i)) := by cases isEmpty_or_nonempty ι exacts [tendsto_of_isEmpty, tendsto_atTop_isLUB h_mono (isLUB_ciSup hbdd)] theorem tendsto_atBot_ciSup (h_anti : Antitone f) (hbdd : BddAbove <\| range f) : Tendsto f atBot (𝓝 (⨆ i, f i)) := by convert tendsto_atTop_ciSup h_anti.dual hbdd.dual using 1 end CiSup section CiInf variable [ConditionallyCompleteLattice α] [InfConvergenceClass α] {f : ι → α} theorem tendsto_atBot_ciInf (h_mono : Monotone f) (hbdd : BddBelow <\| range f) : Tendsto f atBot (𝓝 (⨅ i, f i)) := by convert tendsto_atTop_ciSup h_mono.dual hbdd.dual using 1 theorem tendsto_atTop_ciInf (h_anti : Antitone f) (hbdd : BddBelow <\| range f) : Tendsto f atTop (𝓝 (⨅ i, f i)) := by convert tendsto_atBot_ciSup h_anti.dual hbdd.dual using 1 end CiInf section iSup variable [CompleteLattice α] [SupConvergenceClass α] {f : ι → α} theorem tendsto_atTop_iSup (h_mono : Monotone f) : Tendsto f atTop (𝓝 (⨆ i, f i)) := tendsto_atTop_ciSup h_mono (OrderTop.bddAbove _) theorem tendsto_atBot_iSup (h_anti : Antitone f) : Tendsto f atBot (𝓝 (⨆ i, f i)) := tendsto_atBot_ciSup h_anti (OrderTop.bddAbove _) end iSup section iInf variable [CompleteLattice α] [InfConvergenceClass α] {f : ι → α} theorem tendsto_atBot_iInf (h_mono : Monotone f) : Tendsto f atBot (𝓝 (⨅ i, f i)) := tendsto_atBot_ciInf h_mono (OrderBot.bddBelow _) theorem tendsto_atTop_iInf (h_anti : Antitone f) : Tendsto f atTop (𝓝 (⨅ i, f i)) := tendsto_atTop_ciInf h_anti (OrderBot.bddBelow _) end iInf end instance Prod.supConvergenceClass [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [SupConvergenceClass α] [SupConvergenceClass β] : SupConvergenceClass (α × β) := by constructor rintro ⟨a, b⟩ s h rw [isLUB_prod, ← range_restrict, ← range_restrict] at h have A : Tendsto (fun x : s => (x : α × β).1) atTop (𝓝 a) := tendsto_atTop_isLUB (monotone_fst.restrict s) h.1 have B : Tendsto (fun x : s => (x : α × β).2) atTop (𝓝 b) := tendsto_atTop_isLUB (monotone_snd.restrict s) h.2 exact A.prodMk_nhds B instance [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [InfConvergenceClass α] [InfConvergenceClass β] : InfConvergenceClass (α × β) := show InfConvergenceClass (αᵒᵈ × βᵒᵈ)ᵒᵈ from OrderDual.infConvergenceClass instance Pi.supConvergenceClass {ι : Type} {α : ι → Type} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)] [∀ i, SupConvergenceClass (α i)] : SupConvergenceClass (∀ i, α i) := by refine ⟨fun f s h => ?_⟩ simp only [isLUB_pi, ← range_restrict] at h exact tendsto_pi_nhds.2 fun i => tendsto_atTop_isLUB ((monotone_eval _).restrict _) (h i) instance Pi.infConvergenceClass {ι : Type} {α : ι → Type} [∀ i, Preorder (α i)] [∀ i, TopologicalSpace (α i)] [∀ i, InfConvergenceClass (α i)] : InfConvergenceClass (∀ i, α i) := show InfConvergenceClass (∀ i, (α i)ᵒᵈ)ᵒᵈ from OrderDual.infConvergenceClass instance Pi.supConvergenceClass' {ι : Type} [Preorder α] [TopologicalSpace α] [SupConvergenceClass α] : SupConvergenceClass (ι → α) := supConvergenceClass instance Pi.infConvergenceClass' {ι : Type} [Preorder α] [TopologicalSpace α] [InfConvergenceClass α] : InfConvergenceClass (ι → α) := Pi.infConvergenceClass theorem tendsto_of_monotone {ι α : Type} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Monotone f) : Tendsto f atTop atTop ∨ ∃ l, Tendsto f atTop (𝓝 l) := by classical exact if H : BddAbove (range f) then Or.inr ⟨_, tendsto_atTop_ciSup h_mono H⟩ else Or.inl <\| tendsto_atTop_atTop_of_monotone' h_mono H theorem tendsto_of_antitone {ι α : Type} [Preorder ι] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] {f : ι → α} (h_mono : Antitone f) : Tendsto f atTop atBot ∨ ∃ l, Tendsto f atTop (𝓝 l) := @tendsto_of_monotone ι αᵒᵈ _ _ _ _ _ h_mono theorem tendsto_iff_tendsto_subseq_of_monotone {ι₁ ι₂ α : Type} [SemilatticeSup ι₁] [Preorder ι₂] [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [NoMaxOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Monotone f) (hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) := by constructor <;> intro h · exact h.comp hg · rcases tendsto_of_monotone hf with (h' \| ⟨l', hl'⟩) · exact (not_tendsto_atTop_of_tendsto_nhds h (h'.comp hg)).elim · rwa [tendsto_nhds_unique h (hl'.comp hg)] theorem tendsto_iff_tendsto_subseq_of_antitone {ι₁ ι₂ α : Type} [SemilatticeSup ι₁] [Preorder ι₂] [Nonempty ι₁] [TopologicalSpace α] [ConditionallyCompleteLinearOrder α] [OrderTopology α] [NoMinOrder α] {f : ι₂ → α} {φ : ι₁ → ι₂} {l : α} (hf : Antitone f) (hg : Tendsto φ atTop atTop) : Tendsto f atTop (𝓝 l) ↔ Tendsto (f ∘ φ) atTop (𝓝 l) := tendsto_iff_tendsto_subseq_of_monotone (α := αᵒᵈ) hf hg /-! The next family of results, such as `isLUB_of_tendsto_atTop` and `iSup_eq_of_tendsto`, are converses to the standard fact that bounded monotone functions converge. They state, that if a monotone function `f` tends to `a` along `Filter.atTop`, then that value `a` is a least upper bound for the range of `f`. Related theorems above (`IsLUB.isLUB_of_tendsto`, `IsGLB.isGLB_of_tendsto` etc) cover the case when `f x` tends to `a` as `x` tends to some point `b` in the domain. -/ theorem Monotone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) : f b ≤ a := haveI : Nonempty β := Nonempty.intro b _root_.ge_of_tendsto ha ((eventually_ge_atTop b).mono fun _ hxy => hf hxy) theorem Monotone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) : a ≤ f b := hf.dual.ge_of_tendsto ha b theorem Antitone.le_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atTop (𝓝 a)) (b : β) : a ≤ f b := hf.dual_right.ge_of_tendsto ha b theorem Antitone.ge_of_tendsto [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atBot (𝓝 a)) (b : β) : f b ≤ a := hf.dual_right.le_of_tendsto ha b theorem isLUB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atTop (𝓝 a)) : IsLUB (Set.range f) a := by constructor · rintro _ ⟨b, rfl⟩ exact hf.ge_of_tendsto ha b · exact fun _ hb => le_of_tendsto' ha fun x => hb (Set.mem_range_self x) theorem isGLB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Monotone f) (ha : Tendsto f atBot (𝓝 a)) : IsGLB (Set.range f) a := @isLUB_of_tendsto_atTop αᵒᵈ βᵒᵈ _ _ _ _ _ _ _ hf.dual ha theorem isLUB_of_tendsto_atBot [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeInf β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atBot (𝓝 a)) : IsLUB (Set.range f) a := @isLUB_of_tendsto_atTop α βᵒᵈ _ _ _ _ _ _ _ hf.dual_left ha theorem isGLB_of_tendsto_atTop [TopologicalSpace α] [Preorder α] [OrderClosedTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) (ha : Tendsto f atTop (𝓝 a)) : IsGLB (Set.range f) a := @isGLB_of_tendsto_atBot α βᵒᵈ _ _ _ _ _ _ _ hf.dual_left ha theorem iSup_eq_of_tendsto {α β} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Monotone f) : Tendsto f atTop (𝓝 a) → iSup f = a := tendsto_nhds_unique (tendsto_atTop_iSup hf) theorem iInf_eq_of_tendsto {α} [TopologicalSpace α] [CompleteLinearOrder α] [OrderTopology α] [Nonempty β] [SemilatticeSup β] {f : β → α} {a : α} (hf : Antitone f) : Tendsto f atTop (𝓝 a) → iInf f = a := tendsto_nhds_unique (tendsto_atTop_iInf hf) theorem iSup_eq_iSup_subseq_of_monotone {ι₁ ι₂ α : Type} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f) (hφ : Tendsto φ l atTop) : ⨆ i, f i = ⨆ i, f (φ i) := le_antisymm (iSup_mono' fun i => Exists.imp (fun j (hj : i ≤ φ j) => hf hj) (hφ.eventually <\| eventually_ge_atTop i).exists) (iSup_mono' fun i => ⟨φ i, le_rfl⟩) theorem iSup_eq_iSup_subseq_of_antitone {ι₁ ι₂ α : Type} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f) (hφ : Tendsto φ l atBot) : ⨆ i, f i = ⨆ i, f (φ i) := le_antisymm (iSup_mono' fun i => Exists.imp (fun j (hj : φ j ≤ i) => hf hj) (hφ.eventually <\| eventually_le_atBot i).exists) (iSup_mono' fun i => ⟨φ i, le_rfl⟩) theorem iInf_eq_iInf_subseq_of_monotone {ι₁ ι₂ α : Type} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Monotone f) (hφ : Tendsto φ l atBot) : ⨅ i, f i = ⨅ i, f (φ i) := iSup_eq_iSup_subseq_of_monotone hf.dual hφ theorem iInf_eq_iInf_subseq_of_antitone {ι₁ ι₂ α : Type} [Preorder ι₂] [CompleteLattice α] {l : Filter ι₁} [l.NeBot] {f : ι₂ → α} {φ : ι₁ → ι₂} (hf : Antitone f) (hφ : Tendsto φ l atTop) : ⨅ i, f i = ⨅ i, f (φ i) := iSup_eq_iSup_subseq_of_antitone hf.dual hφ
.lake/packages/mathlib/Mathlib/Topology/Order/NhdsSet.lean	import Mathlib.Topology.Order.Basic /-! # Set neighborhoods of intervals In this file we prove basic theorems about `𝓝ˢ s`, where `s` is one of the intervals `Set.Ici`, `Set.Iic`, `Set.Ioi`, `Set.Iio`, `Set.Ico`, `Set.Ioc`, `Set.Ioo`, and `Set.Icc`. First, we prove lemmas in terms of filter equalities. Then we prove lemmas about `s ∈ 𝓝ˢ t`, where both `s` and `t` are intervals. Finally, we prove a few lemmas about filter bases of `𝓝ˢ (Iic a)` and `𝓝ˢ (Ici a)`. -/ open Set Filter OrderDual open scoped Topology section OrderClosedTopology variable {α : Type} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {a b c d : α} /-! ### Formulae for `𝓝ˢ` of intervals -/ @[simp] theorem nhdsSet_Ioi : 𝓝ˢ (Ioi a) = 𝓟 (Ioi a) := isOpen_Ioi.nhdsSet_eq @[simp] theorem nhdsSet_Iio : 𝓝ˢ (Iio a) = 𝓟 (Iio a) := isOpen_Iio.nhdsSet_eq @[simp] theorem nhdsSet_Ioo : 𝓝ˢ (Ioo a b) = 𝓟 (Ioo a b) := isOpen_Ioo.nhdsSet_eq theorem nhdsSet_Ici : 𝓝ˢ (Ici a) = 𝓝 a ⊔ 𝓟 (Ioi a) := by rw [← Ioi_insert, nhdsSet_insert, nhdsSet_Ioi] theorem nhdsSet_Iic : 𝓝ˢ (Iic a) = 𝓝 a ⊔ 𝓟 (Iio a) := nhdsSet_Ici (α := αᵒᵈ) theorem nhdsSet_Ico (h : a < b) : 𝓝ˢ (Ico a b) = 𝓝 a ⊔ 𝓟 (Ioo a b) := by rw [← Ioo_insert_left h, nhdsSet_insert, nhdsSet_Ioo] theorem nhdsSet_Ioc (h : a < b) : 𝓝ˢ (Ioc a b) = 𝓝 b ⊔ 𝓟 (Ioo a b) := by rw [← Ioo_insert_right h, nhdsSet_insert, nhdsSet_Ioo] theorem nhdsSet_Icc (h : a ≤ b) : 𝓝ˢ (Icc a b) = 𝓝 a ⊔ 𝓝 b ⊔ 𝓟 (Ioo a b) := by rcases h.eq_or_lt with rfl \| hlt · simp · rw [← Ioc_insert_left h, nhdsSet_insert, nhdsSet_Ioc hlt, sup_assoc] /-! ### Lemmas about `Ixi _ ∈ 𝓝ˢ (Set.Ici _)` -/ @[simp] theorem Ioi_mem_nhdsSet_Ici_iff : Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b := by rw [isOpen_Ioi.mem_nhdsSet, Ici_subset_Ioi] alias ⟨_, Ioi_mem_nhdsSet_Ici⟩ := Ioi_mem_nhdsSet_Ici_iff theorem Ici_mem_nhdsSet_Ici (h : a < b) : Ici a ∈ 𝓝ˢ (Ici b) := mem_of_superset (Ioi_mem_nhdsSet_Ici h) Ioi_subset_Ici_self /-! ### Lemmas about `Iix _ ∈ 𝓝ˢ (Set.Iic _)` -/ theorem Iio_mem_nhdsSet_Iic_iff : Iio b ∈ 𝓝ˢ (Iic a) ↔ a < b := Ioi_mem_nhdsSet_Ici_iff (α := αᵒᵈ) alias ⟨_, Iio_mem_nhdsSet_Iic⟩ := Iio_mem_nhdsSet_Iic_iff theorem Iic_mem_nhdsSet_Iic (h : a < b) : Iic b ∈ 𝓝ˢ (Iic a) := Ici_mem_nhdsSet_Ici (α := αᵒᵈ) h /-! ### Lemmas about `Ixx _ ?_ ∈ 𝓝ˢ (Set.Icc _ _)` -/ theorem Ioi_mem_nhdsSet_Icc (h : a < b) : Ioi a ∈ 𝓝ˢ (Icc b c) := nhdsSet_mono Icc_subset_Ici_self <\| Ioi_mem_nhdsSet_Ici h theorem Ici_mem_nhdsSet_Icc (h : a < b) : Ici a ∈ 𝓝ˢ (Icc b c) := mem_of_superset (Ioi_mem_nhdsSet_Icc h) Ioi_subset_Ici_self theorem Iio_mem_nhdsSet_Icc (h : b < c) : Iio c ∈ 𝓝ˢ (Icc a b) := nhdsSet_mono Icc_subset_Iic_self <\| Iio_mem_nhdsSet_Iic h theorem Iic_mem_nhdsSet_Icc (h : b < c) : Iic c ∈ 𝓝ˢ (Icc a b) := mem_of_superset (Iio_mem_nhdsSet_Icc h) Iio_subset_Iic_self theorem Ioo_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Icc b c) := inter_mem (Ioi_mem_nhdsSet_Icc h) (Iio_mem_nhdsSet_Icc h') theorem Ico_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Icc b c) := inter_mem (Ici_mem_nhdsSet_Icc h) (Iio_mem_nhdsSet_Icc h') theorem Ioc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Icc b c) := inter_mem (Ioi_mem_nhdsSet_Icc h) (Iic_mem_nhdsSet_Icc h') theorem Icc_mem_nhdsSet_Icc (h : a < b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Icc b c) := inter_mem (Ici_mem_nhdsSet_Icc h) (Iic_mem_nhdsSet_Icc h') /-! ### Lemmas about `Ixx _ ?_ ∈ 𝓝ˢ (Set.Ico _ _)` -/ theorem Ici_mem_nhdsSet_Ico (h : a < b) : Ici a ∈ 𝓝ˢ (Ico b c) := nhdsSet_mono Ico_subset_Icc_self <\| Ici_mem_nhdsSet_Icc h theorem Ioi_mem_nhdsSet_Ico (h : a < b) : Ioi a ∈ 𝓝ˢ (Ico b c) := nhdsSet_mono Ico_subset_Icc_self <\| Ioi_mem_nhdsSet_Icc h theorem Iio_mem_nhdsSet_Ico (h : b ≤ c) : Iio c ∈ 𝓝ˢ (Ico a b) := nhdsSet_mono Ico_subset_Iio_self <\| by simpa theorem Iic_mem_nhdsSet_Ico (h : b ≤ c) : Iic c ∈ 𝓝ˢ (Ico a b) := mem_of_superset (Iio_mem_nhdsSet_Ico h) Iio_subset_Iic_self theorem Ioo_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioo a d ∈ 𝓝ˢ (Ico b c) := inter_mem (Ioi_mem_nhdsSet_Ico h) (Iio_mem_nhdsSet_Ico h') theorem Icc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Icc a d ∈ 𝓝ˢ (Ico b c) := inter_mem (Ici_mem_nhdsSet_Ico h) (Iic_mem_nhdsSet_Ico h') theorem Ioc_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ioc a d ∈ 𝓝ˢ (Ico b c) := inter_mem (Ioi_mem_nhdsSet_Ico h) (Iic_mem_nhdsSet_Ico h') theorem Ico_mem_nhdsSet_Ico (h : a < b) (h' : c ≤ d) : Ico a d ∈ 𝓝ˢ (Ico b c) := inter_mem (Ici_mem_nhdsSet_Ico h) (Iio_mem_nhdsSet_Ico h') /-! ### Lemmas about `Ixx _ ?_ ∈ 𝓝ˢ (Set.Ioc _ _)` -/ theorem Ioi_mem_nhdsSet_Ioc (h : a ≤ b) : Ioi a ∈ 𝓝ˢ (Ioc b c) := nhdsSet_mono Ioc_subset_Ioi_self <\| by simpa theorem Iio_mem_nhdsSet_Ioc (h : b < c) : Iio c ∈ 𝓝ˢ (Ioc a b) := nhdsSet_mono Ioc_subset_Icc_self <\| Iio_mem_nhdsSet_Icc h theorem Ici_mem_nhdsSet_Ioc (h : a ≤ b) : Ici a ∈ 𝓝ˢ (Ioc b c) := mem_of_superset (Ioi_mem_nhdsSet_Ioc h) Ioi_subset_Ici_self theorem Iic_mem_nhdsSet_Ioc (h : b < c) : Iic c ∈ 𝓝ˢ (Ioc a b) := nhdsSet_mono Ioc_subset_Icc_self <\| Iic_mem_nhdsSet_Icc h theorem Ioo_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioo a d ∈ 𝓝ˢ (Ioc b c) := inter_mem (Ioi_mem_nhdsSet_Ioc h) (Iio_mem_nhdsSet_Ioc h') theorem Icc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Icc a d ∈ 𝓝ˢ (Ioc b c) := inter_mem (Ici_mem_nhdsSet_Ioc h) (Iic_mem_nhdsSet_Ioc h') theorem Ioc_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ioc a d ∈ 𝓝ˢ (Ioc b c) := inter_mem (Ioi_mem_nhdsSet_Ioc h) (Iic_mem_nhdsSet_Ioc h') theorem Ico_mem_nhdsSet_Ioc (h : a ≤ b) (h' : c < d) : Ico a d ∈ 𝓝ˢ (Ioc b c) := inter_mem (Ici_mem_nhdsSet_Ioc h) (Iio_mem_nhdsSet_Ioc h') end OrderClosedTopology /-! ### Filter bases of `𝓝ˢ (Iic a)` and `𝓝ˢ (Ici a)` -/ variable {α : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] theorem hasBasis_nhdsSet_Iic_Iio (a : α) [h : Nonempty (Ioi a)] : HasBasis (𝓝ˢ (Iic a)) (a < ·) Iio := by refine ⟨fun s ↦ ⟨fun hs ↦ ?_, fun ⟨b, hab, hb⟩ ↦ mem_of_superset (Iio_mem_nhdsSet_Iic hab) hb⟩⟩ rw [nhdsSet_Iic, mem_sup, mem_principal] at hs rcases exists_Ico_subset_of_mem_nhds hs.1 (Set.nonempty_coe_sort.1 h) with ⟨b, hab, hbs⟩ exact ⟨b, hab, Iio_subset_Iio_union_Ico.trans (union_subset hs.2 hbs)⟩ theorem hasBasis_nhdsSet_Iic_Iic (a : α) [NeBot (𝓝[>] a)] : HasBasis (𝓝ˢ (Iic a)) (a < ·) Iic := by have : Nonempty (Ioi a) := (Filter.nonempty_of_mem (self_mem_nhdsWithin : Ioi a ∈ 𝓝[>] a)).to_subtype refine (hasBasis_nhdsSet_Iic_Iio _).to_hasBasis (fun c hc ↦ ?_) (fun _ h ↦ ⟨_, h, Iio_subset_Iic_self⟩) simpa only [Iic_subset_Iio] using Filter.nonempty_of_mem (Ioo_mem_nhdsGT hc) @[simp] theorem Iic_mem_nhdsSet_Iic_iff {a b : α} [NeBot (𝓝[>] b)] : Iic a ∈ 𝓝ˢ (Iic b) ↔ b < a := (hasBasis_nhdsSet_Iic_Iic b).mem_iff.trans ⟨fun ⟨_c, hbc, hca⟩ ↦ hbc.trans_le (Iic_subset_Iic.1 hca), fun h ↦ ⟨_, h, Subset.rfl⟩⟩ theorem hasBasis_nhdsSet_Ici_Ioi (a : α) [Nonempty (Iio a)] : HasBasis (𝓝ˢ (Ici a)) (· < a) Ioi := have : Nonempty (Ioi (toDual a)) := ‹_›; hasBasis_nhdsSet_Iic_Iio (toDual a) theorem hasBasis_nhdsSet_Ici_Ici (a : α) [NeBot (𝓝[<] a)] : HasBasis (𝓝ˢ (Ici a)) (· < a) Ici := have : NeBot (𝓝[>] (toDual a)) := ‹_›; hasBasis_nhdsSet_Iic_Iic (toDual a) @[simp] theorem Ici_mem_nhdsSet_Ici_iff {a b : α} [NeBot (𝓝[<] b)] : Ici a ∈ 𝓝ˢ (Ici b) ↔ a < b := have : NeBot (𝓝[>] (toDual b)) := ‹_›; Iic_mem_nhdsSet_Iic_iff (a := toDual a) (b := toDual b)
.lake/packages/mathlib/Mathlib/Topology/Order/UpperLowerSetTopology.lean	import Mathlib.Logic.Lemmas import Mathlib.Topology.AlexandrovDiscrete import Mathlib.Topology.ContinuousMap.Basic import Mathlib.Topology.Order.LowerUpperTopology /-! # Upper and lower sets topologies This file introduces the upper set topology on a preorder as the topology where the open sets are the upper sets and the lower set topology on a preorder as the topology where the open sets are the lower sets. In general the upper set topology does not coincide with the upper topology and the lower set topology does not coincide with the lower topology. ## Main statements - `Topology.IsUpperSet.toAlexandrovDiscrete`: The upper set topology is Alexandrov-discrete. - `Topology.IsUpperSet.isClosed_iff_isLower` - a set is closed if and only if it is a Lower set - `Topology.IsUpperSet.closure_eq_lowerClosure` - topological closure coincides with lower closure - `Topology.IsUpperSet.monotone_iff_continuous` - the continuous functions are the monotone functions - `IsUpperSet.monotone_to_upperTopology_continuous`: A monotone map from a preorder with the upper set topology to a preorder with the upper topology is continuous. We provide the upper set topology in three ways (and similarly for the lower set topology): * `Topology.upperSet`: The upper set topology as a `TopologicalSpace α` * `Topology.IsUpperSet`: Prop-valued mixin typeclass stating that an existing topology is the upper set topology. * `Topology.WithUpperSet`: Type synonym equipping a preorder with its upper set topology. ## Motivation An Alexandrov topology is a topology where the intersection of any collection of open sets is open. The upper set topology is an Alexandrov topology and, given any Alexandrov topological space, we can equip it with a preorder (namely the specialization preorder) whose upper set topology coincides with the original topology. See `Topology.Specialization`. ## Tags upper set topology, lower set topology, preorder, Alexandrov -/ open Set TopologicalSpace Filter variable {α β γ : Type} namespace Topology /-- Topology whose open sets are upper sets. Note: In general the upper set topology does not coincide with the upper topology. -/ def upperSet (α : Type) [Preorder α] : TopologicalSpace α where IsOpen := IsUpperSet isOpen_univ := isUpperSet_univ isOpen_inter _ _ := IsUpperSet.inter isOpen_sUnion _ := isUpperSet_sUnion /-- Topology whose open sets are lower sets. Note: In general the lower set topology does not coincide with the lower topology. -/ def lowerSet (α : Type) [Preorder α] : TopologicalSpace α where IsOpen := IsLowerSet isOpen_univ := isLowerSet_univ isOpen_inter _ _ := IsLowerSet.inter isOpen_sUnion _ := isLowerSet_sUnion /-- Type synonym for a preorder equipped with the upper set topology. -/ def WithUpperSet (α : Type) := α namespace WithUpperSet /-- `toUpperSet` is the identity function to the `WithUpperSet` of a type. -/ @[match_pattern] def toUpperSet : α ≃ WithUpperSet α := Equiv.refl _ /-- `ofUpperSet` is the identity function from the `WithUpperSet` of a type. -/ @[match_pattern] def ofUpperSet : WithUpperSet α ≃ α := Equiv.refl _ @[simp] lemma toUpperSet_symm : (@toUpperSet α).symm = ofUpperSet := rfl @[simp] lemma ofUpperSet_symm : (@ofUpperSet α).symm = toUpperSet := rfl @[simp] lemma toUpperSet_ofUpperSet (a : WithUpperSet α) : toUpperSet (ofUpperSet a) = a := rfl @[simp] lemma ofUpperSet_toUpperSet (a : α) : ofUpperSet (toUpperSet a) = a := rfl lemma toUpperSet_inj {a b : α} : toUpperSet a = toUpperSet b ↔ a = b := Iff.rfl lemma ofUpperSet_inj {a b : WithUpperSet α} : ofUpperSet a = ofUpperSet b ↔ a = b := Iff.rfl /-- A recursor for `WithUpperSet`. Use as `induction x`. -/ @[elab_as_elim, cases_eliminator, induction_eliminator] protected def rec {β : WithUpperSet α → Sort} (h : ∀ a, β (toUpperSet a)) : ∀ a, β a := fun a => h (ofUpperSet a) instance [Nonempty α] : Nonempty (WithUpperSet α) := ‹Nonempty α› instance [Inhabited α] : Inhabited (WithUpperSet α) := ‹Inhabited α› variable [Preorder α] [Preorder β] instance : Preorder (WithUpperSet α) := ‹Preorder α› instance : TopologicalSpace (WithUpperSet α) := upperSet α lemma ofUpperSet_le_iff {a b : WithUpperSet α} : ofUpperSet a ≤ ofUpperSet b ↔ a ≤ b := Iff.rfl lemma toUpperSet_le_iff {a b : α} : toUpperSet a ≤ toUpperSet b ↔ a ≤ b := Iff.rfl /-- `ofUpperSet` as an `OrderIso` -/ def ofUpperSetOrderIso : WithUpperSet α ≃o α where toEquiv := ofUpperSet map_rel_iff' := ofUpperSet_le_iff /-- `toUpperSet` as an `OrderIso` -/ def toUpperSetOrderIso : α ≃o WithUpperSet α where toEquiv := toUpperSet map_rel_iff' := toUpperSet_le_iff end WithUpperSet /-- Type synonym for a preorder equipped with the lower set topology. -/ def WithLowerSet (α : Type) := α namespace WithLowerSet /-- `toLowerSet` is the identity function to the `WithLowerSet` of a type. -/ @[match_pattern] def toLowerSet : α ≃ WithLowerSet α := Equiv.refl _ /-- `ofLowerSet` is the identity function from the `WithLowerSet` of a type. -/ @[match_pattern] def ofLowerSet : WithLowerSet α ≃ α := Equiv.refl _ @[simp] lemma toLowerSet_symm : (@toLowerSet α).symm = ofLowerSet := rfl @[simp] lemma ofLowerSet_symm : (@ofLowerSet α).symm = toLowerSet := rfl @[simp] lemma toLowerSet_ofLowerSet (a : WithLowerSet α) : toLowerSet (ofLowerSet a) = a := rfl @[simp] lemma ofLowerSet_toLowerSet (a : α) : ofLowerSet (toLowerSet a) = a := rfl lemma toLowerSet_inj {a b : α} : toLowerSet a = toLowerSet b ↔ a = b := Iff.rfl lemma ofLowerSet_inj {a b : WithLowerSet α} : ofLowerSet a = ofLowerSet b ↔ a = b := Iff.rfl /-- A recursor for `WithLowerSet`. Use as `induction x`. -/ @[elab_as_elim, cases_eliminator, induction_eliminator] protected def rec {β : WithLowerSet α → Sort} (h : ∀ a, β (toLowerSet a)) : ∀ a, β a := fun a => h (ofLowerSet a) instance [Nonempty α] : Nonempty (WithLowerSet α) := ‹Nonempty α› instance [Inhabited α] : Inhabited (WithLowerSet α) := ‹Inhabited α› variable [Preorder α] instance : Preorder (WithLowerSet α) := ‹Preorder α› instance : TopologicalSpace (WithLowerSet α) := lowerSet α lemma ofLowerSet_le_iff {a b : WithLowerSet α} : ofLowerSet a ≤ ofLowerSet b ↔ a ≤ b := Iff.rfl lemma toLowerSet_le_iff {a b : α} : toLowerSet a ≤ toLowerSet b ↔ a ≤ b := Iff.rfl /-- `ofLowerSet` as an `OrderIso` -/ def ofLowerSetOrderIso : WithLowerSet α ≃o α where toEquiv := ofLowerSet map_rel_iff' := ofLowerSet_le_iff /-- `toLowerSet` as an `OrderIso` -/ def toLowerSetOrderIso : α ≃o WithLowerSet α where toEquiv := toLowerSet map_rel_iff' := toLowerSet_le_iff end WithLowerSet /-- The Upper Set topology is homeomorphic to the Lower Set topology on the dual order -/ def WithUpperSet.toDualHomeomorph [Preorder α] : WithUpperSet α ≃ₜ WithLowerSet αᵒᵈ where toFun := OrderDual.toDual invFun := OrderDual.ofDual left_inv := OrderDual.toDual_ofDual right_inv := OrderDual.ofDual_toDual continuous_toFun := continuous_coinduced_rng continuous_invFun := continuous_coinduced_rng /-- Prop-valued mixin for an ordered topological space to be The upper set topology is the topology where the open sets are the upper sets. In general the upper set topology does not coincide with the upper topology. -/ protected class IsUpperSet (α : Type) [t : TopologicalSpace α] [Preorder α] : Prop where topology_eq_upperSetTopology : t = upperSet α attribute [nolint docBlame] IsUpperSet.topology_eq_upperSetTopology instance [Preorder α] : Topology.IsUpperSet (WithUpperSet α) := ⟨rfl⟩ instance [Preorder α] : @Topology.IsUpperSet α (upperSet α) _ := by letI := upperSet α exact ⟨rfl⟩ /-- The lower set topology is the topology where the open sets are the lower sets. In general the lower set topology does not coincide with the lower topology. -/ protected class IsLowerSet (α : Type) [t : TopologicalSpace α] [Preorder α] : Prop where topology_eq_lowerSetTopology : t = lowerSet α attribute [nolint docBlame] IsLowerSet.topology_eq_lowerSetTopology instance [Preorder α] : Topology.IsLowerSet (WithLowerSet α) := ⟨rfl⟩ instance [Preorder α] : @Topology.IsLowerSet α (lowerSet α) _ := by letI := lowerSet α exact ⟨rfl⟩ namespace IsUpperSet section Preorder variable (α) variable [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] {s : Set α} lemma topology_eq : ‹_› = upperSet α := topology_eq_upperSetTopology variable {α} instance _root_.OrderDual.instIsLowerSet [Preorder α] [TopologicalSpace α] [Topology.IsUpperSet α] : Topology.IsLowerSet αᵒᵈ where topology_eq_lowerSetTopology := by ext; rw [IsUpperSet.topology_eq α] /-- If `α` is equipped with the upper set topology, then it is homeomorphic to `WithUpperSet α`. -/ def WithUpperSetHomeomorph : WithUpperSet α ≃ₜ α := WithUpperSet.ofUpperSet.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ lemma isOpen_iff_isUpperSet : IsOpen s ↔ IsUpperSet s := by rw [topology_eq α] rfl instance toAlexandrovDiscrete : AlexandrovDiscrete α where isOpen_sInter S := by simpa only [isOpen_iff_isUpperSet] using isUpperSet_sInter (α := α) -- c.f. isClosed_iff_lower_and_subset_implies_LUB_mem lemma isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s := by rw [← isOpen_compl_iff, isOpen_iff_isUpperSet, isLowerSet_compl.symm, compl_compl] lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s := by rw [subset_antisymm_iff] refine ⟨?_, lowerClosure_min subset_closure (isClosed_iff_isLower.1 isClosed_closure)⟩ · apply closure_minimal subset_lowerClosure _ rw [isClosed_iff_isLower] exact LowerSet.lower (lowerClosure s) /-- The closure of a singleton `{a}` in the upper set topology is the right-closed left-infinite interval (-∞,a]. -/ @[simp] lemma closure_singleton {a : α} : closure {a} = Iic a := by rw [closure_eq_lowerClosure, lowerClosure_singleton] rfl lemma specializes_iff_le {a b : α} : a ⤳ b ↔ b ≤ a := by simp only [specializes_iff_closure_subset, closure_singleton, Iic_subset_Iic] lemma nhdsKer_eq_upperClosure (s : Set α) : nhdsKer s = ↑(upperClosure s) := by ext; simp [mem_nhdsKer_iff_specializes, specializes_iff_le] @[simp] lemma nhdsKer_singleton (a : α) : nhdsKer {a} = Ici a := by rw [nhdsKer_eq_upperClosure, upperClosure_singleton, UpperSet.coe_Ici] lemma nhds_eq_principal_Ici (a : α) : 𝓝 a = 𝓟 (Ici a) := by rw [← principal_nhdsKer_singleton, nhdsKer_singleton] lemma nhdsSet_eq_principal_upperClosure (s : Set α) : 𝓝ˢ s = 𝓟 ↑(upperClosure s) := by rw [← principal_nhdsKer, nhdsKer_eq_upperClosure] end Preorder protected lemma _root_.Topology.isUpperSet_iff_nhds {α : Type} [TopologicalSpace α] [Preorder α] : Topology.IsUpperSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Ici a)) where mp _ a := nhds_eq_principal_Ici a mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Ici]⟩ instance : Topology.IsUpperSet Prop := by simp [Topology.isUpperSet_iff_nhds, Prop.forall] section maps variable [Preorder α] [Preorder β] open Topology protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β] [Topology.IsUpperSet α] [Topology.IsUpperSet β] {f : α → β} : Monotone f ↔ Continuous f := by constructor · intro hf simp_rw [continuous_def, isOpen_iff_isUpperSet] exact fun _ hs ↦ IsUpperSet.preimage hs hf · intro hf a b hab rw [← mem_Iic, ← closure_singleton] at hab ⊢ apply Continuous.closure_preimage_subset hf {f b} apply mem_of_mem_of_subset hab apply closure_mono rw [singleton_subset_iff, mem_preimage, mem_singleton_iff] lemma monotone_to_upperTopology_continuous [TopologicalSpace α] [TopologicalSpace β] [Topology.IsUpperSet α] [IsUpper β] {f : α → β} (hf : Monotone f) : Continuous f := by simp_rw [continuous_def, isOpen_iff_isUpperSet] intro s hs exact (IsUpper.isUpperSet_of_isOpen hs).preimage hf lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _] [@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ := fun s hs => by rw [@isOpen_iff_isUpperSet α _ t₁] exact IsUpper.isUpperSet_of_isOpen hs end maps end IsUpperSet namespace IsLowerSet section Preorder variable (α) variable [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] {s : Set α} lemma topology_eq : ‹_› = lowerSet α := topology_eq_lowerSetTopology variable {α} instance _root_.OrderDual.instIsUpperSet [Preorder α] [TopologicalSpace α] [Topology.IsLowerSet α] : Topology.IsUpperSet αᵒᵈ where topology_eq_upperSetTopology := by ext; rw [IsLowerSet.topology_eq α] /-- If `α` is equipped with the lower set topology, then it is homeomorphic to `WithLowerSet α`. -/ def WithLowerSetHomeomorph : WithLowerSet α ≃ₜ α := WithLowerSet.ofLowerSet.toHomeomorphOfIsInducing ⟨topology_eq α ▸ induced_id.symm⟩ lemma isOpen_iff_isLowerSet : IsOpen s ↔ IsLowerSet s := by rw [topology_eq α]; rfl instance toAlexandrovDiscrete : AlexandrovDiscrete α := IsUpperSet.toAlexandrovDiscrete (α := αᵒᵈ) lemma isClosed_iff_isUpper : IsClosed s ↔ IsUpperSet s := by rw [← isOpen_compl_iff, isOpen_iff_isLowerSet, isUpperSet_compl.symm, compl_compl] lemma closure_eq_upperClosure {s : Set α} : closure s = upperClosure s := IsUpperSet.closure_eq_lowerClosure (α := αᵒᵈ) /-- The closure of a singleton `{a}` in the lower set topology is the right-closed left-infinite interval (-∞,a]. -/ @[simp] lemma closure_singleton {a : α} : closure {a} = Ici a := by rw [closure_eq_upperClosure, upperClosure_singleton] rfl lemma specializes_iff_le {a b : α} : a ⤳ b ↔ a ≤ b := by simp only [specializes_iff_closure_subset, closure_singleton, Ici_subset_Ici] lemma nhdsKer_eq_lowerClosure (s : Set α) : nhdsKer s = ↑(lowerClosure s) := by ext; simp [mem_nhdsKer_iff_specializes, specializes_iff_le] @[simp] lemma nhdsKer_singleton (a : α) : nhdsKer {a} = Iic a := by rw [nhdsKer_eq_lowerClosure, lowerClosure_singleton, LowerSet.coe_Iic] lemma nhds_eq_principal_Iic (a : α) : 𝓝 a = 𝓟 (Iic a) := by rw [← principal_nhdsKer_singleton, nhdsKer_singleton] lemma nhdsSet_eq_principal_lowerClosure (s : Set α) : 𝓝ˢ s = 𝓟 ↑(lowerClosure s) := by rw [← principal_nhdsKer, nhdsKer_eq_lowerClosure] end Preorder protected lemma _root_.Topology.isLowerSet_iff_nhds {α : Type*} [TopologicalSpace α] [Preorder α] : Topology.IsLowerSet α ↔ (∀ a : α, 𝓝 a = 𝓟 (Iic a)) where mp _ a := nhds_eq_principal_Iic a mpr hα := ⟨by simp [TopologicalSpace.ext_iff_nhds, hα, nhds_eq_principal_Iic]⟩ section maps variable [Preorder α] [Preorder β] open Topology open OrderDual protected lemma monotone_iff_continuous [TopologicalSpace α] [TopologicalSpace β] [Topology.IsLowerSet α] [Topology.IsLowerSet β] {f : α → β} : Monotone f ↔ Continuous f := by rw [← monotone_dual_iff] exact IsUpperSet.monotone_iff_continuous (α := αᵒᵈ) (β := βᵒᵈ) (f := (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ)) lemma monotone_to_lowerTopology_continuous [TopologicalSpace α] [TopologicalSpace β] [Topology.IsLowerSet α] [IsLower β] {f : α → β} (hf : Monotone f) : Continuous f := IsUpperSet.monotone_to_upperTopology_continuous (α := αᵒᵈ) (β := βᵒᵈ) hf.dual lemma lowerSet_le_lower {t₁ t₂ : TopologicalSpace α} [@Topology.IsLowerSet α t₁ _] [@IsLower α t₂ _] : t₁ ≤ t₂ := fun s hs => by rw [@isOpen_iff_isLowerSet α _ t₁] exact IsLower.isLowerSet_of_isOpen hs end maps end IsLowerSet lemma isUpperSet_orderDual [Preorder α] [TopologicalSpace α] : Topology.IsUpperSet αᵒᵈ ↔ Topology.IsLowerSet α := by constructor · apply OrderDual.instIsLowerSet · apply OrderDual.instIsUpperSet lemma isLowerSet_orderDual [Preorder α] [TopologicalSpace α] : Topology.IsLowerSet αᵒᵈ ↔ Topology.IsUpperSet α := isUpperSet_orderDual.symm namespace WithUpperSet variable [Preorder α] [Preorder β] [Preorder γ] /-- A monotone map between preorders spaces induces a continuous map between themselves considered with the upper set topology. -/ def map (f : α →o β) : C(WithUpperSet α, WithUpperSet β) where toFun := toUpperSet ∘ f ∘ ofUpperSet continuous_toFun := continuous_def.2 fun _s hs ↦ IsUpperSet.preimage hs f.monotone @[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ := rfl @[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) := rfl @[simp] lemma toUpperSet_specializes_toUpperSet {a b : α} : toUpperSet a ⤳ toUpperSet b ↔ b ≤ a := by simp_rw [specializes_iff_closure_subset, IsUpperSet.closure_singleton, Iic_subset_Iic, toUpperSet_le_iff] @[simp] lemma ofUpperSet_le_ofUpperSet {a b : WithUpperSet α} : ofUpperSet a ≤ ofUpperSet b ↔ b ⤳ a := toUpperSet_specializes_toUpperSet.symm @[simp] lemma isUpperSet_toUpperSet_preimage {s : Set (WithUpperSet α)} : IsUpperSet (toUpperSet ⁻¹' s) ↔ IsOpen s := Iff.rfl @[simp] lemma isOpen_ofUpperSet_preimage {s : Set α} : IsOpen (ofUpperSet ⁻¹' s) ↔ IsUpperSet s := isUpperSet_toUpperSet_preimage.symm end WithUpperSet namespace WithLowerSet variable [Preorder α] [Preorder β] [Preorder γ] /-- A monotone map between preorders spaces induces a continuous map between themselves considered with the lower set topology. -/ def map (f : α →o β) : C(WithLowerSet α, WithLowerSet β) where toFun := toLowerSet ∘ f ∘ ofLowerSet continuous_toFun := continuous_def.2 fun _s hs ↦ IsLowerSet.preimage hs f.monotone @[simp] lemma map_id : map (OrderHom.id : α →o α) = ContinuousMap.id _ := rfl @[simp] lemma map_comp (g : β →o γ) (f : α →o β) : map (g.comp f) = (map g).comp (map f) := rfl @[simp] lemma toLowerSet_specializes_toLowerSet {a b : α} : toLowerSet a ⤳ toLowerSet b ↔ a ≤ b := by simp_rw [specializes_iff_closure_subset, IsLowerSet.closure_singleton, Ici_subset_Ici, toLowerSet_le_iff] @[simp] lemma ofLowerSet_le_ofLowerSet {a b : WithLowerSet α} : ofLowerSet a ≤ ofLowerSet b ↔ a ⤳ b := toLowerSet_specializes_toLowerSet.symm @[simp] lemma isLowerSet_toLowerSet_preimage {s : Set (WithLowerSet α)} : IsLowerSet (toLowerSet ⁻¹' s) ↔ IsOpen s := Iff.rfl @[simp] lemma isOpen_ofLowerSet_preimage {s : Set α} : IsOpen (ofLowerSet ⁻¹' s) ↔ IsLowerSet s := isLowerSet_toLowerSet_preimage.symm end WithLowerSet end Topology
.lake/packages/mathlib/Mathlib/Topology/Order/LeftRightNhds.lean	import Mathlib.Algebra.Ring.Pointwise.Set import Mathlib.Order.Filter.AtTopBot.CompleteLattice import Mathlib.Order.Filter.AtTopBot.Group import Mathlib.Topology.Order.Basic /-! # Neighborhoods to the left and to the right on an `OrderTopology` We've seen some properties of left and right neighborhood of a point in an `OrderClosedTopology`. In an `OrderTopology`, such neighborhoods can be characterized as the sets containing suitable intervals to the right or to the left of `a`. We give now these characterizations. -/ open Set Filter TopologicalSpace Topology Function open OrderDual (toDual ofDual) variable {α β γ : Type} section LinearOrder variable [TopologicalSpace α] [LinearOrder α] section OrderTopology variable [OrderTopology α] open List in /-- The following statements are equivalent: 0. `s` is a neighborhood of `a` within `(a, +∞)`; 1. `s` is a neighborhood of `a` within `(a, b]`; 2. `s` is a neighborhood of `a` within `(a, b)`; 3. `s` includes `(a, u)` for some `u ∈ (a, b]`; 4. `s` includes `(a, u)` for some `u > a`. -/ theorem TFAE_mem_nhdsGT {a b : α} (hab : a < b) (s : Set α) : TFAE [s ∈ 𝓝[>] a, s ∈ 𝓝[Ioc a b] a, s ∈ 𝓝[Ioo a b] a, ∃ u ∈ Ioc a b, Ioo a u ⊆ s, ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by tfae_have 1 ↔ 2 := by rw [nhdsWithin_Ioc_eq_nhdsGT hab] tfae_have 1 ↔ 3 := by rw [nhdsWithin_Ioo_eq_nhdsGT hab] tfae_have 4 → 5 := fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩ tfae_have 5 → 1 \| ⟨u, hau, hu⟩ => mem_of_superset (Ioo_mem_nhdsGT hau) hu tfae_have 1 → 4 \| h => by rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩ rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩ exact ⟨u, au, fun x hx => hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, hx.1⟩⟩ tfae_finish theorem mem_nhdsGT_iff_exists_mem_Ioc_Ioo_subset {a u' : α} {s : Set α} (hu' : a < u') : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioc a u', Ioo a u ⊆ s := (TFAE_mem_nhdsGT hu' s).out 0 3 /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)` with `a < u < u'`, provided `a` is not a top element. -/ theorem mem_nhdsGT_iff_exists_Ioo_subset' {a u' : α} {s : Set α} (hu' : a < u') : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s := (TFAE_mem_nhdsGT hu' s).out 0 4 theorem nhdsGT_basis_of_exists_gt {a : α} (h : ∃ b, a < b) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) := let ⟨_, h⟩ := h ⟨fun _ => mem_nhdsGT_iff_exists_Ioo_subset' h⟩ lemma nhdsGT_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) := nhdsGT_basis_of_exists_gt <\| exists_gt a theorem nhdsGT_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by by_cases ha : IsTop a · simp [ha, ha.isMax.Ioi_eq] · simp only [ha, false_or] rw [isTop_iff_isMax, not_isMax_iff] at ha simp only [(nhdsGT_basis_of_exists_gt ha).eq_bot_iff, covBy_iff_Ioo_eq] /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u)` with `a < u`. -/ theorem mem_nhdsGT_iff_exists_Ioo_subset [NoMaxOrder α] {a : α} {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioo a u ⊆ s := let ⟨_u', hu'⟩ := exists_gt a mem_nhdsGT_iff_exists_Ioo_subset' hu' /-- The set of points which are isolated on the right is countable when the space is second-countable. -/ theorem countable_setOf_isolated_right [SecondCountableTopology α] : { x : α \| 𝓝[>] x = ⊥ }.Countable := by simp only [nhdsGT_eq_bot_iff, setOf_or] exact (subsingleton_isTop α).countable.union countable_setOf_covBy_right /-- The set of points which are isolated on the left is countable when the space is second-countable. -/ theorem countable_setOf_isolated_left [SecondCountableTopology α] : { x : α \| 𝓝[<] x = ⊥ }.Countable := countable_setOf_isolated_right (α := αᵒᵈ) /-- The set of points in a set which are isolated on the right in this set is countable when the space is second-countable. -/ theorem countable_setOf_isolated_right_within [SecondCountableTopology α] {s : Set α} : { x ∈ s \| 𝓝[s ∩ Ioi x] x = ⊥ }.Countable := by /- This does not follow from `countable_setOf_isolated_right`, which gives the result when `s` is the whole space, as one cannot use it inside the subspace since it doesn't have the order topology. Instead, we follow the main steps of its proof. -/ let t := { x ∈ s \| 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬ IsTop x} suffices H : t.Countable by have : { x ∈ s \| 𝓝[s ∩ Ioi x] x = ⊥ } ⊆ t ∪ {x \| IsTop x} := by intro x hx by_cases h'x : IsTop x · simp [h'x] · simpa [-sep_and, t, h'x] apply Countable.mono this simp [H, (subsingleton_isTop α).countable] have (x) (hx : x ∈ t) : ∃ y > x, s ∩ Ioo x y = ∅ := by simp only [← empty_mem_iff_bot, mem_nhdsWithin_iff_exists_mem_nhds_inter, subset_empty_iff, IsTop, not_forall, not_le, mem_setOf_eq, t] at hx rcases hx.2.1 with ⟨u, hu, h'u⟩ obtain ⟨y, hxy, hy⟩ : ∃ y, x < y ∧ Ico x y ⊆ u := exists_Ico_subset_of_mem_nhds hu hx.2.2 refine ⟨y, hxy, ?_⟩ contrapose! h'u apply h'u.mono intro z hz exact ⟨hy ⟨hz.2.1.le, hz.2.2⟩, hz.1, hz.2.1⟩ choose! y hy h'y using this apply Set.PairwiseDisjoint.countable_of_Ioo (y := y) _ hy simp only [PairwiseDisjoint, Set.Pairwise, Function.onFun] intro a ha b hb hab wlog H : a < b generalizing a b with h · have : b < a := lt_of_le_of_ne (not_lt.1 H) hab.symm exact (h hb ha hab.symm this).symm have : y a ≤ b := by by_contra! have : b ∈ s ∩ Ioo a (y a) := by simp [hb.1, H, this] simp [h'y a ha] at this rw [disjoint_iff_forall_ne] exact fun u hu v hv ↦ ((hu.2.trans_le this).trans hv.1).ne /-- The set of points in a set which are isolated on the left in this set is countable when the space is second-countable. -/ theorem countable_setOf_isolated_left_within [SecondCountableTopology α] {s : Set α} : { x ∈ s \| 𝓝[s ∩ Iio x] x = ⊥ }.Countable := countable_setOf_isolated_right_within (α := αᵒᵈ) /-- A set is a neighborhood of `a` within `(a, +∞)` if and only if it contains an interval `(a, u]` with `a < u`. -/ theorem mem_nhdsGT_iff_exists_Ioc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α} : s ∈ 𝓝[>] a ↔ ∃ u ∈ Ioi a, Ioc a u ⊆ s := by rw [mem_nhdsGT_iff_exists_Ioo_subset] constructor · rintro ⟨u, au, as⟩ rcases exists_between au with ⟨v, hv⟩ exact ⟨v, hv.1, fun x hx => as ⟨hx.1, lt_of_le_of_lt hx.2 hv.2⟩⟩ · rintro ⟨u, au, as⟩ exact ⟨u, au, Subset.trans Ioo_subset_Ioc_self as⟩ open List in /-- The following statements are equivalent: 0. `s` is a neighborhood of `b` within `(-∞, b)` 1. `s` is a neighborhood of `b` within `[a, b)` 2. `s` is a neighborhood of `b` within `(a, b)` 3. `s` includes `(l, b)` for some `l ∈ [a, b)` 4. `s` includes `(l, b)` for some `l < b` -/ theorem TFAE_mem_nhdsLT {a b : α} (h : a < b) (s : Set α) : TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)` s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)` s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)` ∃ l ∈ Ico a b, Ioo l b ⊆ s,-- 3 : `s` includes `(l, b)` for some `l ∈ [a, b)` ∃ l ∈ Iio b, Ioo l b ⊆ s] := by-- 4 : `s` includes `(l, b)` for some `l < b` simpa using TFAE_mem_nhdsGT h.dual (ofDual ⁻¹' s) theorem mem_nhdsLT_iff_exists_mem_Ico_Ioo_subset {a l' : α} {s : Set α} (hl' : l' < a) : s ∈ 𝓝[<] a ↔ ∃ l ∈ Ico l' a, Ioo l a ⊆ s := (TFAE_mem_nhdsLT hl' s).out 0 3 /-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)` with `l < a`, provided `a` is not a bottom element. -/ theorem mem_nhdsLT_iff_exists_Ioo_subset' {a l' : α} {s : Set α} (hl' : l' < a) : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s := (TFAE_mem_nhdsLT hl' s).out 0 4 /-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `(l, a)` with `l < a`. -/ theorem mem_nhdsLT_iff_exists_Ioo_subset [NoMinOrder α] {a : α} {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ioo l a ⊆ s := let ⟨_, h⟩ := exists_lt a mem_nhdsLT_iff_exists_Ioo_subset' h /-- A set is a neighborhood of `a` within `(-∞, a)` if and only if it contains an interval `[l, a)` with `l < a`. -/ theorem mem_nhdsLT_iff_exists_Ico_subset [NoMinOrder α] [DenselyOrdered α] {a : α} {s : Set α} : s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s := by have : ofDual ⁻¹' s ∈ 𝓝[>] toDual a ↔ _ := mem_nhdsGT_iff_exists_Ioc_subset simpa using this theorem nhdsLT_basis_of_exists_lt {a : α} (h : ∃ b, b < a) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) := let ⟨_, h⟩ := h ⟨fun _ => mem_nhdsLT_iff_exists_Ioo_subset' h⟩ theorem nhdsLT_basis [NoMinOrder α] (a : α) : (𝓝[<] a).HasBasis (· < a) (Ioo · a) := nhdsLT_basis_of_exists_lt <\| exists_lt a theorem nhdsLT_eq_bot_iff {a : α} : 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a := by convert (config := {preTransparency := .default}) nhdsGT_eq_bot_iff (a := OrderDual.toDual a) using 4 exact ofDual_covBy_ofDual_iff open List in /-- The following statements are equivalent: 0. `s` is a neighborhood of `a` within `[a, +∞)`; 1. `s` is a neighborhood of `a` within `[a, b]`; 2. `s` is a neighborhood of `a` within `[a, b)`; 3. `s` includes `[a, u)` for some `u ∈ (a, b]`; 4. `s` includes `[a, u)` for some `u > a`. -/ theorem TFAE_mem_nhdsGE {a b : α} (hab : a < b) (s : Set α) : TFAE [s ∈ 𝓝[≥] a, s ∈ 𝓝[Icc a b] a, s ∈ 𝓝[Ico a b] a, ∃ u ∈ Ioc a b, Ico a u ⊆ s, ∃ u ∈ Ioi a, Ico a u ⊆ s] := by tfae_have 1 ↔ 2 := by rw [nhdsWithin_Icc_eq_nhdsGE hab] tfae_have 1 ↔ 3 := by rw [nhdsWithin_Ico_eq_nhdsGE hab] tfae_have 1 ↔ 5 := (nhdsGE_basis_of_exists_gt ⟨b, hab⟩).mem_iff tfae_have 4 → 5 := fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩ tfae_have 5 → 4 \| ⟨u, hua, hus⟩ => ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩, (Ico_subset_Ico_right <\| min_le_left _ _).trans hus⟩ tfae_finish theorem mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioc a u', Ico a u ⊆ s := (TFAE_mem_nhdsGE hu' s).out 0 3 (by simp) (by simp) /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)` with `a < u < u'`, provided `a` is not a top element. -/ theorem mem_nhdsGE_iff_exists_Ico_subset' {a u' : α} {s : Set α} (hu' : a < u') : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s := (TFAE_mem_nhdsGE hu' s).out 0 4 (by simp) (by simp) /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u)` with `a < u`. -/ theorem mem_nhdsGE_iff_exists_Ico_subset [NoMaxOrder α] {a : α} {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u ∈ Ioi a, Ico a u ⊆ s := let ⟨_, hu'⟩ := exists_gt a mem_nhdsGE_iff_exists_Ico_subset' hu' theorem nhdsGE_basis_Ico [NoMaxOrder α] (a : α) : (𝓝[≥] a).HasBasis (fun u => a < u) (Ico a) := ⟨fun _ => mem_nhdsGE_iff_exists_Ico_subset⟩ /-- The filter of right neighborhoods has a basis of closed intervals. -/ theorem nhdsGE_basis_Icc [NoMaxOrder α] [DenselyOrdered α] {a : α} : (𝓝[≥] a).HasBasis (a < ·) (Icc a) := (nhdsGE_basis _).to_hasBasis (fun _u hu ↦ (exists_between hu).imp fun _v hv ↦ hv.imp_right Icc_subset_Ico_right) fun u hu ↦ ⟨u, hu, Ico_subset_Icc_self⟩ /-- A set is a neighborhood of `a` within `[a, +∞)` if and only if it contains an interval `[a, u]` with `a < u`. -/ theorem mem_nhdsGE_iff_exists_Icc_subset [NoMaxOrder α] [DenselyOrdered α] {a : α} {s : Set α} : s ∈ 𝓝[≥] a ↔ ∃ u, a < u ∧ Icc a u ⊆ s := nhdsGE_basis_Icc.mem_iff open List in /-- The following statements are equivalent: 0. `s` is a neighborhood of `b` within `(-∞, b]` 1. `s` is a neighborhood of `b` within `[a, b]` 2. `s` is a neighborhood of `b` within `(a, b]` 3. `s` includes `(l, b]` for some `l ∈ [a, b)` 4. `s` includes `(l, b]` for some `l < b` -/ theorem TFAE_mem_nhdsLE {a b : α} (h : a < b) (s : Set α) : TFAE [s ∈ 𝓝[≤] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b]` s ∈ 𝓝[Icc a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b]` s ∈ 𝓝[Ioc a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b]` ∃ l ∈ Ico a b, Ioc l b ⊆ s,-- 3 : `s` includes `(l, b]` for some `l ∈ [a, b)` ∃ l ∈ Iio b, Ioc l b ⊆ s] := by-- 4 : `s` includes `(l, b]` for some `l < b` simpa using TFAE_mem_nhdsGE h.dual (ofDual ⁻¹' s) theorem mem_nhdsLE_iff_exists_mem_Ico_Ioc_subset {a l' : α} {s : Set α} (hl' : l' < a) : s ∈ 𝓝[≤] a ↔ ∃ l ∈ Ico l' a, Ioc l a ⊆ s := (TFAE_mem_nhdsLE hl' s).out 0 3 (by simp) (by simp) /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]` with `l < a`, provided `a` is not a bottom element. -/ theorem mem_nhdsLE_iff_exists_Ioc_subset' {a l' : α} {s : Set α} (hl' : l' < a) : s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s := (TFAE_mem_nhdsLE hl' s).out 0 4 (by simp) (by simp) /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `(l, a]` with `l < a`. -/ theorem mem_nhdsLE_iff_exists_Ioc_subset [NoMinOrder α] {a : α} {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l ∈ Iio a, Ioc l a ⊆ s := let ⟨_, hl'⟩ := exists_lt a mem_nhdsLE_iff_exists_Ioc_subset' hl' /-- A set is a neighborhood of `a` within `(-∞, a]` if and only if it contains an interval `[l, a]` with `l < a`. -/ theorem mem_nhdsLE_iff_exists_Icc_subset [NoMinOrder α] [DenselyOrdered α] {a : α} {s : Set α} : s ∈ 𝓝[≤] a ↔ ∃ l, l < a ∧ Icc l a ⊆ s := calc s ∈ 𝓝[≤] a ↔ ofDual ⁻¹' s ∈ 𝓝[≥] (toDual a) := Iff.rfl _ ↔ ∃ u : α, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ofDual ⁻¹' s := mem_nhdsGE_iff_exists_Icc_subset _ ↔ ∃ l, l < a ∧ Icc l a ⊆ s := by simp /-- The filter of left neighborhoods has a basis of closed intervals. -/ theorem nhdsLE_basis_Icc [NoMinOrder α] [DenselyOrdered α] {a : α} : (𝓝[≤] a).HasBasis (· < a) (Icc · a) := ⟨fun _ ↦ mem_nhdsLE_iff_exists_Icc_subset⟩ end OrderTopology end LinearOrder section LinearOrderedCommGroup variable [TopologicalSpace α] [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] [OrderTopology α] variable {l : Filter β} {f g : β → α} @[to_additive] theorem nhds_eq_iInf_mabs_div (a : α) : 𝓝 a = ⨅ r > 1, 𝓟 { b \| \|a / b\|ₘ < r } := by simp only [nhds_eq_order, mabs_lt, setOf_and, ← inf_principal, iInf_inf_eq] refine (congr_arg₂ _ ?_ ?_).trans (inf_comm ..) · refine (Equiv.divLeft a).iInf_congr fun x => ?_; simp [Ioi] · refine (Equiv.divRight a).iInf_congr fun x => ?_; simp [Iio] @[to_additive] theorem orderTopology_of_nhds_mabs {α : Type} [TopologicalSpace α] [CommGroup α] [LinearOrder α] [IsOrderedMonoid α] (h_nhds : ∀ a : α, 𝓝 a = ⨅ r > 1, 𝓟 { b \| \|a / b\|ₘ < r }) : OrderTopology α := by refine ⟨TopologicalSpace.ext_nhds fun a => ?_⟩ rw [h_nhds] letI := Preorder.topology α; letI : OrderTopology α := ⟨rfl⟩ exact (nhds_eq_iInf_mabs_div a).symm @[to_additive] theorem LinearOrderedCommGroup.tendsto_nhds {x : Filter β} {a : α} : Tendsto f x (𝓝 a) ↔ ∀ ε > (1 : α), ∀ᶠ b in x, \|f b / a\|ₘ < ε := by simp [nhds_eq_iInf_mabs_div, mabs_div_comm a] @[to_additive] theorem eventually_mabs_div_lt (a : α) {ε : α} (hε : 1 < ε) : ∀ᶠ x in 𝓝 a, \|x / a\|ₘ < ε := (nhds_eq_iInf_mabs_div a).symm ▸ mem_iInf_of_mem ε (mem_iInf_of_mem hε <\| by simp only [mabs_div_comm, mem_principal_self]) /-- In a linearly ordered commutative group with the order topology, if `f` tends to `C` and `g` tends to `atTop` then `f * g` tends to `atTop`. -/ @[to_additive add_atTop /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C` and `g` tends to `atTop` then `f + g` tends to `atTop`. -/] theorem Filter.Tendsto.mul_atTop' {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atTop := by nontriviality α obtain ⟨C', hC'⟩ : ∃ C', C' < C := exists_lt C refine tendsto_atTop_mul_left_of_le' _ C' ?_ hg exact (hf.eventually (lt_mem_nhds hC')).mono fun x => le_of_lt /-- In a linearly ordered commutative group with the order topology, if `f` tends to `C` and `g` tends to `atBot` then `f * g` tends to `atBot`. -/ @[to_additive add_atBot /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `C` and `g` tends to `atBot` then `f + g` tends to `atBot`. -/] theorem Filter.Tendsto.mul_atBot' {C : α} (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atBot := Filter.Tendsto.mul_atTop' (α := αᵒᵈ) hf hg /-- In a linearly ordered commutative group with the order topology, if `f` tends to `atTop` and `g` tends to `C` then `f * g` tends to `atTop`. -/ @[to_additive atTop_add /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `atTop` and `g` tends to `C` then `f + g` tends to `atTop`. -/] theorem Filter.Tendsto.atTop_mul' {C : α} (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by conv in _ * _ => rw [mul_comm] exact hg.mul_atTop' hf /-- In a linearly ordered commutative group with the order topology, if `f` tends to `atBot` and `g` tends to `C` then `f * g` tends to `atBot`. -/ @[to_additive atBot_add /-- In a linearly ordered additive commutative group with the order topology, if `f` tends to `atBot` and `g` tends to `C` then `f + g` tends to `atBot`. -/] theorem Filter.Tendsto.atBot_mul' {C : α} (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by conv in _ * _ => rw [mul_comm] exact hg.mul_atBot' hf @[to_additive] theorem nhds_basis_mabs_div_lt [NoMaxOrder α] (a : α) : (𝓝 a).HasBasis (fun ε : α => (1 : α) < ε) fun ε => { b \| \|b / a\|ₘ < ε } := by simp only [nhds_eq_iInf_mabs_div, mabs_div_comm (a := a)] refine hasBasis_biInf_principal' (fun x hx y hy => ?_) (exists_gt _) exact ⟨min x y, lt_min hx hy, fun _ hz => hz.trans_le (min_le_left _ _), fun _ hz => hz.trans_le (min_le_right _ _)⟩ @[to_additive] theorem nhds_basis_Ioo_one_lt [NoMaxOrder α] (a : α) : (𝓝 a).HasBasis (fun ε : α => (1 : α) < ε) fun ε => Ioo (a / ε) (a * ε) := by convert nhds_basis_mabs_div_lt a simp only [Ioo, mabs_lt, ← div_lt_iff_lt_mul, inv_lt_div_iff_lt_mul, div_lt_comm] @[to_additive] theorem nhds_basis_Icc_one_lt [NoMaxOrder α] [DenselyOrdered α] (a : α) : (𝓝 a).HasBasis ((1 : α) < ·) fun ε ↦ Icc (a / ε) (a * ε) := (nhds_basis_Ioo_one_lt a).to_hasBasis (fun _ε ε₁ ↦ let ⟨δ, δ₁, δε⟩ := exists_between ε₁ ⟨δ, δ₁, Icc_subset_Ioo (by gcongr) (by gcongr)⟩) (fun ε ε₁ ↦ ⟨ε, ε₁, Ioo_subset_Icc_self⟩) variable (α) in @[to_additive] theorem nhds_basis_one_mabs_lt [NoMaxOrder α] : (𝓝 (1 : α)).HasBasis (fun ε : α => (1 : α) < ε) fun ε => { b \| \|b\|ₘ < ε } := by simpa using nhds_basis_mabs_div_lt (1 : α) /-- If `a > 1`, then open intervals `(a / ε, aε)`, `1 < ε ≤ a`, form a basis of neighborhoods of `a`. This upper bound for `ε` guarantees that all elements of these intervals are greater than one. -/ @[to_additive /-- If `a` is positive, then the intervals `(a - ε, a + ε)`, `0 < ε ≤ a`, form a basis of neighborhoods of `a`. This upper bound for `ε` guarantees that all elements of these intervals are positive. -/] theorem nhds_basis_Ioo_one_lt_of_one_lt [NoMaxOrder α] {a : α} (ha : 1 < a) : (𝓝 a).HasBasis (fun ε : α => (1 : α) < ε ∧ ε ≤ a) fun ε => Ioo (a / ε) (a * ε) := (nhds_basis_Ioo_one_lt a).restrict fun ε hε ↦ ⟨min a ε, lt_min ha hε, min_le_left _ _, by gcongr <;> apply min_le_right⟩ end LinearOrderedCommGroup namespace Set.OrdConnected section ClosedIciTopology variable [TopologicalSpace α] [LinearOrder α] [ClosedIciTopology α] {S : Set α} {x y : α} /-- If `S` is order-connected and contains two points `x < y`, then `S` is a right neighbourhood of `x`. -/ lemma mem_nhdsGE (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[≥] x := mem_of_superset (Icc_mem_nhdsGE hxy) <\| hS.out hx hy /-- If `S` is order-connected and contains two points `x < y`, then `S` is a punctured right neighbourhood of `x`. -/ lemma mem_nhdsGT (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[>] x := nhdsWithin_mono _ Ioi_subset_Ici_self <\| hS.mem_nhdsGE hx hy hxy end ClosedIciTopology variable [TopologicalSpace α] [LinearOrder α] [ClosedIicTopology α] {S : Set α} {x y : α} /-- If `S` is order-connected and contains two points `x < y`, then `S` is a left neighbourhood of `y`. -/ lemma mem_nhdsLE (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[≤] y := hS.dual.mem_nhdsGE hy hx hxy /-- If `S` is order-connected and contains two points `x < y`, then `S` is a punctured left neighbourhood of `y`. -/ lemma mem_nhdsLT (hS : OrdConnected S) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) : S ∈ 𝓝[<] y := hS.dual.mem_nhdsGT hy hx hxy end OrdConnected end Set
.lake/packages/mathlib/Mathlib/Topology/Order/Monotone.lean	import Mathlib.Tactic.Order import Mathlib.Topology.Order.IsLUB /-! # Monotone functions on an order topology This file contains lemmas about limits and continuity for monotone / antitone functions on linearly-ordered sets (with the order topology). For example, we prove that a monotone function has left and right limits at any point (`Monotone.tendsto_nhdsLT`, `Monotone.tendsto_nhdsGT`). -/ open Set Filter TopologicalSpace Topology Function open OrderDual (toDual ofDual) variable {α β : Type} section LinearOrder variable [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [LinearOrder β] {s : Set α} {x : α} {f : α → β} lemma MonotoneOn.insert_of_continuousWithinAt [TopologicalSpace β] [OrderClosedTopology β] (hf : MonotoneOn f s) (hx : ClusterPt x (𝓟 s)) (h'x : ContinuousWithinAt f s x) : MonotoneOn f (insert x s) := by have : (𝓝[s] x).NeBot := hx apply monotoneOn_insert_iff.2 ⟨fun b hb hbx ↦ ?_, fun b hb hxb ↦ ?_, hf⟩ · rcases hbx.eq_or_lt with rfl \| hbx · exact le_rfl simp only [ContinuousWithinAt] at h'x apply ge_of_tendsto h'x have : s ∩ Ioi b ∈ 𝓝[s] x := inter_mem_nhdsWithin _ (Ioi_mem_nhds hbx) filter_upwards [this] with y hy using hf hb hy.1 (le_of_lt hy.2) · rcases hxb.eq_or_lt with rfl \| hxb · exact le_rfl simp only [ContinuousWithinAt] at h'x apply le_of_tendsto h'x have : s ∩ Iio b ∈ 𝓝[s] x := inter_mem_nhdsWithin _ (Iio_mem_nhds hxb) filter_upwards [this] with y hy exact hf hy.1 hb (le_of_lt hy.2) /-- If a function is monotone on a set in a second countable topological space, then there are only countably many points that have several preimages. -/ lemma MonotoneOn.countable_setOf_two_preimages [SecondCountableTopology α] (hf : MonotoneOn f s) : Set.Countable {c \| ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} := by nontriviality α let t := {c \| ∃ x, ∃ y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} have : ∀ c ∈ t, ∃ x, ∃ y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c := fun c hc ↦ hc choose! x y hxs hys hxy hfx hfy using this let u := x '' t suffices H : Set.Countable (x '' t) by have : Set.InjOn x t := by intro c hc d hd hcd have : f (x c) = f (x d) := by simp [hcd] rwa [hfx _ hc, hfx _ hd] at this exact countable_of_injective_of_countable_image this H apply Set.PairwiseDisjoint.countable_of_Ioo (y := fun a ↦ y (f a)); swap · rintro a ⟨c, hc, rfl⟩ rw [hfx _ hc] exact hxy _ hc simp only [PairwiseDisjoint, Set.Pairwise, mem_image, onFun, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] intro c hc d hd hcd wlog H : c < d generalizing c d with h · apply (h d hd c hc hcd.symm ?_).symm have : c ≠ d := fun h ↦ hcd (congrArg x h) order simp only [disjoint_iff_forall_ne, mem_Ioo, ne_eq, and_imp] rintro a xca ayc b xda ayd rfl rw [hfx _ hc] at ayc have : x d ≤ y c := (xda.trans ayc).le have : f (x d) ≤ f (y c) := hf (hxs _ hd) (hys _ hc) this rw [hfx _ hd, hfy _ hc] at this exact not_le.2 H this /-- If a function is monotone in a second countable topological space, then there are only countably many points that have several preimages. -/ lemma Monotone.countable_setOf_two_preimages [SecondCountableTopology α] (hf : Monotone f) : Set.Countable {c \| ∃ x y, x < y ∧ f x = c ∧ f y = c} := by rw [← monotoneOn_univ] at hf simpa using hf.countable_setOf_two_preimages /-- If a function is antitone on a set in a second countable topological space, then there are only countably many points that have several preimages. -/ lemma AntitoneOn.countable_setOf_two_preimages [SecondCountableTopology α] (hf : AntitoneOn f s) : Set.Countable {c \| ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c} := (MonotoneOn.countable_setOf_two_preimages hf.dual_right :) /-- If a function is antitone in a second countable topological space, then there are only countably many points that have several preimages. -/ lemma Antitone.countable_setOf_two_preimages [SecondCountableTopology α] (hf : Antitone f) : Set.Countable {c \| ∃ x y, x < y ∧ f x = c ∧ f y = c} := (Monotone.countable_setOf_two_preimages hf.dual_right :) section Continuity variable [TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] /-- In a second countable space, the set of points where a monotone function is not right-continuous within a set is at most countable. Superseded by `MonotoneOn.countable_not_continuousWithinAt` which gives the two-sided version. -/ theorem MonotoneOn.countable_not_continuousWithinAt_Ioi (hf : MonotoneOn f s) : Set.Countable {x ∈ s \| ¬ContinuousWithinAt f (s ∩ Ioi x) x} := by apply (countable_image_lt_image_Ioi_within s f).mono rintro x ⟨xs, hx : ¬ContinuousWithinAt f (s ∩ Ioi x) x⟩ dsimp only [mem_setOf_eq] contrapose! hx refine tendsto_order.2 ⟨fun m hm => ?_, fun u hu => ?_⟩ · filter_upwards [@self_mem_nhdsWithin _ _ x (s ∩ Ioi x)] with y hy exact hm.trans_le (hf xs hy.1 (le_of_lt hy.2)) rcases hx xs u hu with ⟨v, vs, xv, fvu⟩ have : s ∩ Ioo x v ∈ 𝓝[s ∩ Ioi x] x := by simp [nhdsWithin_inter, mem_inf_of_left, self_mem_nhdsWithin, mem_inf_of_right, Ioo_mem_nhdsGT xv] filter_upwards [this] with y hy exact (hf hy.1 vs hy.2.2.le).trans_lt fvu /-- In a second countable space, the set of points where a monotone function is not left-continuous within a set is at most countable. Superseded by `MonotoneOn.countable_not_continuousWithinAt` which gives the two-sided version. -/ theorem MonotoneOn.countable_not_continuousWithinAt_Iio (hf : MonotoneOn f s) : Set.Countable {x ∈ s \| ¬ContinuousWithinAt f (s ∩ Iio x) x} := hf.dual.countable_not_continuousWithinAt_Ioi /-- In a second countable space, the set of points where a monotone function is not continuous within a set is at most countable. -/ theorem MonotoneOn.countable_not_continuousWithinAt (hf : MonotoneOn f s) : Set.Countable {x ∈ s \| ¬ContinuousWithinAt f s x} := by apply (hf.countable_not_continuousWithinAt_Ioi.union hf.countable_not_continuousWithinAt_Iio).mono refine compl_subset_compl.1 ?_ simp only [compl_union] rintro x ⟨hx, h'x⟩ simp only [mem_compl_iff, mem_setOf_eq, not_and, not_not] at hx h'x ⊢ intro xs exact continuousWithinAt_iff_continuous_left'_right'.2 ⟨h'x xs, hx xs⟩ /-- In a second countable space, the set of points where a monotone function is not continuous is at most countable. -/ theorem Monotone.countable_not_continuousAt (hf : Monotone f) : Set.Countable {x \| ¬ContinuousAt f x} := by simpa [continuousWithinAt_univ] using (hf.monotoneOn univ).countable_not_continuousWithinAt /-- In a second countable space, the set of points where an antitone function is not continuous within a set is at most countable. -/ theorem _root_.AntitoneOn.countable_not_continuousWithinAt {s : Set α} (hf : AntitoneOn f s) : Set.Countable {x ∈ s \| ¬ContinuousWithinAt f s x} := hf.dual_right.countable_not_continuousWithinAt /-- In a second countable space, the set of points where an antitone function is not continuous is at most countable. -/ theorem Antitone.countable_not_continuousAt (hf : Antitone f) : Set.Countable {x \| ¬ContinuousAt f x} := hf.dual_right.countable_not_continuousAt end Continuity end LinearOrder section ConditionallyCompleteLinearOrder variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] /-- A monotone function continuous at the supremum of a nonempty set sends this supremum to the supremum of the image of this set. -/ theorem MonotoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α} (Cf : ContinuousWithinAt f A (sSup A)) (Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sSup (f '' A) := --This is a particular case of the more general `IsLUB.isLUB_of_tendsto` .symm <\| ((isLUB_csSup A_nonemp A_bdd).isLUB_of_tendsto Mf A_nonemp <\| Cf.mono_left fun ⦃_⦄ a ↦ a).csSup_eq (A_nonemp.image f) /-- A monotone function continuous at the supremum of a nonempty set sends this supremum to the supremum of the image of this set. -/ theorem Monotone.map_csSup_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A)) (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sSup (f '' A) := MonotoneOn.map_csSup_of_continuousWithinAt Cf.continuousWithinAt (Mf.monotoneOn _) A_nonemp A_bdd /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed supremum to the indexed supremum of the composition. -/ theorem Monotone.map_ciSup_of_continuousAt {ι : Sort} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨆ i, f (g i) := by rw [iSup, Monotone.map_csSup_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iSup, comp_def] /-- A monotone function continuous at the infimum of a nonempty set sends this infimum to the infimum of the image of this set. -/ theorem MonotoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α} (Cf : ContinuousWithinAt f A (sInf A)) (Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) : f (sInf A) = sInf (f '' A) := MonotoneOn.map_csSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd /-- A monotone function continuous at the infimum of a nonempty set sends this infimum to the infimum of the image of this set. -/ theorem Monotone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A)) (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) : f (sInf A) = sInf (f '' A) := Monotone.map_csSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed infimum to the indexed infimum of the composition. -/ theorem Monotone.map_ciInf_of_continuousAt {ι : Sort} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) := by rw [iInf, Monotone.map_csInf_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iInf, comp_def] /-- An antitone function continuous at the infimum of a nonempty set sends this infimum to the supremum of the image of this set. -/ theorem AntitoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α} (Cf : ContinuousWithinAt f A (sInf A)) (Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) : f (sInf A) = sSup (f '' A) := MonotoneOn.map_csInf_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd /-- An antitone function continuous at the infimum of a nonempty set sends this infimum to the supremum of the image of this set. -/ theorem Antitone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A)) (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) : f (sInf A) = sSup (f '' A) := Monotone.map_csInf_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed infimum to the indexed supremum of the composition. -/ theorem Antitone.map_ciInf_of_continuousAt {ι : Sort} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) := by rw [iInf, Antitone.map_csInf_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iSup, comp_def] /-- An antitone function continuous at the supremum of a nonempty set sends this supremum to the infimum of the image of this set. -/ theorem AntitoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α} (Cf : ContinuousWithinAt f A (sSup A)) (Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sInf (f '' A) := MonotoneOn.map_csSup_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd /-- An antitone function continuous at the supremum of a nonempty set sends this supremum to the infimum of the image of this set. -/ theorem Antitone.map_csSup_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A)) (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sInf (f '' A) := Monotone.map_csSup_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed supremum to the indexed infimum of the composition. -/ theorem Antitone.map_ciSup_of_continuousAt {ι : Sort} [Nonempty ι] {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) := by rw [iSup, Antitone.map_csSup_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iInf, comp_def] end ConditionallyCompleteLinearOrder section CompleteLinearOrder variable [CompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [CompleteLinearOrder β] [TopologicalSpace β] [OrderClosedTopology β] theorem sSup_mem_closure {s : Set α} (hs : s.Nonempty) : sSup s ∈ closure s := (isLUB_sSup s).mem_closure hs theorem sInf_mem_closure {s : Set α} (hs : s.Nonempty) : sInf s ∈ closure s := (isGLB_sInf s).mem_closure hs theorem IsClosed.sSup_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sSup s ∈ s := (isLUB_sSup s).mem_of_isClosed hs hc theorem IsClosed.sInf_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sInf s ∈ s := (isGLB_sInf s).mem_of_isClosed hs hc /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends this supremum to the supremum of the image of this set. -/ theorem MonotoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α} (Cf : ContinuousWithinAt f s (sSup s)) (Mf : MonotoneOn f s) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) := by rcases s.eq_empty_or_nonempty with h \| h · simp [h, fbot] · exact Mf.map_csSup_of_continuousWithinAt Cf h /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends this supremum to the supremum of the image of this set. -/ theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) := MonotoneOn.map_sSup_of_continuousWithinAt Cf.continuousWithinAt (Mf.monotoneOn _) fbot /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/ theorem Monotone.map_iSup_of_continuousAt {ι : Sort} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (⨆ i, g i) = ⨆ i, f (g i) := by rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, ← range_comp, iSup, comp_def] /-- A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends this infimum to the infimum of the image of this set. -/ theorem MonotoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α} (Cf : ContinuousWithinAt f s (sInf s)) (Mf : MonotoneOn f s) (ftop : f ⊤ = ⊤) : f (sInf s) = sInf (f '' s) := MonotoneOn.map_sSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop /-- A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends this infimum to the infimum of the image of this set. -/ theorem Monotone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (sInf s) = sInf (f '' s) := Monotone.map_sSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop /-- If a monotone function sending `top` to `top` is continuous at the indexed infimum over a `Sort`, then it sends this indexed infimum to the indexed infimum of the composition. -/ theorem Monotone.map_iInf_of_continuousAt {ι : Sort} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (iInf g) = iInf (f ∘ g) := Monotone.map_iSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop /-- An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends this supremum to the infimum of the image of this set. -/ theorem AntitoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α} (Cf : ContinuousWithinAt f s (sSup s)) (Af : AntitoneOn f s) (fbot : f ⊥ = ⊤) : f (sSup s) = sInf (f '' s) := MonotoneOn.map_sSup_of_continuousWithinAt (show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sSup s) from Cf) Af fbot /-- An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends this supremum to the infimum of the image of this set. -/ theorem Antitone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s)) (Af : Antitone f) (fbot : f ⊥ = ⊤) : f (sSup s) = sInf (f '' s) := Monotone.map_sSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sSup s) from Cf) Af fbot /-- An antitone function sending `bot` to `top` is continuous at the indexed supremum over a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/ theorem Antitone.map_iSup_of_continuousAt {ι : Sort} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iSup g)) (Af : Antitone f) (fbot : f ⊥ = ⊤) : f (⨆ i, g i) = ⨅ i, f (g i) := Monotone.map_iSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iSup g) from Cf) Af fbot /-- An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends this infimum to the supremum of the image of this set. -/ theorem AntitoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α} (Cf : ContinuousWithinAt f s (sInf s)) (Af : AntitoneOn f s) (ftop : f ⊤ = ⊥) : f (sInf s) = sSup (f '' s) := MonotoneOn.map_sInf_of_continuousWithinAt (show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sInf s) from Cf) Af ftop /-- An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends this infimum to the supremum of the image of this set. -/ theorem Antitone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s)) (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (sInf s) = sSup (f '' s) := Monotone.map_sInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sInf s) from Cf) Af ftop /-- If an antitone function sending `top` to `bot` is continuous at the indexed infimum over a `Sort`, then it sends this indexed infimum to the indexed supremum of the composition. -/ theorem Antitone.map_iInf_of_continuousAt {ι : Sort} {f : α → β} {g : ι → α} (Cf : ContinuousAt f (iInf g)) (Af : Antitone f) (ftop : f ⊤ = ⊥) : f (iInf g) = iSup (f ∘ g) := Monotone.map_iInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iInf g) from Cf) Af ftop end CompleteLinearOrder section ConditionallyCompleteLinearOrder variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] theorem csSup_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddAbove s) : sSup s ∈ closure s := (isLUB_csSup hs B).mem_closure hs theorem csInf_mem_closure {s : Set α} (hs : s.Nonempty) (B : BddBelow s) : sInf s ∈ closure s := (isGLB_csInf hs B).mem_closure hs theorem IsClosed.csSup_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) : sSup s ∈ s := (isLUB_csSup hs B).mem_of_isClosed hs hc theorem IsClosed.csInf_mem {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) : sInf s ∈ s := (isGLB_csInf hs B).mem_of_isClosed hs hc theorem IsClosed.isLeast_csInf {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddBelow s) : IsLeast s (sInf s) := ⟨hc.csInf_mem hs B, (isGLB_csInf hs B).1⟩ theorem IsClosed.isGreatest_csSup {s : Set α} (hc : IsClosed s) (hs : s.Nonempty) (B : BddAbove s) : IsGreatest s (sSup s) := IsClosed.isLeast_csInf (α := αᵒᵈ) hc hs B lemma MonotoneOn.tendsto_nhdsWithin_Ioo_left {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x y : α} (h_nonempty : (Ioo y x).Nonempty) (Mf : MonotoneOn f (Ioo y x)) (h_bdd : BddAbove (f '' Ioo y x)) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Ioo y x))) := by refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩ · obtain ⟨z, ⟨yz, zx⟩, lz⟩ : ∃ a : α, a ∈ Ioo y x ∧ l < f a := by simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using exists_lt_of_lt_csSup (h_nonempty.image _) hl filter_upwards [Ioo_mem_nhdsLT zx] with w hw exact lz.trans_le <\| Mf ⟨yz, zx⟩ ⟨yz.trans_le hw.1.le, hw.2⟩ hw.1.le · rcases h_nonempty with ⟨_, hy, hx⟩ filter_upwards [Ioo_mem_nhdsLT (hy.trans hx)] with w hw exact (le_csSup h_bdd (mem_image_of_mem _ hw)).trans_lt hm lemma MonotoneOn.tendsto_nhdsWithin_Ioo_right {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x y : α} (h_nonempty : (Ioo x y).Nonempty) (Mf : MonotoneOn f (Ioo x y)) (h_bdd : BddBelow (f '' Ioo x y)) : Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioo x y))) := by refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩ · rcases h_nonempty with ⟨p, hy, hx⟩ filter_upwards [Ioo_mem_nhdsGT (hy.trans hx)] with w hw exact hl.trans_le <\| csInf_le h_bdd (mem_image_of_mem _ hw) · obtain ⟨z, ⟨xz, zy⟩, zm⟩ : ∃ a : α, a ∈ Ioo x y ∧ f a < m := by simpa [mem_image, exists_prop, exists_exists_and_eq_and] using exists_lt_of_csInf_lt (h_nonempty.image _) hm filter_upwards [Ioo_mem_nhdsGT xz] with w hw exact (Mf ⟨hw.1, hw.2.trans zy⟩ ⟨xz, zy⟩ hw.2.le).trans_lt zm lemma MonotoneOn.tendsto_nhdsLT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α} (Mf : MonotoneOn f (Iio x)) (h_bdd : BddAbove (f '' Iio x)) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) := by rcases eq_empty_or_nonempty (Iio x) with (h \| h); · simp [h] refine tendsto_order.2 ⟨fun l hl => ?_, fun m hm => ?_⟩ · obtain ⟨z, zx, lz⟩ : ∃ a : α, a < x ∧ l < f a := by simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using exists_lt_of_lt_csSup (h.image _) hl filter_upwards [Ioo_mem_nhdsLT zx] with y hy using lz.trans_le (Mf zx hy.2 hy.1.le) · refine mem_of_superset self_mem_nhdsWithin fun y hy => lt_of_le_of_lt ?_ hm exact le_csSup h_bdd (mem_image_of_mem _ hy) lemma MonotoneOn.tendsto_nhdsGT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α} (Mf : MonotoneOn f (Ioi x)) (h_bdd : BddBelow (f '' Ioi x)) : Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioi x))) := MonotoneOn.tendsto_nhdsLT (α := αᵒᵈ) (β := βᵒᵈ) Mf.dual h_bdd /-- A monotone map has a limit to the left of any point `x`, equal to `sSup (f '' (Iio x))`. -/ theorem Monotone.tendsto_nhdsLT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sSup (f '' Iio x))) := MonotoneOn.tendsto_nhdsLT (Mf.monotoneOn _) (Mf.map_bddAbove bddAbove_Iio) /-- A monotone map has a limit to the right of any point `x`, equal to `sInf (f '' (Ioi x))`. -/ theorem Monotone.tendsto_nhdsGT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Mf : Monotone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sInf (f '' Ioi x))) := Monotone.tendsto_nhdsLT (α := αᵒᵈ) (β := βᵒᵈ) Mf.dual x lemma AntitoneOn.tendsto_nhdsWithin_Ioo_left {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x y : α} (h_nonempty : (Ioo y x).Nonempty) (Af : AntitoneOn f (Ioo y x)) (h_bdd : BddBelow (f '' Ioo y x)) : Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Ioo y x))) := MonotoneOn.tendsto_nhdsWithin_Ioo_left h_nonempty Af.dual_right h_bdd lemma AntitoneOn.tendsto_nhdsWithin_Ioo_right {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x y : α} (h_nonempty : (Ioo x y).Nonempty) (Af : AntitoneOn f (Ioo x y)) (h_bdd : BddAbove (f '' Ioo x y)) : Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioo x y))) := MonotoneOn.tendsto_nhdsWithin_Ioo_right h_nonempty Af.dual_right h_bdd lemma AntitoneOn.tendsto_nhdsLT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α} (Af : AntitoneOn f (Iio x)) (h_bdd : BddBelow (f '' Iio x)) : Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) := MonotoneOn.tendsto_nhdsLT Af.dual_right h_bdd lemma AntitoneOn.tendsto_nhdsGT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} {x : α} (Af : AntitoneOn f (Ioi x)) (h_bdd : BddAbove (f '' Ioi x)) : Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioi x))) := MonotoneOn.tendsto_nhdsGT Af.dual_right h_bdd /-- An antitone map has a limit to the left of any point `x`, equal to `sInf (f '' (Iio x))`. -/ theorem Antitone.tendsto_nhdsLT {α β : Type} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Af : Antitone f) (x : α) : Tendsto f (𝓝[<] x) (𝓝 (sInf (f '' Iio x))) := Monotone.tendsto_nhdsLT Af.dual_right x /-- An antitone map has a limit to the right of any point `x`, equal to `sSup (f '' (Ioi x))`. -/ theorem Antitone.tendsto_nhdsGT {α β : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β] {f : α → β} (Af : Antitone f) (x : α) : Tendsto f (𝓝[>] x) (𝓝 (sSup (f '' Ioi x))) := Monotone.tendsto_nhdsGT Af.dual_right x end ConditionallyCompleteLinearOrder
.lake/packages/mathlib/Mathlib/Topology/Order/Lattice.lean	import Mathlib.Topology.Constructions import Mathlib.Topology.Order.OrderClosed /-! # Topological lattices In this file we define mixin classes `ContinuousInf` and `ContinuousSup`. We define the class `TopologicalLattice` as a topological space and lattice `L` extending `ContinuousInf` and `ContinuousSup`. ## References * [Gierz et al, A Compendium of Continuous Lattices][GierzEtAl1980] ## Tags topological, lattice -/ open Filter open Topology /-- Let `L` be a topological space and let `L×L` be equipped with the product topology and let `⊓:L×L → L` be an infimum. Then `L` is said to have (jointly) continuous infimum if the map `⊓:L×L → L` is continuous. -/ class ContinuousInf (L : Type) [TopologicalSpace L] [Min L] : Prop where /-- The infimum is continuous -/ continuous_inf : Continuous fun p : L × L => p.1 ⊓ p.2 /-- Let `L` be a topological space and let `L×L` be equipped with the product topology and let `⊓:L×L → L` be a supremum. Then `L` is said to have (jointly) continuous supremum* if the map `⊓:L×L → L` is continuous. -/ class ContinuousSup (L : Type) [TopologicalSpace L] [Max L] : Prop where /-- The supremum is continuous -/ continuous_sup : Continuous fun p : L × L => p.1 ⊔ p.2 -- see Note [lower instance priority] instance (priority := 100) OrderDual.continuousSup (L : Type) [TopologicalSpace L] [Min L] [ContinuousInf L] : ContinuousSup Lᵒᵈ where continuous_sup := @ContinuousInf.continuous_inf L _ _ _ -- see Note [lower instance priority] instance (priority := 100) OrderDual.continuousInf (L : Type) [TopologicalSpace L] [Max L] [ContinuousSup L] : ContinuousInf Lᵒᵈ where continuous_inf := @ContinuousSup.continuous_sup L _ _ _ /-- Let `L` be a lattice equipped with a topology such that `L` has continuous infimum and supremum. Then `L` is said to be a topological lattice. -/ class TopologicalLattice (L : Type) [TopologicalSpace L] [Lattice L] : Prop extends ContinuousInf L, ContinuousSup L -- see Note [lower instance priority] instance (priority := 100) OrderDual.topologicalLattice (L : Type) [TopologicalSpace L] [Lattice L] [TopologicalLattice L] : TopologicalLattice Lᵒᵈ where -- see Note [lower instance priority] instance (priority := 100) LinearOrder.topologicalLattice {L : Type} [TopologicalSpace L] [LinearOrder L] [OrderClosedTopology L] : TopologicalLattice L where continuous_inf := continuous_min continuous_sup := continuous_max variable {L X : Type} [TopologicalSpace L] [TopologicalSpace X] @[continuity] theorem continuous_inf [Min L] [ContinuousInf L] : Continuous fun p : L × L => p.1 ⊓ p.2 := ContinuousInf.continuous_inf @[continuity, fun_prop] theorem Continuous.inf [Min L] [ContinuousInf L] {f g : X → L} (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x ⊓ g x := continuous_inf.comp (hf.prodMk hg :) @[continuity] theorem continuous_sup [Max L] [ContinuousSup L] : Continuous fun p : L × L => p.1 ⊔ p.2 := ContinuousSup.continuous_sup @[continuity, fun_prop] theorem Continuous.sup [Max L] [ContinuousSup L] {f g : X → L} (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x ⊔ g x := continuous_sup.comp (hf.prodMk hg :) namespace Filter.Tendsto section SupInf variable {α : Type} {l : Filter α} {f g : α → L} {x y : L} lemma sup_nhds' [Max L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) : Tendsto (f ⊔ g) l (𝓝 (x ⊔ y)) := (continuous_sup.tendsto _).comp (hf.prodMk_nhds hg) lemma sup_nhds [Max L] [ContinuousSup L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) : Tendsto (fun i => f i ⊔ g i) l (𝓝 (x ⊔ y)) := hf.sup_nhds' hg lemma inf_nhds' [Min L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) : Tendsto (f ⊓ g) l (𝓝 (x ⊓ y)) := (continuous_inf.tendsto _).comp (hf.prodMk_nhds hg) lemma inf_nhds [Min L] [ContinuousInf L] (hf : Tendsto f l (𝓝 x)) (hg : Tendsto g l (𝓝 y)) : Tendsto (fun i => f i ⊓ g i) l (𝓝 (x ⊓ y)) := hf.inf_nhds' hg end SupInf open Finset variable {ι α : Type} {s : Finset ι} {f : ι → α → L} {l : Filter α} {g : ι → L} lemma finset_sup'_nhds [SemilatticeSup L] [ContinuousSup L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.sup' hne f) l (𝓝 (s.sup' hne g)) := by induction hne using Finset.Nonempty.cons_induction with \| singleton => simpa using hs \| cons a s ha hne ihs => rw [forall_mem_cons] at hs simp only [sup'_cons, hne] exact hs.1.sup_nhds (ihs hs.2) lemma finset_sup'_nhds_apply [SemilatticeSup L] [ContinuousSup L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (fun a ↦ s.sup' hne (f · a)) l (𝓝 (s.sup' hne g)) := by simpa only [← Finset.sup'_apply] using finset_sup'_nhds hne hs lemma finset_inf'_nhds [SemilatticeInf L] [ContinuousInf L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.inf' hne f) l (𝓝 (s.inf' hne g)) := finset_sup'_nhds (L := Lᵒᵈ) hne hs lemma finset_inf'_nhds_apply [SemilatticeInf L] [ContinuousInf L] (hne : s.Nonempty) (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (fun a ↦ s.inf' hne (f · a)) l (𝓝 (s.inf' hne g)) := finset_sup'_nhds_apply (L := Lᵒᵈ) hne hs lemma finset_sup_nhds [SemilatticeSup L] [OrderBot L] [ContinuousSup L] (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.sup f) l (𝓝 (s.sup g)) := by rcases s.eq_empty_or_nonempty with rfl \| hne · simpa using tendsto_const_nhds · simp only [← sup'_eq_sup hne] exact finset_sup'_nhds hne hs lemma finset_sup_nhds_apply [SemilatticeSup L] [OrderBot L] [ContinuousSup L] (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (fun a ↦ s.sup (f · a)) l (𝓝 (s.sup g)) := by simpa only [← Finset.sup_apply] using finset_sup_nhds hs lemma finset_inf_nhds [SemilatticeInf L] [OrderTop L] [ContinuousInf L] (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (s.inf f) l (𝓝 (s.inf g)) := finset_sup_nhds (L := Lᵒᵈ) hs lemma finset_inf_nhds_apply [SemilatticeInf L] [OrderTop L] [ContinuousInf L] (hs : ∀ i ∈ s, Tendsto (f i) l (𝓝 (g i))) : Tendsto (fun a ↦ s.inf (f · a)) l (𝓝 (s.inf g)) := finset_sup_nhds_apply (L := Lᵒᵈ) hs end Filter.Tendsto section Sup variable [Max L] [ContinuousSup L] {f g : X → L} {s : Set X} {x : X} lemma ContinuousAt.sup' (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f ⊔ g) x := hf.sup_nhds' hg @[fun_prop] lemma ContinuousAt.sup (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun a ↦ f a ⊔ g a) x := hf.sup' hg lemma ContinuousWithinAt.sup' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (f ⊔ g) s x := hf.sup_nhds' hg lemma ContinuousWithinAt.sup (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun a ↦ f a ⊔ g a) s x := hf.sup' hg lemma ContinuousOn.sup' (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (f ⊔ g) s := fun x hx ↦ (hf x hx).sup' (hg x hx) @[fun_prop] lemma ContinuousOn.sup (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun a ↦ f a ⊔ g a) s := hf.sup' hg lemma Continuous.sup' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊔ g) := hf.sup hg end Sup section Inf variable [Min L] [ContinuousInf L] {f g : X → L} {s : Set X} {x : X} lemma ContinuousAt.inf' (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (f ⊓ g) x := hf.inf_nhds' hg @[fun_prop] lemma ContinuousAt.inf (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun a ↦ f a ⊓ g a) x := hf.inf' hg lemma ContinuousWithinAt.inf' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (f ⊓ g) s x := hf.inf_nhds' hg lemma ContinuousWithinAt.inf (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) : ContinuousWithinAt (fun a ↦ f a ⊓ g a) s x := hf.inf' hg lemma ContinuousOn.inf' (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (f ⊓ g) s := fun x hx ↦ (hf x hx).inf' (hg x hx) @[fun_prop] lemma ContinuousOn.inf (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun a ↦ f a ⊓ g a) s := hf.inf' hg lemma Continuous.inf' (hf : Continuous f) (hg : Continuous g) : Continuous (f ⊓ g) := hf.inf hg end Inf section FinsetSup' variable {ι : Type} [SemilatticeSup L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} lemma ContinuousAt.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (fun a ↦ s.sup' hne (f · a)) x := Tendsto.finset_sup'_nhds_apply hne hs lemma ContinuousAt.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (s.sup' hne f) x := by simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs lemma ContinuousWithinAt.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (fun a ↦ s.sup' hne (f · a)) t x := Tendsto.finset_sup'_nhds_apply hne hs lemma ContinuousWithinAt.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup' hne f) t x := by simpa only [← Finset.sup'_apply] using finset_sup'_apply hne hs lemma ContinuousOn.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (fun a ↦ s.sup' hne (f · a)) t := fun x hx ↦ ContinuousWithinAt.finset_sup'_apply hne fun i hi ↦ hs i hi x hx lemma ContinuousOn.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (s.sup' hne f) t := fun x hx ↦ ContinuousWithinAt.finset_sup' hne fun i hi ↦ hs i hi x hx lemma Continuous.finset_sup'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (fun a ↦ s.sup' hne (f · a)) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup'_apply _ fun i hi ↦ (hs i hi).continuousAt lemma Continuous.finset_sup' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup' hne f) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup' _ fun i hi ↦ (hs i hi).continuousAt end FinsetSup' section FinsetSup variable {ι : Type} [SemilatticeSup L] [OrderBot L] [ContinuousSup L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} lemma ContinuousAt.finset_sup_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (fun a ↦ s.sup (f · a)) x := Tendsto.finset_sup_nhds_apply hs lemma ContinuousAt.finset_sup (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (s.sup f) x := by simpa only [← Finset.sup_apply] using finset_sup_apply hs lemma ContinuousWithinAt.finset_sup_apply (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (fun a ↦ s.sup (f · a)) t x := Tendsto.finset_sup_nhds_apply hs lemma ContinuousWithinAt.finset_sup (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.sup f) t x := by simpa only [← Finset.sup_apply] using finset_sup_apply hs lemma ContinuousOn.finset_sup_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (fun a ↦ s.sup (f · a)) t := fun x hx ↦ ContinuousWithinAt.finset_sup_apply fun i hi ↦ hs i hi x hx lemma ContinuousOn.finset_sup (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (s.sup f) t := fun x hx ↦ ContinuousWithinAt.finset_sup fun i hi ↦ hs i hi x hx lemma Continuous.finset_sup_apply (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (fun a ↦ s.sup (f · a)) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup_apply fun i hi ↦ (hs i hi).continuousAt lemma Continuous.finset_sup (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.sup f) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_sup fun i hi ↦ (hs i hi).continuousAt end FinsetSup section FinsetInf' variable {ι : Type} [SemilatticeInf L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} lemma ContinuousAt.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (fun a ↦ s.inf' hne (f · a)) x := Tendsto.finset_inf'_nhds_apply hne hs lemma ContinuousAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (s.inf' hne f) x := by simpa only [← Finset.inf'_apply] using finset_inf'_apply hne hs lemma ContinuousWithinAt.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (fun a ↦ s.inf' hne (f · a)) t x := Tendsto.finset_inf'_nhds_apply hne hs lemma ContinuousWithinAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf' hne f) t x := by simpa only [← Finset.inf'_apply] using finset_inf'_apply hne hs lemma ContinuousOn.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (fun a ↦ s.inf' hne (f · a)) t := fun x hx ↦ ContinuousWithinAt.finset_inf'_apply hne fun i hi ↦ hs i hi x hx lemma ContinuousOn.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (s.inf' hne f) t := fun x hx ↦ ContinuousWithinAt.finset_inf' hne fun i hi ↦ hs i hi x hx lemma Continuous.finset_inf'_apply (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (fun a ↦ s.inf' hne (f · a)) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf'_apply _ fun i hi ↦ (hs i hi).continuousAt lemma Continuous.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.inf' hne f) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf' _ fun i hi ↦ (hs i hi).continuousAt end FinsetInf' section FinsetInf variable {ι : Type*} [SemilatticeInf L] [OrderTop L] [ContinuousInf L] {s : Finset ι} {f : ι → X → L} {t : Set X} {x : X} lemma ContinuousAt.finset_inf_apply (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (fun a ↦ s.inf (f · a)) x := Tendsto.finset_inf_nhds_apply hs lemma ContinuousAt.finset_inf (hs : ∀ i ∈ s, ContinuousAt (f i) x) : ContinuousAt (s.inf f) x := by simpa only [← Finset.inf_apply] using finset_inf_apply hs lemma ContinuousWithinAt.finset_inf_apply (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (fun a ↦ s.inf (f · a)) t x := Tendsto.finset_inf_nhds_apply hs lemma ContinuousWithinAt.finset_inf (hs : ∀ i ∈ s, ContinuousWithinAt (f i) t x) : ContinuousWithinAt (s.inf f) t x := by simpa only [← Finset.inf_apply] using finset_inf_apply hs lemma ContinuousOn.finset_inf_apply (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (fun a ↦ s.inf (f · a)) t := fun x hx ↦ ContinuousWithinAt.finset_inf_apply fun i hi ↦ hs i hi x hx lemma ContinuousOn.finset_inf (hs : ∀ i ∈ s, ContinuousOn (f i) t) : ContinuousOn (s.inf f) t := fun x hx ↦ ContinuousWithinAt.finset_inf fun i hi ↦ hs i hi x hx lemma Continuous.finset_inf_apply (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (fun a ↦ s.inf (f · a)) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf_apply fun i hi ↦ (hs i hi).continuousAt lemma Continuous.finset_inf (hs : ∀ i ∈ s, Continuous (f i)) : Continuous (s.inf f) := continuous_iff_continuousAt.2 fun _ ↦ ContinuousAt.finset_inf fun i hi ↦ (hs i hi).continuousAt end FinsetInf
.lake/packages/mathlib/Mathlib/Topology/Order/HullKernel.lean	import Mathlib.Data.Set.Subset import Mathlib.Order.Irreducible import Mathlib.Topology.Order.LowerUpperTopology import Mathlib.Topology.Sets.Closeds /-! # Hull-Kernel Topology Let `α` be a `CompleteLattice` and let `T` be a subset of `α`. The pair of maps `S → sInf (Subtype.val '' S)` and `a → T ↓∩ Ici a` are often referred to as the `kernel` and the `hull` respectively. They form an antitone Galois connection between the subsets of `T` and `α`. When `α` can be generated from `T` by taking infs, this becomes a Galois insertion and the relative topology (`Topology.lower`) on `T` takes on a particularly simple form: the relative-open sets are exactly the sets of the form `(hull T a)ᶜ` for some `a` in `α`. The topological closure coincides with the closure arising from the Galois insertion. For this reason the relative lower topology on `T` is often referred to as the "hull-kernel topology". The names "Jacobson topology" and "structure topology" also occur in the literature. ## Main statements - `PrimitiveSpectrum.isTopologicalBasis_relativeLower` - the sets `(hull a)ᶜ` form a basis for the relative lower topology on `T`. - `PrimitiveSpectrum.isOpen_iff` - for a complete lattice, the sets `(hull a)ᶜ` are the relative topology. - `PrimitiveSpectrum.gc` - the `kernel` and the `hull` form a Galois connection - `PrimitiveSpectrum.gi` - when `T` generates `α`, the Galois connection becomes an insertion. ## Implementation notes The antitone Galois connection from `Set T` to `α` is implemented as a monotone Galois connection between `Set T` to `αᵒᵈ`. ## Motivation The motivating example for the study of a set `T` of prime elements which generate `α` is the primitive spectrum of the lattice of M-ideals of a Banach space. ## References * [Gierz et al, A Compendium of Continuous Lattices][GierzEtAl1980] * [Henriksen et al, Joincompact spaces, continuous lattices and C-algebras*][henriksen_et_al1997] ## Tags lower topology, hull-kernel topology, Jacobson topology, structure topology, primitive spectrum -/ variable {α} open TopologicalSpace open Topology open Set open Set.Notation section SemilatticeInf variable [SemilatticeInf α] namespace PrimitiveSpectrum /-- For `a` of type `α` the set of element of `T` which dominate `a` is the `hull` of `a` in `T`. -/ abbrev hull (T : Set α) (a : α) := T ↓∩ Ici a variable {T : Set α} /- The set of relative-closed sets of the form `hull T a` for some `a` in `α` is closed under pairwise union. -/ lemma hull_inf (hT : ∀ p ∈ T, InfPrime p) (a b : α) : hull T (a ⊓ b) = hull T a ∪ hull T b := by ext p constructor <;> intro h · exact (hT p p.2).2 h · rcases h with (h1 \| h3) · exact inf_le_of_left_le h1 · exact inf_le_of_right_le h3 variable [OrderTop α] /- Every relative-closed set of the form `T ↓∩ (↑(upperClosure F))` for `F` finite is a relative-closed set of the form `hull T a` where `a = ⨅ F`. -/ open Finset in lemma hull_finsetInf (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) : hull T (inf F id) = T ↓∩ upperClosure (F : Set α) := by rw [coe_upperClosure] induction F using Finset.cons_induction with \| empty => simp only [coe_empty, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.preimage_empty, inf_empty] by_contra hf rw [← Set.not_nonempty_iff_eq_empty, not_not] at hf obtain ⟨x, hx⟩ := hf exact (hT x (Subtype.coe_prop x)).1 (isMax_iff_eq_top.mpr (eq_top_iff.mpr hx)) \| cons a F' _ I4 => simp [hull_inf hT, I4] /- Every relative-open set of the form `T ↓∩ (↑(upperClosure F))ᶜ` for `F` finite is a relative-open set of the form `(hull T a)ᶜ` where `a = ⨅ F`. -/ open Finset in lemma preimage_upperClosure_compl_finset (hT : ∀ p ∈ T, InfPrime p) (F : Finset α) : T ↓∩ (upperClosure (F : Set α))ᶜ = (hull T (inf F id))ᶜ := by rw [Set.preimage_compl, (hull_finsetInf hT)] variable [TopologicalSpace α] [IsLower α] /- The relative-open sets of the form `(hull T a)ᶜ` for `a` in `α` form a basis for the relative Lower topology. -/ lemma isTopologicalBasis_relativeLower (hT : ∀ p ∈ T, InfPrime p) : IsTopologicalBasis { S : Set T \| ∃ (a : α), (hull T a)ᶜ = S } := by convert isTopologicalBasis_subtype Topology.IsLower.isTopologicalBasis T ext R simp only [preimage_compl, mem_setOf_eq, IsLower.lowerBasis, mem_image, exists_exists_and_eq_and] constructor <;> intro ha · obtain ⟨a, ha'⟩ := ha use {a} rw [← (Function.Injective.preimage_image Subtype.val_injective R), ← ha'] simp only [finite_singleton, upperClosure_singleton, UpperSet.coe_Ici, image_val_compl, Subtype.image_preimage_coe, diff_self_inter, preimage_diff, Subtype.coe_preimage_self, true_and] exact compl_eq_univ_diff (Subtype.val ⁻¹' Ici a) · obtain ⟨F, hF⟩ := ha lift F to Finset α using hF.1 use Finset.inf F id ext simp [hull_finsetInf hT, ← hF.2] end PrimitiveSpectrum end SemilatticeInf namespace PrimitiveSpectrum variable [CompleteLattice α] {T : Set α} universe v lemma hull_iSup {ι : Sort v} (s : ι → α) : hull T (iSup s) = ⋂ i, hull T (s i) := by aesop lemma hull_sSup (S : Set α) : hull T (sSup S) = ⋂₀ { hull T a \| a ∈ S } := by aesop /- When `α` is complete, a set is Lower topology relative-open if and only if it is of the form `(hull T a)ᶜ` for some `a` in `α`.-/ lemma isOpen_iff [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p) (S : Set T) : IsOpen S ↔ ∃ (a : α), S = (hull T a)ᶜ := by constructor <;> intro h · let R := {a : α \| (hull T a)ᶜ ⊆ S} use sSup R rw [IsTopologicalBasis.open_eq_sUnion' (isTopologicalBasis_relativeLower hT) h] aesop · obtain ⟨a, ha⟩ := h exact ⟨(Ici a)ᶜ, isClosed_Ici.isOpen_compl, ha.symm⟩ /- When `α` is complete, a set is closed in the relative lower topology if and only if it is of the form `hull T a` for some `a` in `α`.-/ lemma isClosed_iff [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p) {S : Set T} : IsClosed S ↔ ∃ (a : α), S = hull T a := by simp only [← isOpen_compl_iff, isOpen_iff hT, compl_inj_iff] /-- For a subset `S` of `T`, `kernel S` is the infimum of `S` (considered as a set of `α`) -/ abbrev kernel (S : Set T) := sInf (Subtype.val '' S) /- The pair of maps `kernel` and `hull` form an antitone Galois connection between the subsets of `T` and `α`. -/ open OrderDual in theorem gc : GaloisConnection (α := Set T) (β := αᵒᵈ) (fun S => toDual (kernel S)) (fun a => hull T (ofDual a)) := fun S a => by simp [Set.subset_def] lemma gc_closureOperator (S : Set T) : gc.closureOperator S = hull T (kernel S) := by simp only [toDual_sInf, GaloisConnection.closureOperator_apply, ofDual_sSup] rw [← preimage_comp, ← OrderDual.toDual_symm_eq, Equiv.symm_comp_self, preimage_id_eq, id_eq] variable (T) /-- `T` order generates `α` if, for every `a` in `α`, there exists a subset of `T` such that `a` is the `kernel` of `S`. -/ def OrderGenerates := ∀ (a : α), ∃ (S : Set T), a = kernel S variable {T} /-- When `T` is order generating, the `kernel` and the `hull` form a Galois insertion -/ def gi (hG : OrderGenerates T) : GaloisInsertion (α := Set T) (β := αᵒᵈ) (OrderDual.toDual ∘ kernel) (hull T ∘ OrderDual.ofDual) := gc.toGaloisInsertion fun a ↦ by obtain ⟨S, rfl⟩ := hG a rw [OrderDual.le_toDual, kernel, kernel] exact sInf_le_sInf <\| image_val_mono fun c hcS => by rw [hull, mem_preimage, mem_Ici] exact CompleteSemilatticeInf.sInf_le _ _ (mem_image_of_mem Subtype.val hcS) lemma kernel_hull (hG : OrderGenerates T) (a : α) : kernel (hull T a) = a := by conv_rhs => rw [← OrderDual.ofDual_toDual a, ← (gi hG).l_u_eq a] rfl lemma hull_kernel_of_isClosed [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) {C : Set T} (h : IsClosed C) : hull T (kernel C) = C := by obtain ⟨a, ha⟩ := (isClosed_iff hT).mp h rw [ha, kernel_hull hG] lemma closedsGC_closureOperator [TopologicalSpace α] [IsLower α] (hT : ∀ p ∈ T, InfPrime p) (hG : OrderGenerates T) (S : Set T) : (TopologicalSpace.Closeds.gc (α := T)).closureOperator S = hull T (kernel S) := by simp only [GaloisConnection.closureOperator_apply, Closeds.coe_closure, closure, le_antisymm_iff] constructor · exact fun ⦃a⦄ a ↦ a (hull T (kernel S)) ⟨(isClosed_iff hT).mpr ⟨kernel S, rfl⟩, image_subset_iff.mp (fun _ hbS => CompleteSemilatticeInf.sInf_le _ _ hbS)⟩ · simp_rw [le_eq_subset, subset_sInter_iff] intro R hR rw [← (hull_kernel_of_isClosed hT hG hR.1), ← gc_closureOperator] exact ClosureOperator.monotone _ hR.2 end PrimitiveSpectrum
.lake/packages/mathlib/Mathlib/Topology/Order/Rolle.lean	import Mathlib.Topology.Order.ExtendFrom import Mathlib.Topology.Order.Compact import Mathlib.Topology.Order.LocalExtr import Mathlib.Topology.Order.T5 /-! # Rolle's Theorem (topological part) In this file we prove the purely topological part of Rolle's Theorem: a function that is continuous on an interval $[a, b]$, $a < b$, has a local extremum at a point $x ∈ (a, b)$ provided that $f(a)=f(b)$. We also prove several variations of this statement. In `Mathlib/Analysis/Calculus/LocalExtr/Rolle` we use these lemmas to prove several versions of Rolle's Theorem from calculus. ## Keywords local minimum, local maximum, extremum, Rolle's Theorem -/ open Filter Set Topology variable {X Y : Type*} [ConditionallyCompleteLinearOrder X] [DenselyOrdered X] [TopologicalSpace X] [OrderTopology X] [LinearOrder Y] [TopologicalSpace Y] [OrderTopology Y] {f : X → Y} {a b : X} {l : Y} /-- A continuous function on a closed interval with `f a = f b` takes either its maximum or its minimum value at a point in the interior of the interval. -/ theorem exists_Ioo_extr_on_Icc (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : ∃ c ∈ Ioo a b, IsExtrOn f (Icc a b) c := by have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab) -- Consider absolute min and max points obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x := isCompact_Icc.exists_isMinOn ne hfc obtain ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f C := isCompact_Icc.exists_isMaxOn ne hfc by_cases hc : f c = f a · by_cases hC : f C = f a · have : ∀ x ∈ Icc a b, f x = f a := fun x hx => le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx) -- `f` is a constant, so we can take any point in `Ioo a b` rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩ refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩ simp only [mem_setOf_eq, this x hx, this c' (Ioo_subset_Icc_self hc'), le_rfl] · refine ⟨C, ⟨lt_of_le_of_ne Cmem.1 <\| mt ?_ hC, lt_of_le_of_ne Cmem.2 <\| mt ?_ hC⟩, Or.inr Cge⟩ exacts [fun h => by rw [h], fun h => by rw [h, hfI]] · refine ⟨c, ⟨lt_of_le_of_ne cmem.1 <\| mt ?_ hc, lt_of_le_of_ne cmem.2 <\| mt ?_ hc⟩, Or.inl cle⟩ exacts [fun h => by rw [h], fun h => by rw [h, hfI]] /-- A continuous function on a closed interval with `f a = f b` has a local extremum at some point of the corresponding open interval. -/ theorem exists_isLocalExtr_Ioo (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : ∃ c ∈ Ioo a b, IsLocalExtr f c := let ⟨c, cmem, hc⟩ := exists_Ioo_extr_on_Icc hab hfc hfI ⟨c, cmem, hc.isLocalExtr <\| Icc_mem_nhds cmem.1 cmem.2⟩ /-- If a function `f` is continuous on an open interval and tends to the same value at its endpoints, then it has an extremum on this open interval. -/ lemma exists_isExtrOn_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) : ∃ c ∈ Ioo a b, IsExtrOn f (Ioo a b) c := by have h : EqOn (extendFrom (Ioo a b) f) f (Ioo a b) := extendFrom_extends hfc obtain ⟨c, hc, hfc⟩ : ∃ c ∈ Ioo a b, IsExtrOn (extendFrom (Ioo a b) f) (Icc a b) c := exists_Ioo_extr_on_Icc hab (continuousOn_Icc_extendFrom_Ioo hab.ne hfc ha hb) ((eq_lim_at_left_extendFrom_Ioo hab ha).trans (eq_lim_at_right_extendFrom_Ioo hab hb).symm) exact ⟨c, hc, (hfc.on_subset Ioo_subset_Icc_self).congr h (h hc)⟩ /-- If a function `f` is continuous on an open interval and tends to the same value at its endpoints, then it has a local extremum on this open interval. -/ lemma exists_isLocalExtr_Ioo_of_tendsto (hab : a < b) (hfc : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 l)) (hb : Tendsto f (𝓝[<] b) (𝓝 l)) : ∃ c ∈ Ioo a b, IsLocalExtr f c := let ⟨c, cmem, hc⟩ := exists_isExtrOn_Ioo_of_tendsto hab hfc ha hb ⟨c, cmem, hc.isLocalExtr <\| Ioo_mem_nhds cmem.1 cmem.2⟩
.lake/packages/mathlib/Mathlib/Topology/Order/LocalExtr.lean	import Mathlib.Order.Filter.Extr import Mathlib.Topology.ContinuousOn /-! # Local extrema of functions on topological spaces ## Main definitions This file defines special versions of `IsFilter f a l`, `=Min/Max/Extr`, from `Mathlib/Order/Filter/Extr.lean` for two kinds of filters: `nhdsWithin` and `nhds`. These versions are called `IsLocalOn` and `IsLocal`, respectively. ## Main statements Many lemmas in this file restate those from `Mathlib/Order/Filter/Extr.lean`, and you can find detailed documentation there. These convenience lemmas are provided only to make the dot notation return propositions of expected types, not just `IsFilter`. Here is the list of statements specific to these two types of filters: `IsLocal.on`, `IsLocalOn.on_subset`: restrict to a subset; * `IsLocalOn.inter` : intersect the set with another one; `IsOn.localize` : a global extremum is a local extremum too. `Is[Local]On.isLocal` : if we have `IsLocalOn f s a` and `s ∈ 𝓝 a`, then we have `IsLocal f a`. -/ universe u v w x variable {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [TopologicalSpace α] open Set Filter Topology section Preorder variable [Preorder β] [Preorder γ] (f : α → β) (s : Set α) (a : α) /-- `IsLocalMinOn f s a` means that `f a ≤ f x` for all `x ∈ s` in some neighborhood of `a`. -/ def IsLocalMinOn := IsMinFilter f (𝓝[s] a) a /-- `IsLocalMaxOn f s a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. -/ def IsLocalMaxOn := IsMaxFilter f (𝓝[s] a) a /-- `IsLocalExtrOn f s a` means `IsLocalMinOn f s a ∨ IsLocalMaxOn f s a`. -/ def IsLocalExtrOn := IsExtrFilter f (𝓝[s] a) a /-- `IsLocalMin f a` means that `f a ≤ f x` for all `x` in some neighborhood of `a`. -/ def IsLocalMin := IsMinFilter f (𝓝 a) a /-- `IsLocalMax f a` means that `f x ≤ f a` for all `x ∈ s` in some neighborhood of `a`. -/ def IsLocalMax := IsMaxFilter f (𝓝 a) a /-- `IsLocalExtr f s a` means `IsLocalMin f s a ∨ IsLocalMax f s a`. -/ def IsLocalExtr := IsExtrFilter f (𝓝 a) a variable {f s a} theorem IsLocalExtrOn.elim {p : Prop} : IsLocalExtrOn f s a → (IsLocalMinOn f s a → p) → (IsLocalMaxOn f s a → p) → p := Or.elim theorem IsLocalExtr.elim {p : Prop} : IsLocalExtr f a → (IsLocalMin f a → p) → (IsLocalMax f a → p) → p := Or.elim /-! ### Restriction to (sub)sets -/ theorem IsLocalMin.on (h : IsLocalMin f a) (s) : IsLocalMinOn f s a := h.filter_inf _ theorem IsLocalMax.on (h : IsLocalMax f a) (s) : IsLocalMaxOn f s a := h.filter_inf _ theorem IsLocalExtr.on (h : IsLocalExtr f a) (s) : IsLocalExtrOn f s a := h.filter_inf _ theorem IsLocalMinOn.on_subset {t : Set α} (hf : IsLocalMinOn f t a) (h : s ⊆ t) : IsLocalMinOn f s a := hf.filter_mono <\| nhdsWithin_mono a h theorem IsLocalMaxOn.on_subset {t : Set α} (hf : IsLocalMaxOn f t a) (h : s ⊆ t) : IsLocalMaxOn f s a := hf.filter_mono <\| nhdsWithin_mono a h theorem IsLocalExtrOn.on_subset {t : Set α} (hf : IsLocalExtrOn f t a) (h : s ⊆ t) : IsLocalExtrOn f s a := hf.filter_mono <\| nhdsWithin_mono a h theorem IsLocalMinOn.inter (hf : IsLocalMinOn f s a) (t) : IsLocalMinOn f (s ∩ t) a := hf.on_subset inter_subset_left theorem IsLocalMaxOn.inter (hf : IsLocalMaxOn f s a) (t) : IsLocalMaxOn f (s ∩ t) a := hf.on_subset inter_subset_left theorem IsLocalExtrOn.inter (hf : IsLocalExtrOn f s a) (t) : IsLocalExtrOn f (s ∩ t) a := hf.on_subset inter_subset_left theorem IsMinOn.localize (hf : IsMinOn f s a) : IsLocalMinOn f s a := hf.filter_mono <\| inf_le_right theorem IsMaxOn.localize (hf : IsMaxOn f s a) : IsLocalMaxOn f s a := hf.filter_mono <\| inf_le_right theorem IsExtrOn.localize (hf : IsExtrOn f s a) : IsLocalExtrOn f s a := hf.filter_mono <\| inf_le_right theorem IsLocalMinOn.isLocalMin (hf : IsLocalMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a := have : 𝓝 a ≤ 𝓟 s := le_principal_iff.2 hs hf.filter_mono <\| le_inf le_rfl this theorem IsLocalMaxOn.isLocalMax (hf : IsLocalMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a := have : 𝓝 a ≤ 𝓟 s := le_principal_iff.2 hs hf.filter_mono <\| le_inf le_rfl this theorem IsLocalExtrOn.isLocalExtr (hf : IsLocalExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a := hf.elim (fun hf => (hf.isLocalMin hs).isExtr) fun hf => (hf.isLocalMax hs).isExtr lemma isLocalMinOn_univ_iff : IsLocalMinOn f univ a ↔ IsLocalMin f a := by simp only [IsLocalMinOn, IsLocalMin, nhdsWithin_univ] lemma isLocalMaxOn_univ_iff : IsLocalMaxOn f univ a ↔ IsLocalMax f a := by simp only [IsLocalMaxOn, IsLocalMax, nhdsWithin_univ] lemma isLocalExtrOn_univ_iff : IsLocalExtrOn f univ a ↔ IsLocalExtr f a := isLocalMinOn_univ_iff.or isLocalMaxOn_univ_iff theorem IsMinOn.isLocalMin (hf : IsMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a := hf.localize.isLocalMin hs theorem IsMaxOn.isLocalMax (hf : IsMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a := hf.localize.isLocalMax hs theorem IsExtrOn.isLocalExtr (hf : IsExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a := hf.localize.isLocalExtr hs theorem IsLocalMinOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalMinOn f s a) [NeBot (𝓝[<] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) := fun hle => have : ∀ᶠ y in 𝓝[<] f a, f a ≤ y := (eventually_map.2 hf).filter_mono (inf_le_left.trans hle) let ⟨_y, hy⟩ := (this.and self_mem_nhdsWithin).exists hy.1.not_gt hy.2 theorem IsLocalMaxOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalMaxOn f s a) [NeBot (𝓝[>] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) := @IsLocalMinOn.not_nhds_le_map α βᵒᵈ _ _ _ _ _ ‹_› hf ‹_› theorem IsLocalExtrOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalExtrOn f s a) [NeBot (𝓝[<] f a)] [NeBot (𝓝[>] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) := hf.elim (fun h => h.not_nhds_le_map) fun h => h.not_nhds_le_map /-! ### Constant -/ theorem isLocalMinOn_const {b : β} : IsLocalMinOn (fun _ => b) s a := isMinFilter_const theorem isLocalMaxOn_const {b : β} : IsLocalMaxOn (fun _ => b) s a := isMaxFilter_const theorem isLocalExtrOn_const {b : β} : IsLocalExtrOn (fun _ => b) s a := isExtrFilter_const theorem isLocalMin_const {b : β} : IsLocalMin (fun _ => b) a := isMinFilter_const theorem isLocalMax_const {b : β} : IsLocalMax (fun _ => b) a := isMaxFilter_const theorem isLocalExtr_const {b : β} : IsLocalExtr (fun _ => b) a := isExtrFilter_const /-! ### Composition with (anti)monotone functions -/ nonrec theorem IsLocalMin.comp_mono (hf : IsLocalMin f a) {g : β → γ} (hg : Monotone g) : IsLocalMin (g ∘ f) a := hf.comp_mono hg nonrec theorem IsLocalMax.comp_mono (hf : IsLocalMax f a) {g : β → γ} (hg : Monotone g) : IsLocalMax (g ∘ f) a := hf.comp_mono hg nonrec theorem IsLocalExtr.comp_mono (hf : IsLocalExtr f a) {g : β → γ} (hg : Monotone g) : IsLocalExtr (g ∘ f) a := hf.comp_mono hg nonrec theorem IsLocalMin.comp_antitone (hf : IsLocalMin f a) {g : β → γ} (hg : Antitone g) : IsLocalMax (g ∘ f) a := hf.comp_antitone hg nonrec theorem IsLocalMax.comp_antitone (hf : IsLocalMax f a) {g : β → γ} (hg : Antitone g) : IsLocalMin (g ∘ f) a := hf.comp_antitone hg nonrec theorem IsLocalExtr.comp_antitone (hf : IsLocalExtr f a) {g : β → γ} (hg : Antitone g) : IsLocalExtr (g ∘ f) a := hf.comp_antitone hg nonrec theorem IsLocalMinOn.comp_mono (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Monotone g) : IsLocalMinOn (g ∘ f) s a := hf.comp_mono hg nonrec theorem IsLocalMaxOn.comp_mono (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Monotone g) : IsLocalMaxOn (g ∘ f) s a := hf.comp_mono hg nonrec theorem IsLocalExtrOn.comp_mono (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Monotone g) : IsLocalExtrOn (g ∘ f) s a := hf.comp_mono hg nonrec theorem IsLocalMinOn.comp_antitone (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Antitone g) : IsLocalMaxOn (g ∘ f) s a := hf.comp_antitone hg nonrec theorem IsLocalMaxOn.comp_antitone (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Antitone g) : IsLocalMinOn (g ∘ f) s a := hf.comp_antitone hg nonrec theorem IsLocalExtrOn.comp_antitone (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Antitone g) : IsLocalExtrOn (g ∘ f) s a := hf.comp_antitone hg open scoped Relator nonrec theorem IsLocalMin.bicomp_mono [Preorder δ] {op : β → γ → δ} (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMin f a) {g : α → γ} (hg : IsLocalMin g a) : IsLocalMin (fun x => op (f x) (g x)) a := hf.bicomp_mono hop hg nonrec theorem IsLocalMax.bicomp_mono [Preorder δ] {op : β → γ → δ} (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMax f a) {g : α → γ} (hg : IsLocalMax g a) : IsLocalMax (fun x => op (f x) (g x)) a := hf.bicomp_mono hop hg nonrec theorem IsLocalMinOn.bicomp_mono [Preorder δ] {op : β → γ → δ} (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMinOn f s a) {g : α → γ} (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => op (f x) (g x)) s a := hf.bicomp_mono hop hg nonrec theorem IsLocalMaxOn.bicomp_mono [Preorder δ] {op : β → γ → δ} (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMaxOn f s a) {g : α → γ} (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => op (f x) (g x)) s a := hf.bicomp_mono hop hg /-! ### Composition with `ContinuousAt` -/ theorem IsLocalMin.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalMin f (g b)) (hg : ContinuousAt g b) : IsLocalMin (f ∘ g) b := hg hf theorem IsLocalMax.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalMax f (g b)) (hg : ContinuousAt g b) : IsLocalMax (f ∘ g) b := hg hf theorem IsLocalExtr.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalExtr f (g b)) (hg : ContinuousAt g b) : IsLocalExtr (f ∘ g) b := hf.comp_tendsto hg theorem IsLocalMin.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ} (hf : IsLocalMin f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMinOn (f ∘ g) s b := hf.comp_tendsto (hg b hb) theorem IsLocalMax.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ} (hf : IsLocalMax f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMaxOn (f ∘ g) s b := hf.comp_tendsto (hg b hb) theorem IsLocalExtr.comp_continuousOn [TopologicalSpace δ] {s : Set δ} (g : δ → α) {b : δ} (hf : IsLocalExtr f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalExtrOn (f ∘ g) s b := hf.elim (fun hf => (hf.comp_continuousOn hg hb).isExtr) fun hf => (IsLocalMax.comp_continuousOn hf hg hb).isExtr theorem IsLocalMinOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} {g : δ → α} {b : δ} (hf : IsLocalMinOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMinOn (f ∘ g) s b := hf.comp_tendsto (tendsto_nhdsWithin_mono_right (image_subset_iff.mpr hst) (ContinuousWithinAt.tendsto_nhdsWithin_image (hg b hb))) theorem IsLocalMaxOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} {g : δ → α} {b : δ} (hf : IsLocalMaxOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMaxOn (f ∘ g) s b := hf.comp_tendsto (tendsto_nhdsWithin_mono_right (image_subset_iff.mpr hst) (ContinuousWithinAt.tendsto_nhdsWithin_image (hg b hb))) theorem IsLocalExtrOn.comp_continuousOn [TopologicalSpace δ] {t : Set α} {s : Set δ} (g : δ → α) {b : δ} (hf : IsLocalExtrOn f t (g b)) (hst : s ⊆ g ⁻¹' t) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalExtrOn (f ∘ g) s b := hf.elim (fun hf => (hf.comp_continuousOn hst hg hb).isExtr) fun hf => (IsLocalMaxOn.comp_continuousOn hf hst hg hb).isExtr end Preorder /-! ### Pointwise addition -/ section OrderedAddCommMonoid variable [AddCommMonoid β] [PartialOrder β] [IsOrderedAddMonoid β] {f g : α → β} {a : α} {s : Set α} {l : Filter α} nonrec theorem IsLocalMin.add (hf : IsLocalMin f a) (hg : IsLocalMin g a) : IsLocalMin (fun x => f x + g x) a := hf.add hg nonrec theorem IsLocalMax.add (hf : IsLocalMax f a) (hg : IsLocalMax g a) : IsLocalMax (fun x => f x + g x) a := hf.add hg nonrec theorem IsLocalMinOn.add (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => f x + g x) s a := hf.add hg nonrec theorem IsLocalMaxOn.add (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => f x + g x) s a := hf.add hg end OrderedAddCommMonoid /-! ### Pointwise negation and subtraction -/ section OrderedAddCommGroup variable [AddCommGroup β] [PartialOrder β] [IsOrderedAddMonoid β] {f g : α → β} {a : α} {s : Set α} {l : Filter α} nonrec theorem IsLocalMin.neg (hf : IsLocalMin f a) : IsLocalMax (fun x => -f x) a := hf.neg nonrec theorem IsLocalMax.neg (hf : IsLocalMax f a) : IsLocalMin (fun x => -f x) a := hf.neg nonrec theorem IsLocalExtr.neg (hf : IsLocalExtr f a) : IsLocalExtr (fun x => -f x) a := hf.neg nonrec theorem IsLocalMinOn.neg (hf : IsLocalMinOn f s a) : IsLocalMaxOn (fun x => -f x) s a := hf.neg nonrec theorem IsLocalMaxOn.neg (hf : IsLocalMaxOn f s a) : IsLocalMinOn (fun x => -f x) s a := hf.neg nonrec theorem IsLocalExtrOn.neg (hf : IsLocalExtrOn f s a) : IsLocalExtrOn (fun x => -f x) s a := hf.neg nonrec theorem IsLocalMin.sub (hf : IsLocalMin f a) (hg : IsLocalMax g a) : IsLocalMin (fun x => f x - g x) a := hf.sub hg nonrec theorem IsLocalMax.sub (hf : IsLocalMax f a) (hg : IsLocalMin g a) : IsLocalMax (fun x => f x - g x) a := hf.sub hg nonrec theorem IsLocalMinOn.sub (hf : IsLocalMinOn f s a) (hg : IsLocalMaxOn g s a) : IsLocalMinOn (fun x => f x - g x) s a := hf.sub hg nonrec theorem IsLocalMaxOn.sub (hf : IsLocalMaxOn f s a) (hg : IsLocalMinOn g s a) : IsLocalMaxOn (fun x => f x - g x) s a := hf.sub hg end OrderedAddCommGroup /-! ### Pointwise `sup`/`inf` -/ section SemilatticeSup variable [SemilatticeSup β] {f g : α → β} {a : α} {s : Set α} {l : Filter α} nonrec theorem IsLocalMin.sup (hf : IsLocalMin f a) (hg : IsLocalMin g a) : IsLocalMin (fun x => f x ⊔ g x) a := hf.sup hg nonrec theorem IsLocalMax.sup (hf : IsLocalMax f a) (hg : IsLocalMax g a) : IsLocalMax (fun x => f x ⊔ g x) a := hf.sup hg nonrec theorem IsLocalMinOn.sup (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => f x ⊔ g x) s a := hf.sup hg nonrec theorem IsLocalMaxOn.sup (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => f x ⊔ g x) s a := hf.sup hg end SemilatticeSup section SemilatticeInf variable [SemilatticeInf β] {f g : α → β} {a : α} {s : Set α} {l : Filter α} nonrec theorem IsLocalMin.inf (hf : IsLocalMin f a) (hg : IsLocalMin g a) : IsLocalMin (fun x => f x ⊓ g x) a := hf.inf hg nonrec theorem IsLocalMax.inf (hf : IsLocalMax f a) (hg : IsLocalMax g a) : IsLocalMax (fun x => f x ⊓ g x) a := hf.inf hg nonrec theorem IsLocalMinOn.inf (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => f x ⊓ g x) s a := hf.inf hg nonrec theorem IsLocalMaxOn.inf (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => f x ⊓ g x) s a := hf.inf hg end SemilatticeInf /-! ### Pointwise `min`/`max` -/ section LinearOrder variable [LinearOrder β] {f g : α → β} {a : α} {s : Set α} {l : Filter α} nonrec theorem IsLocalMin.min (hf : IsLocalMin f a) (hg : IsLocalMin g a) : IsLocalMin (fun x => min (f x) (g x)) a := hf.min hg nonrec theorem IsLocalMax.min (hf : IsLocalMax f a) (hg : IsLocalMax g a) : IsLocalMax (fun x => min (f x) (g x)) a := hf.min hg nonrec theorem IsLocalMinOn.min (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => min (f x) (g x)) s a := hf.min hg nonrec theorem IsLocalMaxOn.min (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => min (f x) (g x)) s a := hf.min hg nonrec theorem IsLocalMin.max (hf : IsLocalMin f a) (hg : IsLocalMin g a) : IsLocalMin (fun x => max (f x) (g x)) a := hf.max hg nonrec theorem IsLocalMax.max (hf : IsLocalMax f a) (hg : IsLocalMax g a) : IsLocalMax (fun x => max (f x) (g x)) a := hf.max hg nonrec theorem IsLocalMinOn.max (hf : IsLocalMinOn f s a) (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => max (f x) (g x)) s a := hf.max hg nonrec theorem IsLocalMaxOn.max (hf : IsLocalMaxOn f s a) (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => max (f x) (g x)) s a := hf.max hg end LinearOrder section Eventually /-! ### Relation with `eventually` comparisons of two functions -/ variable [Preorder β] {s : Set α} theorem Filter.EventuallyLE.isLocalMaxOn {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝[s] a] f) (hfga : f a = g a) (h : IsLocalMaxOn f s a) : IsLocalMaxOn g s a := hle.isMaxFilter hfga h nonrec theorem IsLocalMaxOn.congr {f g : α → β} {a : α} (h : IsLocalMaxOn f s a) (heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMaxOn g s a := h.congr heq <\| heq.eq_of_nhdsWithin hmem theorem Filter.EventuallyEq.isLocalMaxOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMaxOn f s a ↔ IsLocalMaxOn g s a := heq.isMaxFilter_iff <\| heq.eq_of_nhdsWithin hmem theorem Filter.EventuallyLE.isLocalMinOn {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝[s] a] g) (hfga : f a = g a) (h : IsLocalMinOn f s a) : IsLocalMinOn g s a := hle.isMinFilter hfga h nonrec theorem IsLocalMinOn.congr {f g : α → β} {a : α} (h : IsLocalMinOn f s a) (heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMinOn g s a := h.congr heq <\| heq.eq_of_nhdsWithin hmem nonrec theorem Filter.EventuallyEq.isLocalMinOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalMinOn f s a ↔ IsLocalMinOn g s a := heq.isMinFilter_iff <\| heq.eq_of_nhdsWithin hmem nonrec theorem IsLocalExtrOn.congr {f g : α → β} {a : α} (h : IsLocalExtrOn f s a) (heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalExtrOn g s a := h.congr heq <\| heq.eq_of_nhdsWithin hmem theorem Filter.EventuallyEq.isLocalExtrOn_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : IsLocalExtrOn f s a ↔ IsLocalExtrOn g s a := heq.isExtrFilter_iff <\| heq.eq_of_nhdsWithin hmem theorem Filter.EventuallyLE.isLocalMax {f g : α → β} {a : α} (hle : g ≤ᶠ[𝓝 a] f) (hfga : f a = g a) (h : IsLocalMax f a) : IsLocalMax g a := hle.isMaxFilter hfga h nonrec theorem IsLocalMax.congr {f g : α → β} {a : α} (h : IsLocalMax f a) (heq : f =ᶠ[𝓝 a] g) : IsLocalMax g a := h.congr heq heq.eq_of_nhds theorem Filter.EventuallyEq.isLocalMax_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) : IsLocalMax f a ↔ IsLocalMax g a := heq.isMaxFilter_iff heq.eq_of_nhds theorem Filter.EventuallyLE.isLocalMin {f g : α → β} {a : α} (hle : f ≤ᶠ[𝓝 a] g) (hfga : f a = g a) (h : IsLocalMin f a) : IsLocalMin g a := hle.isMinFilter hfga h nonrec theorem IsLocalMin.congr {f g : α → β} {a : α} (h : IsLocalMin f a) (heq : f =ᶠ[𝓝 a] g) : IsLocalMin g a := h.congr heq heq.eq_of_nhds theorem Filter.EventuallyEq.isLocalMin_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) : IsLocalMin f a ↔ IsLocalMin g a := heq.isMinFilter_iff heq.eq_of_nhds nonrec theorem IsLocalExtr.congr {f g : α → β} {a : α} (h : IsLocalExtr f a) (heq : f =ᶠ[𝓝 a] g) : IsLocalExtr g a := h.congr heq heq.eq_of_nhds theorem Filter.EventuallyEq.isLocalExtr_iff {f g : α → β} {a : α} (heq : f =ᶠ[𝓝 a] g) : IsLocalExtr f a ↔ IsLocalExtr g a := heq.isExtrFilter_iff heq.eq_of_nhds end Eventually /-- If `f` is monotone to the left and antitone to the right, then it has a local maximum. -/ lemma isLocalMax_of_mono_anti' {α : Type} [TopologicalSpace α] [LinearOrder α] {β : Type} [Preorder β] {b : α} {f : α → β} {a : Set α} (ha : a ∈ 𝓝[≤] b) {c : Set α} (hc : c ∈ 𝓝[≥] b) (h₀ : MonotoneOn f a) (h₁ : AntitoneOn f c) : IsLocalMax f b := have : b ∈ a := mem_of_mem_nhdsWithin (by simp) ha have : b ∈ c := mem_of_mem_nhdsWithin (by simp) hc mem_of_superset (nhds_of_Ici_Iic ha hc) (fun x _ => by rcases le_total x b <;> aesop) /-- If `f` is antitone to the left and monotone to the right, then it has a local minimum. -/ lemma isLocalMin_of_anti_mono' {α : Type} [TopologicalSpace α] [LinearOrder α] {β : Type} [Preorder β] {b : α} {f : α → β} {a : Set α} (ha : a ∈ 𝓝[≤] b) {c : Set α} (hc : c ∈ 𝓝[≥] b) (h₀ : AntitoneOn f a) (h₁ : MonotoneOn f c) : IsLocalMin f b := have : b ∈ a := mem_of_mem_nhdsWithin (by simp) ha have : b ∈ c := mem_of_mem_nhdsWithin (by simp) hc mem_of_superset (nhds_of_Ici_Iic ha hc) (fun x _ => by rcases le_total x b <;> aesop)
.lake/packages/mathlib/Mathlib/Topology/Order/ExtendFrom.lean	import Mathlib.Topology.ExtendFrom import Mathlib.Topology.Order.DenselyOrdered /-! # Lemmas about `extendFrom` in an order topology. -/ open Filter Set Topology variable {α β : Type*} theorem continuousOn_Icc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la lb : β} (hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : ContinuousOn (extendFrom (Ioo a b) f) (Icc a b) := by apply continuousOn_extendFrom · rw [closure_Ioo hab] · intro x x_in rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl \| rfl \| h) · exact ⟨la, ha.mono_left <\| nhdsWithin_mono _ Ioo_subset_Ioi_self⟩ · exact ⟨lb, hb.mono_left <\| nhdsWithin_mono _ Ioo_subset_Iio_self⟩ · exact ⟨f x, hf x h⟩ theorem eq_lim_at_left_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {la : β} (hab : a < b) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) : extendFrom (Ioo a b) f a = la := by apply extendFrom_eq · rw [closure_Ioo hab.ne] simp only [le_of_lt hab, left_mem_Icc] · simpa [hab] theorem eq_lim_at_right_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : extendFrom (Ioo a b) f b = lb := by apply extendFrom_eq · rw [closure_Ioo hab.ne] simp only [le_of_lt hab, right_mem_Icc] · simpa [hab] theorem continuousOn_Ico_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {la : β} (hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la)) : ContinuousOn (extendFrom (Ioo a b) f) (Ico a b) := by apply continuousOn_extendFrom · rw [closure_Ioo hab.ne] exact Ico_subset_Icc_self · intro x x_in rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl \| h) · use la simpa [hab] · exact ⟨f x, hf x h⟩ theorem continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α] [OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β} (hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) := by have := @continuousOn_Ico_extendFrom_Ioo αᵒᵈ _ _ _ _ _ _ _ f _ _ lb hab erw [Ico_toDual, Ioi_toDual, Ioo_toDual] at this exact this hf hb
.lake/packages/mathlib/Mathlib/Topology/Order/Real.lean	import Mathlib.Data.EReal.Basic import Mathlib.Topology.Order.T5 /-! # The reals are equipped with their order topology This file contains results related to the order topology on (extended) (non-negative) real numbers. We - prove that `ℝ` and `ℝ≥0` are equipped with the order topology and bornology, - endow `EReal` with the order topology (and prove some very basic lemmas), - define the topology `ℝ≥0∞` (which is the order topology, not the `EMetricSpace` topology) -/ assert_not_exists IsTopologicalRing UniformSpace open Set namespace EReal /-! ### Topological structure on `EReal` We endow `EReal` with the order topology. Most proofs are adapted from the corresponding proofs on `ℝ≥0∞`. -/ instance : TopologicalSpace EReal := Preorder.topology EReal instance : OrderTopology EReal := ⟨rfl⟩ instance : T5Space EReal := inferInstance instance : T2Space EReal := inferInstance lemma denseRange_ratCast : DenseRange (fun r : ℚ ↦ ((r : ℝ) : EReal)) := dense_of_exists_between fun _ _ h => exists_range_iff.2 <\| exists_rat_btwn_of_lt h end EReal namespace ENNReal /-- Topology on `ℝ≥0∞`. Note: this is different from the `EMetricSpace` topology. The `EMetricSpace` topology has `IsOpen {∞}`, while this topology doesn't have singleton elements. -/ instance : TopologicalSpace ℝ≥0∞ := Preorder.topology ℝ≥0∞ instance : OrderTopology ℝ≥0∞ := ⟨rfl⟩ -- short-circuit type class inference instance : T2Space ℝ≥0∞ := inferInstance instance : T5Space ℝ≥0∞ := inferInstance instance : T4Space ℝ≥0∞ := inferInstance end ENNReal
.lake/packages/mathlib/Mathlib/Topology/Order/Hom/Esakia.lean	import Mathlib.Order.Hom.Bounded import Mathlib.Topology.Order.Hom.Basic /-! # Esakia morphisms This file defines pseudo-epimorphisms and Esakia morphisms. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types. ## Types of morphisms * `PseudoEpimorphism`: Pseudo-epimorphisms. Maps `f` such that `f a ≤ b` implies the existence of `a'` such that `a ≤ a'` and `f a' = b`. * `EsakiaHom`: Esakia morphisms. Continuous pseudo-epimorphisms. ## Typeclasses * `PseudoEpimorphismClass` * `EsakiaHomClass` ## References * [Wikipedia, Esakia space](https://en.wikipedia.org/wiki/Esakia_space) -/ open Function variable {F α β γ δ : Type} /-- The type of pseudo-epimorphisms, aka p-morphisms, aka bounded maps, from `α` to `β`. -/ structure PseudoEpimorphism (α β : Type) [Preorder α] [Preorder β] extends α →o β where exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b /-- The type of Esakia morphisms, aka continuous pseudo-epimorphisms, from `α` to `β`. -/ structure EsakiaHom (α β : Type) [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] extends α →Co β where exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b section /-- `PseudoEpimorphismClass F α β` states that `F` is a type of `⊔`-preserving morphisms. You should extend this class when you extend `PseudoEpimorphism`. -/ class PseudoEpimorphismClass (F : Type) (α β : outParam Type) [Preorder α] [Preorder β] [FunLike F α β] : Prop extends RelHomClass F ((· ≤ ·) : α → α → Prop) ((· ≤ ·) : β → β → Prop) where exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b /-- `EsakiaHomClass F α β` states that `F` is a type of lattice morphisms. You should extend this class when you extend `EsakiaHom`. -/ class EsakiaHomClass (F : Type) (α β : outParam Type*) [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [FunLike F α β] : Prop extends ContinuousOrderHomClass F α β where exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b end export PseudoEpimorphismClass (exists_map_eq_of_map_le) section Hom variable [FunLike F α β] -- See note [lower instance priority] instance (priority := 100) PseudoEpimorphismClass.toTopHomClass [PartialOrder α] [OrderTop α] [Preorder β] [OrderTop β] [PseudoEpimorphismClass F α β] : TopHomClass F α β where map_top f := by let ⟨b, h⟩ := exists_map_eq_of_map_le f (@le_top _ _ _ <\| f ⊤) rw [← top_le_iff.1 h.1, h.2] -- See note [lower instance priority] instance (priority := 100) EsakiaHomClass.toPseudoEpimorphismClass [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [EsakiaHomClass F α β] : PseudoEpimorphismClass F α β := { ‹EsakiaHomClass F α β› with map_rel := ContinuousOrderHomClass.map_monotone } instance [Preorder α] [Preorder β] [PseudoEpimorphismClass F α β] : CoeTC F (PseudoEpimorphism α β) := ⟨fun f => ⟨f, exists_map_eq_of_map_le f⟩⟩ instance [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [EsakiaHomClass F α β] : CoeTC F (EsakiaHom α β) := ⟨fun f => ⟨f, exists_map_eq_of_map_le f⟩⟩ end Hom -- See note [lower instance priority] instance (priority := 100) OrderIsoClass.toPseudoEpimorphismClass [Preorder α] [Preorder β] [EquivLike F α β] [OrderIsoClass F α β] : PseudoEpimorphismClass F α β where exists_map_eq_of_map_le f _a b h := ⟨EquivLike.inv f b, (le_map_inv_iff f).2 h, EquivLike.right_inv _ _⟩ /-! ### Pseudo-epimorphisms -/ namespace PseudoEpimorphism variable [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] instance instFunLike : FunLike (PseudoEpimorphism α β) α β where coe f := f.toFun coe_injective' f g h := by obtain ⟨⟨_, _⟩, _⟩ := f obtain ⟨⟨_, _⟩, _⟩ := g congr instance : PseudoEpimorphismClass (PseudoEpimorphism α β) α β where map_rel f _ _ h := f.monotone' h exists_map_eq_of_map_le := PseudoEpimorphism.exists_map_eq_of_map_le' @[simp] theorem toOrderHom_eq_coe (f : PseudoEpimorphism α β) : ⇑f.toOrderHom = f := rfl theorem toFun_eq_coe {f : PseudoEpimorphism α β} : f.toFun = (f : α → β) := rfl @[ext] theorem ext {f g : PseudoEpimorphism α β} (h : ∀ a, f a = g a) : f = g := DFunLike.ext f g h /-- Copy of a `PseudoEpimorphism` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : PseudoEpimorphism α β := ⟨f.toOrderHom.copy f' h, by simpa only [h.symm, toFun_eq_coe] using f.exists_map_eq_of_map_le'⟩ @[simp] theorem coe_copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl theorem copy_eq (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : f.copy f' h = f := DFunLike.ext' h variable (α) /-- `id` as a `PseudoEpimorphism`. -/ protected def id : PseudoEpimorphism α α := ⟨OrderHom.id, fun _ b h => ⟨b, h, rfl⟩⟩ instance : Inhabited (PseudoEpimorphism α α) := ⟨PseudoEpimorphism.id α⟩ @[simp, norm_cast] theorem coe_id : ⇑(PseudoEpimorphism.id α) = id := rfl @[simp, norm_cast] theorem coe_id_orderHom : (PseudoEpimorphism.id α : α →o α) = OrderHom.id := rfl variable {α} @[simp] theorem id_apply (a : α) : PseudoEpimorphism.id α a = a := rfl /-- Composition of `PseudoEpimorphism`s as a `PseudoEpimorphism`. -/ def comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : PseudoEpimorphism α γ := ⟨g.toOrderHom.comp f.toOrderHom, fun a b h₀ => by obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀ obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁ exact ⟨b, h₂, rfl⟩⟩ @[simp] theorem coe_comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : (g.comp f : α → γ) = g ∘ f := rfl @[simp] theorem coe_comp_orderHom (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : (g.comp f : α →o γ) = (g : β →o γ).comp f := rfl @[simp] theorem comp_apply (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) (a : α) : (g.comp f) a = g (f a) := rfl @[simp] theorem comp_assoc (h : PseudoEpimorphism γ δ) (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : (h.comp g).comp f = h.comp (g.comp f) := rfl @[simp] theorem comp_id (f : PseudoEpimorphism α β) : f.comp (PseudoEpimorphism.id α) = f := ext fun _ => rfl @[simp] theorem id_comp (f : PseudoEpimorphism α β) : (PseudoEpimorphism.id β).comp f = f := ext fun _ => rfl @[simp] theorem cancel_right {g₁ g₂ : PseudoEpimorphism β γ} {f : PseudoEpimorphism α β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <\| hf.forall.2 <\| DFunLike.ext_iff.1 h, congr_arg (comp · f)⟩ @[simp] theorem cancel_left {g : PseudoEpimorphism β γ} {f₁ f₂ : PseudoEpimorphism α β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun a => hg <\| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ end PseudoEpimorphism /-! ### Esakia morphisms -/ namespace EsakiaHom variable [TopologicalSpace α] [Preorder α] [TopologicalSpace β] [Preorder β] [TopologicalSpace γ] [Preorder γ] [TopologicalSpace δ] [Preorder δ] def toPseudoEpimorphism (f : EsakiaHom α β) : PseudoEpimorphism α β := { f with } instance instFunLike : FunLike (EsakiaHom α β) α β where coe f := f.toFun coe_injective' f g h := by obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g congr instance : EsakiaHomClass (EsakiaHom α β) α β where map_monotone f := f.monotone' map_continuous f := f.continuous_toFun exists_map_eq_of_map_le f := f.exists_map_eq_of_map_le' @[simp] theorem toContinuousOrderHom_coe {f : EsakiaHom α β} : f.toContinuousOrderHom = (f : α → β) := rfl theorem toFun_eq_coe {f : EsakiaHom α β} : f.toFun = (f : α → β) := rfl @[ext] theorem ext {f g : EsakiaHom α β} (h : ∀ a, f a = g a) : f = g := DFunLike.ext f g h /-- Copy of an `EsakiaHom` with a new `toFun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : EsakiaHom α β := ⟨f.toContinuousOrderHom.copy f' h, by simpa only [h.symm, toFun_eq_coe] using f.exists_map_eq_of_map_le'⟩ @[simp] theorem coe_copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl theorem copy_eq (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f := DFunLike.ext' h variable (α) /-- `id` as an `EsakiaHom`. -/ protected def id : EsakiaHom α α := ⟨ContinuousOrderHom.id α, fun _ b h => ⟨b, h, rfl⟩⟩ instance : Inhabited (EsakiaHom α α) := ⟨EsakiaHom.id α⟩ @[simp, norm_cast] theorem coe_id : ⇑(EsakiaHom.id α) = id := rfl @[simp, norm_cast] theorem coe_id_pseudoEpimorphism : (EsakiaHom.id α : PseudoEpimorphism α α) = PseudoEpimorphism.id α := rfl variable {α} @[simp] theorem id_apply (a : α) : EsakiaHom.id α a = a := rfl @[simp, norm_cast] theorem coe_id_continuousOrderHom : (EsakiaHom.id α : α →Co α) = ContinuousOrderHom.id α := rfl /-- Composition of `EsakiaHom`s as an `EsakiaHom`. -/ def comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : EsakiaHom α γ := ⟨g.toContinuousOrderHom.comp f.toContinuousOrderHom, fun a b h₀ => by obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀ obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁ exact ⟨b, h₂, rfl⟩⟩ @[simp] theorem coe_comp_continuousOrderHom (g : EsakiaHom β γ) (f : EsakiaHom α β) : (g.comp f : α →Co γ) = (g : β →Co γ).comp f := rfl @[simp] theorem coe_comp_pseudoEpimorphism (g : EsakiaHom β γ) (f : EsakiaHom α β) : (g.comp f : PseudoEpimorphism α γ) = (g : PseudoEpimorphism β γ).comp f := rfl @[simp] theorem coe_comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : (g.comp f : α → γ) = g ∘ f := rfl @[simp] theorem comp_apply (g : EsakiaHom β γ) (f : EsakiaHom α β) (a : α) : (g.comp f) a = g (f a) := rfl @[simp] theorem comp_assoc (h : EsakiaHom γ δ) (g : EsakiaHom β γ) (f : EsakiaHom α β) : (h.comp g).comp f = h.comp (g.comp f) := rfl @[simp] theorem comp_id (f : EsakiaHom α β) : f.comp (EsakiaHom.id α) = f := ext fun _ => rfl @[simp] theorem id_comp (f : EsakiaHom α β) : (EsakiaHom.id β).comp f = f := ext fun _ => rfl @[simp] theorem cancel_right {g₁ g₂ : EsakiaHom β γ} {f : EsakiaHom α β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <\| hf.forall.2 <\| DFunLike.ext_iff.1 h, congr_arg (comp · f)⟩ @[simp] theorem cancel_left {g : EsakiaHom β γ} {f₁ f₂ : EsakiaHom α β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun a => hg <\| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ end EsakiaHom
.lake/packages/mathlib/Mathlib/Topology/Order/Hom/Basic.lean	import Mathlib.Topology.Continuous import Mathlib.Topology.ContinuousMap.Defs /-! # Continuous order homomorphisms This file defines continuous order homomorphisms, that is maps which are both continuous and monotone. They are also called Priestley homomorphisms because they are the morphisms of the category of Priestley spaces. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types. ## Types of morphisms * `ContinuousOrderHom`: Continuous monotone functions, aka Priestley homomorphisms. ## Typeclasses * `ContinuousOrderHomClass` -/ open Function variable {F α β γ δ : Type} /-- The type of continuous monotone maps from `α` to `β`, aka Priestley homomorphisms. -/ structure ContinuousOrderHom (α β : Type) [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] extends OrderHom α β where continuous_toFun : Continuous toFun @[inherit_doc] infixr:25 " →Co " => ContinuousOrderHom section /-- `ContinuousOrderHomClass F α β` states that `F` is a type of continuous monotone maps. You should extend this class when you extend `ContinuousOrderHom`. -/ class ContinuousOrderHomClass (F : Type) (α β : outParam Type) [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where map_monotone (f : F) : Monotone f namespace ContinuousOrderHomClass variable [Preorder α] [Preorder β] [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousOrderHomClass F α β] -- See note [lower instance priority] instance (priority := 100) toOrderHomClass : OrderHomClass F α β := { ‹ContinuousOrderHomClass F α β› with map_rel := ContinuousOrderHomClass.map_monotone } /-- Turn an element of a type `F` satisfying `ContinuousOrderHomClass F α β` into an actual `ContinuousOrderHom`. This is declared as the default coercion from `F` to `α →Co β`. -/ @[coe] def toContinuousOrderHom (f : F) : α →Co β := { toFun := f monotone' := ContinuousOrderHomClass.map_monotone f continuous_toFun := map_continuous f } instance : CoeTC F (α →Co β) := ⟨toContinuousOrderHom⟩ end ContinuousOrderHomClass /-! ### Top homomorphisms -/ namespace ContinuousOrderHom variable [TopologicalSpace α] [Preorder α] [TopologicalSpace β] section Preorder variable [Preorder β] [TopologicalSpace γ] [Preorder γ] [TopologicalSpace δ] [Preorder δ] /-- Reinterpret a `ContinuousOrderHom` as a `ContinuousMap`. -/ def toContinuousMap (f : α →Co β) : C(α, β) := { f with } instance instFunLike : FunLike (α →Co β) α β where coe f := f.toFun coe_injective' f g h := by obtain ⟨⟨_, _⟩, _⟩ := f obtain ⟨⟨_, _⟩, _⟩ := g congr instance : ContinuousOrderHomClass (α →Co β) α β where map_monotone f := f.monotone' map_continuous f := f.continuous_toFun @[simp] theorem coe_toOrderHom (f : α →Co β) : ⇑f.toOrderHom = f := rfl theorem toFun_eq_coe {f : α →Co β} : f.toFun = (f : α → β) := rfl @[ext] theorem ext {f g : α →Co β} (h : ∀ a, f a = g a) : f = g := DFunLike.ext f g h /-- Copy of a `ContinuousOrderHom` with a new `ContinuousMap` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →Co β) (f' : α → β) (h : f' = f) : α →Co β := ⟨f.toOrderHom.copy f' h, h.symm.subst f.continuous_toFun⟩ @[simp] theorem coe_copy (f : α →Co β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl theorem copy_eq (f : α →Co β) (f' : α → β) (h : f' = f) : f.copy f' h = f := DFunLike.ext' h variable (α) /-- `id` as a `ContinuousOrderHom`. -/ protected def id : α →Co α := ⟨OrderHom.id, continuous_id⟩ instance : Inhabited (α →Co α) := ⟨ContinuousOrderHom.id _⟩ @[simp, norm_cast] theorem coe_id : ⇑(ContinuousOrderHom.id α) = id := rfl variable {α} @[simp] theorem id_apply (a : α) : ContinuousOrderHom.id α a = a := rfl /-- Composition of `ContinuousOrderHom`s as a `ContinuousOrderHom`. -/ def comp (f : β →Co γ) (g : α →Co β) : ContinuousOrderHom α γ := ⟨f.toOrderHom.comp g.toOrderHom, f.continuous_toFun.comp g.continuous_toFun⟩ @[simp] theorem coe_comp (f : β →Co γ) (g : α →Co β) : (f.comp g : α → γ) = f ∘ g := rfl @[simp] theorem comp_apply (f : β →Co γ) (g : α →Co β) (a : α) : (f.comp g) a = f (g a) := rfl @[simp] theorem comp_assoc (f : γ →Co δ) (g : β →Co γ) (h : α →Co β) : (f.comp g).comp h = f.comp (g.comp h) := rfl @[simp] theorem comp_id (f : α →Co β) : f.comp (ContinuousOrderHom.id α) = f := ext fun _ => rfl @[simp] theorem id_comp (f : α →Co β) : (ContinuousOrderHom.id β).comp f = f := ext fun _ => rfl @[simp] theorem cancel_right {g₁ g₂ : β →Co γ} {f : α →Co β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <\| hf.forall.2 <\| DFunLike.ext_iff.1 h, fun h => congr_arg₂ _ h rfl⟩ @[simp] theorem cancel_left {g : β →Co γ} {f₁ f₂ : α →Co β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun a => hg <\| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ instance : Preorder (α →Co β) := Preorder.lift ((↑) : (α →Co β) → α → β) end Preorder instance [PartialOrder β] : PartialOrder (α →Co β) := PartialOrder.lift ((↑) : (α →Co β) → α → β) DFunLike.coe_injective end ContinuousOrderHom end
.lake/packages/mathlib/Mathlib/Topology/Order/Category/FrameAdjunction.lean	import Mathlib.Topology.Category.Locale /-! # Adjunction between Locales and Topological Spaces This file defines the point functor from the category of locales to topological spaces and proves that it is right adjoint to the forgetful functor from topological spaces to locales. ## Main declarations * `Locale.pt`: the points functor from the category of locales to the category of topological spaces. * `Locale.adjunctionTopToLocalePT`: the adjunction between the functors `topToLocale` and `pt`. ## Motivation This adjunction provides a framework in which several Stone-type dualities fit. ## Implementation notes * In naming the various functions below, we follow common terminology and reserve the word point for an inhabitant of a type `X` which is a topological space, while we use the word element for an inhabitant of a type `L` which is a locale. ## References * [J. Picado and A. Pultr, Frames and Locales: topology without points][picado2011frames] ## Tags topological space, frame, locale, Stone duality, adjunction, points -/ open CategoryTheory Order Set Topology TopologicalSpace namespace Locale /-! ### Definition of the points functor `pt` -/ section pt_definition variable (L : Type) [CompleteLattice L] /-- The type of points of a complete lattice `L`, where a point* of a complete lattice is, by definition, a frame homomorphism from `L` to `Prop`. -/ abbrev PT := FrameHom L Prop /-- The frame homomorphism from a complete lattice `L` to the complete lattice of sets of points of `L`. -/ @[simps] def openOfElementHom : FrameHom L (Set (PT L)) where toFun u := {x \| x u} map_inf' a b := by simp [Set.setOf_and] map_top' := by simp map_sSup' S := by ext; simp [Prop.exists_iff] namespace PT /-- The topology on the set of points of the complete lattice `L`. -/ instance instTopologicalSpace : TopologicalSpace (PT L) where IsOpen s := ∃ u, {x \| x u} = s isOpen_univ := ⟨⊤, by simp⟩ isOpen_inter := by rintro s t ⟨u, rfl⟩ ⟨v, rfl⟩; use u ⊓ v; simp_rw [map_inf]; rfl isOpen_sUnion S hS := by choose f hf using hS use ⨆ t, ⨆ ht, f t ht simp_rw [map_iSup, iSup_Prop_eq, setOf_exists, hf, sUnion_eq_biUnion] /-- Characterization of when a subset of the space of points is open. -/ lemma isOpen_iff (U : Set (PT L)) : IsOpen U ↔ ∃ u : L, {x \| x u} = U := Iff.rfl end PT /-- The covariant functor `pt` from the category of locales to the category of topological spaces, which sends a locale `L` to the topological space `PT L` of homomorphisms from `L` to `Prop` and a locale homomorphism `f` to a continuous function between the spaces of points. -/ def pt : Locale ⥤ TopCat where obj L := .of (PT L.unop) map f := TopCat.ofHom ⟨fun p ↦ p.comp f.unop.hom, continuous_def.2 <\| by rintro s ⟨u, rfl⟩; use f.unop u; rfl⟩ end pt_definition section locale_top_adjunction variable (X : Type*) [TopologicalSpace X] (L : Locale) /-- The unit of the adjunction between locales and topological spaces, which associates with a point `x` of the space `X` a point of the locale of opens of `X`. -/ @[simps] def localePointOfSpacePoint (x : X) : PT (Opens X) where toFun := (x ∈ ·) map_inf' _ _ := rfl map_top' := rfl map_sSup' S := by simp [Prop.exists_iff] /-- The counit is a frame homomorphism. -/ def counitAppCont : FrameHom L (Opens <\| PT L) where toFun u := ⟨openOfElementHom L u, u, rfl⟩ map_inf' a b := by simp map_top' := by simp map_sSup' S := by ext; simp /-- The forgetful functor `topToLocale` is left adjoint to the functor `pt`. -/ def adjunctionTopToLocalePT : topToLocale ⊣ pt where unit := { app := fun X ↦ TopCat.ofHom ⟨localePointOfSpacePoint X, continuous_def.2 <\| by rintro _ ⟨u, rfl⟩; simpa using u.2⟩ } counit := { app := fun L ↦ ⟨Frm.ofHom (counitAppCont L)⟩ } end locale_top_adjunction end Locale
.lake/packages/mathlib/Mathlib/Topology/Order/Category/AlexDisc.lean	import Mathlib.Topology.Specialization /-! # Category of Alexandrov-discrete topological spaces This defines `AlexDisc`, the category of Alexandrov-discrete topological spaces with continuous maps, and proves it's equivalent to the category of preorders. -/ open CategoryTheory Topology /-- The category of Alexandrov-discrete spaces. -/ structure AlexDisc extends TopCat where [is_alexandrovDiscrete : AlexandrovDiscrete carrier] namespace AlexDisc attribute [instance] is_alexandrovDiscrete instance : CoeSort AlexDisc (Type _) := ⟨fun X => X.toTopCat⟩ instance category : Category AlexDisc := InducedCategory.category toTopCat instance concreteCategory : ConcreteCategory AlexDisc (C(·, ·)) := InducedCategory.concreteCategory toTopCat instance instHasForgetToTop : HasForget₂ AlexDisc TopCat := InducedCategory.hasForget₂ toTopCat -- TODO: generalize to `InducedCategory.forget₂_full`? instance forgetToTop_full : (forget₂ AlexDisc TopCat).Full where map_surjective f := ⟨f, rfl⟩ instance forgetToTop_faithful : (forget₂ AlexDisc TopCat).Faithful where /-- Construct a bundled `AlexDisc` from the underlying topological space. -/ abbrev of (X : Type) [TopologicalSpace X] [AlexandrovDiscrete X] : AlexDisc where toTopCat := TopCat.of X lemma coe_of (α : Type) [TopologicalSpace α] [AlexandrovDiscrete α] : ↥(of α) = α := rfl @[simp] lemma forgetToTop_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] : (forget₂ AlexDisc TopCat).obj (of α) = TopCat.of α := rfl @[simp] lemma coe_forgetToTop (X : AlexDisc) : ↥((forget₂ _ TopCat).obj X) = X := rfl /-- Constructs an equivalence between preorders from an order isomorphism between them. -/ @[simps] def Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β where hom := TopCat.ofHom (e : ContinuousMap α β) inv := TopCat.ofHom (e.symm : ContinuousMap β α) hom_inv_id := by ext; apply e.symm_apply_apply inv_hom_id := by ext; apply e.apply_symm_apply end AlexDisc /-- Sends a topological space to its specialisation order. -/ @[simps] def alexDiscEquivPreord : AlexDisc ≌ Preord where functor := forget₂ _ _ ⋙ topToPreord inverse.obj X := AlexDisc.of (WithUpperSet X) inverse.map f := TopCat.ofHom (WithUpperSet.map f.hom) unitIso := NatIso.ofComponents fun X ↦ AlexDisc.Iso.mk <\| by dsimp; exact homeoWithUpperSetTopologyorderIso X counitIso := NatIso.ofComponents fun X ↦ Preord.Iso.mk <\| by dsimp; exact (orderIsoSpecializationWithUpperSetTopology X).symm
.lake/packages/mathlib/Mathlib/Topology/Hom/ContinuousEval.lean	import Mathlib.Topology.Hom.ContinuousEvalConst import Mathlib.Topology.ContinuousMap.Defs /-! # Bundled maps with evaluation continuous in both variables In this file we define a class `ContinuousEval F X Y` saying that `F` is a bundled morphism class (in the sense of `FunLike`) with a topology such that `fun (f, x) : F × X ↦ f x` is a continuous function. -/ open scoped Topology open Filter /-- A typeclass saying that `F` is a bundled morphism class (in the sense of `FunLike`) with a topology such that `fun (f, x) : F × X ↦ f x` is a continuous function. -/ class ContinuousEval (F : Type) (X Y : outParam Type) [FunLike F X Y] [TopologicalSpace F] [TopologicalSpace X] [TopologicalSpace Y] : Prop where /-- Evaluation of a bundled morphism at a point is continuous in both variables. -/ continuous_eval : Continuous fun fx : F × X ↦ fx.1 fx.2 export ContinuousEval (continuous_eval) variable {F X Y Z : Type} [FunLike F X Y] [TopologicalSpace F] [TopologicalSpace X] [TopologicalSpace Y] [ContinuousEval F X Y] [TopologicalSpace Z] {f : Z → F} {g : Z → X} {s : Set Z} {z : Z} @[continuity, fun_prop] protected theorem Continuous.eval (hf : Continuous f) (hg : Continuous g) : Continuous fun z ↦ f z (g z) := continuous_eval.comp (hf.prodMk hg) /-- If a type `F'` of bundled morphisms admits a continuous projection to a type satisfying `ContinuousEval`, then `F'` satisfies this predicate too. The word "forget" in the name is motivated by the term "forgetful functor". -/ theorem ContinuousEval.of_continuous_forget {F' : Type} [FunLike F' X Y] [TopologicalSpace F'] {f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g := by intro; rfl) : ContinuousEval F' X Y where continuous_eval := by simpa only [← hf] using hc.fst'.eval continuous_snd instance (priority := 100) ContinuousEval.toContinuousMapClass : ContinuousMapClass F X Y where map_continuous _ := continuous_const.eval continuous_id instance (priority := 100) ContinuousEval.toContinuousEvalConst : ContinuousEvalConst F X Y where continuous_eval_const _ := continuous_id.eval continuous_const protected theorem Filter.Tendsto.eval {α : Type*} {l : Filter α} {f : α → F} {f₀ : F} {g : α → X} {x₀ : X} (hf : Tendsto f l (𝓝 f₀)) (hg : Tendsto g l (𝓝 x₀)) : Tendsto (fun a ↦ f a (g a)) l (𝓝 (f₀ x₀)) := (ContinuousEval.continuous_eval.tendsto _).comp (hf.prodMk_nhds hg) protected nonrec theorem ContinuousAt.eval (hf : ContinuousAt f z) (hg : ContinuousAt g z) : ContinuousAt (fun z ↦ f z (g z)) z := hf.eval hg protected nonrec theorem ContinuousWithinAt.eval (hf : ContinuousWithinAt f s z) (hg : ContinuousWithinAt g s z) : ContinuousWithinAt (fun z ↦ f z (g z)) s z := hf.eval hg protected theorem ContinuousOn.eval (hf : ContinuousOn f s) (hg : ContinuousOn g s) : ContinuousOn (fun z ↦ f z (g z)) s := fun z hz ↦ (hf z hz).eval (hg z hz)
.lake/packages/mathlib/Mathlib/Topology/Hom/ContinuousEvalConst.lean	import Mathlib.Topology.Constructions /-! # Bundled morphisms with continuous evaluation at a point In this file we define a typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`) with a topology on `F` such that evaluation at a point is continuous in `f : F`. ## Implementation Notes For now, we define the typeclass for non-dependent bundled functions only. Whenever we add a type of bundled dependent functions with a topology having this property, we may decide to generalize from `FunLike` to `DFunLike`. -/ open scoped Topology open Filter /-- A typeclass saying that `F` is a type of bundled morphisms (in the sense of `DFunLike`) with a topology on `F` such that evaluation at a point is continuous in `f : F`. -/ class ContinuousEvalConst (F : Type) (α X : outParam Type) [FunLike F α X] [TopologicalSpace F] [TopologicalSpace X] : Prop where continuous_eval_const (x : α) : Continuous fun f : F ↦ f x export ContinuousEvalConst (continuous_eval_const) section ContinuousEvalConst variable {F α X Z : Type} [FunLike F α X] [TopologicalSpace F] [TopologicalSpace X] [ContinuousEvalConst F α X] [TopologicalSpace Z] {f : Z → F} {s : Set Z} {z : Z} /-- If a type `F'` of bundled morphisms admits a continuous projection to a type satisfying `ContinuousEvalConst`, then `F'` satisfies this predicate too. The word "forget" in the name is motivated by the term "forgetful functor". -/ theorem ContinuousEvalConst.of_continuous_forget {F' : Type} [FunLike F' α X] [TopologicalSpace F'] {f : F' → F} (hc : Continuous f) (hf : ∀ g, ⇑(f g) = g := by intro; rfl) : ContinuousEvalConst F' α X where continuous_eval_const x := by simpa only [← hf] using (continuous_eval_const x).comp hc @[continuity, fun_prop] protected theorem Continuous.eval_const (hf : Continuous f) (x : α) : Continuous (f · x) := (continuous_eval_const x).comp hf theorem continuous_coeFun : Continuous (DFunLike.coe : F → α → X) := continuous_pi continuous_eval_const protected theorem Continuous.coeFun (hf : Continuous f) : Continuous fun z ↦ ⇑(f z) := continuous_pi hf.eval_const protected theorem Filter.Tendsto.eval_const {ι : Type} {l : Filter ι} {f : ι → F} {g : F} (hf : Tendsto f l (𝓝 g)) (a : α) : Tendsto (f · a) l (𝓝 (g a)) := ((continuous_id.eval_const a).tendsto _).comp hf protected theorem Filter.Tendsto.coeFun {ι : Type} {l : Filter ι} {f : ι → F} {g : F} (hf : Tendsto f l (𝓝 g)) : Tendsto (fun i ↦ ⇑(f i)) l (𝓝 ⇑g) := (continuous_id.coeFun.tendsto _).comp hf protected nonrec theorem ContinuousAt.eval_const (hf : ContinuousAt f z) (x : α) : ContinuousAt (f · x) z := hf.eval_const x protected nonrec theorem ContinuousAt.coeFun (hf : ContinuousAt f z) : ContinuousAt (fun z ↦ ⇑(f z)) z := hf.coeFun protected nonrec theorem ContinuousWithinAt.eval_const (hf : ContinuousWithinAt f s z) (x : α) : ContinuousWithinAt (f · x) s z := hf.eval_const x protected nonrec theorem ContinuousWithinAt.coeFun (hf : ContinuousWithinAt f s z) : ContinuousWithinAt (fun z ↦ ⇑(f z)) s z := hf.coeFun protected theorem ContinuousOn.eval_const (hf : ContinuousOn f s) (x : α) : ContinuousOn (f · x) s := fun z hz ↦ (hf z hz).eval_const x protected theorem ContinuousOn.coeFun (hf : ContinuousOn f s) (x : α) : ContinuousOn (f · x) s := fun z hz ↦ (hf z hz).eval_const x end ContinuousEvalConst
.lake/packages/mathlib/Mathlib/Topology/Hom/Open.lean	import Mathlib.Topology.ContinuousMap.Basic /-! # Continuous open maps This file defines bundled continuous open maps. We use the `DFunLike` design, so each type of morphisms has a companion typeclass which is meant to be satisfied by itself and all stricter types. ## Types of morphisms * `ContinuousOpenMap`: Continuous open maps. ## Typeclasses * `ContinuousOpenMapClass` -/ open Function variable {F α β γ δ : Type} /-- The type of continuous open maps from `α` to `β`, aka Priestley homomorphisms. -/ structure ContinuousOpenMap (α β : Type) [TopologicalSpace α] [TopologicalSpace β] extends ContinuousMap α β where map_open' : IsOpenMap toFun @[inherit_doc] infixr:25 " →CO " => ContinuousOpenMap section /-- `ContinuousOpenMapClass F α β` states that `F` is a type of continuous open maps. You should extend this class when you extend `ContinuousOpenMap`. -/ class ContinuousOpenMapClass (F : Type) (α β : outParam Type) [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where map_open (f : F) : IsOpenMap f end export ContinuousOpenMapClass (map_open) instance [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [ContinuousOpenMapClass F α β] : CoeTC F (α →CO β) := ⟨fun f => ⟨f, map_open f⟩⟩ /-! ### Continuous open maps -/ namespace ContinuousOpenMap variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] instance instFunLike : FunLike (α →CO β) α β where coe f := f.toFun coe_injective' f g h := by obtain ⟨⟨_, _⟩, _⟩ := f obtain ⟨⟨_, _⟩, _⟩ := g congr instance : ContinuousOpenMapClass (α →CO β) α β where map_continuous f := f.continuous_toFun map_open f := f.map_open' theorem toFun_eq_coe {f : α →CO β} : f.toFun = (f : α → β) := rfl /-- `simp`-normal form of `toFun_eq_coe`. -/ @[simp] theorem coe_toContinuousMap (f : α →CO β) : (f.toContinuousMap : α → β) = f := rfl @[ext] theorem ext {f g : α →CO β} (h : ∀ a, f a = g a) : f = g := DFunLike.ext f g h /-- Copy of a `ContinuousOpenMap` with a new `ContinuousMap` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : α →CO β) (f' : α → β) (h : f' = f) : α →CO β := ⟨f.toContinuousMap.copy f' <\| h, h.symm.subst f.map_open'⟩ @[simp] theorem coe_copy (f : α →CO β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl theorem copy_eq (f : α →CO β) (f' : α → β) (h : f' = f) : f.copy f' h = f := DFunLike.ext' h variable (α) /-- `id` as a `ContinuousOpenMap`. -/ protected def id : α →CO α := ⟨ContinuousMap.id _, IsOpenMap.id⟩ instance : Inhabited (α →CO α) := ⟨ContinuousOpenMap.id _⟩ @[simp, norm_cast] theorem coe_id : ⇑(ContinuousOpenMap.id α) = id := rfl variable {α} @[simp] theorem id_apply (a : α) : ContinuousOpenMap.id α a = a := rfl /-- Composition of `ContinuousOpenMap`s as a `ContinuousOpenMap`. -/ def comp (f : β →CO γ) (g : α →CO β) : ContinuousOpenMap α γ := ⟨f.toContinuousMap.comp g.toContinuousMap, f.map_open'.comp g.map_open'⟩ @[simp] theorem coe_comp (f : β →CO γ) (g : α →CO β) : (f.comp g : α → γ) = f ∘ g := rfl @[simp] theorem comp_apply (f : β →CO γ) (g : α →CO β) (a : α) : (f.comp g) a = f (g a) := rfl @[simp] theorem comp_assoc (f : γ →CO δ) (g : β →CO γ) (h : α →CO β) : (f.comp g).comp h = f.comp (g.comp h) := rfl @[simp] theorem comp_id (f : α →CO β) : f.comp (ContinuousOpenMap.id α) = f := ext fun _ => rfl @[simp] theorem id_comp (f : α →CO β) : (ContinuousOpenMap.id β).comp f = f := ext fun _ => rfl @[simp] theorem cancel_right {g₁ g₂ : β →CO γ} {f : α →CO β} (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ := ⟨fun h => ext <\| hf.forall.2 <\| DFunLike.ext_iff.1 h, fun h => congr_arg₂ _ h rfl⟩ @[simp] theorem cancel_left {g : β →CO γ} {f₁ f₂ : α →CO β} (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ := ⟨fun h => ext fun a => hg <\| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ end ContinuousOpenMap
.lake/packages/mathlib/Mathlib/Topology/CWComplex/Abstract/Basic.lean	import Mathlib.Topology.Category.TopCat.Limits.Basic import Mathlib.Topology.Category.TopCat.Sphere import Mathlib.AlgebraicTopology.RelativeCellComplex.Basic /-! # CW-complexes This file defines (relative) CW-complexes using a categorical approach. ## Main definitions * `RelativeCWComplex`: A relative CW-complex is the colimit of an expanding sequence of subspaces `sk i` (called the $(i-1)$-skeleton) for `i ≥ 0`, where `sk 0` (i.e., the $(-1)$-skeleton) is an arbitrary topological space, and each `sk (n + 1)` (i.e., the $n$-skeleton) is obtained from `sk n` (i.e., the $(n-1)$-skeleton) by attaching `n`-disks. * `CWComplex`: A CW-complex is a relative CW-complex whose `sk 0` (i.e., $(-1)$-skeleton) is empty. ## Implementation Notes This file provides a categorical approach to CW complexes, defining them via colimits and transfinite compositions. For a classical approach that defines CW complexes via explicit cells and attaching maps, see `Mathlib/Topology/CWComplex/Classical/Basic.lean`. The two approaches are equivalent but serve different purposes: * This approach is more suitable for categorical arguments and generalizations * The classical approach is more convenient for concrete geometric arguments ## References * [R. Fritsch and R. Piccinini, Cellular Structures in Topology][fritsch-piccinini1990] ## TODO * Prove the equivalence between this categorical approach and the classical approach in `Mathlib/Topology/CWComplex/Classical/Basic.lean`. Currently there is no way to move between the two definitions. -/ open TopCat universe u open CategoryTheory Limits HomotopicalAlgebra namespace TopCat namespace RelativeCWComplex /-- For each `n : ℕ`, this is the family of morphisms which sends the unique element of `Unit` to `diskBoundaryInclusion n : ∂𝔻 n ⟶ 𝔻 n`. -/ @[nolint unusedArguments] abbrev basicCell (n : ℕ) (_ : Unit) : ∂𝔻 n ⟶ 𝔻 n := diskBoundaryInclusion n end RelativeCWComplex open RelativeCWComplex in /-- A relative CW-complex is a morphism `f : X ⟶ Y` equipped with data expressing that `Y` identifies to the colimit of a functor `F : ℕ ⥤ TopCat` with that `F.obj 0 ≅ X` and for any `n : ℕ`, `F.obj (n + 1)` is obtained from `F.obj n` by attaching `n`-disks. -/ abbrev RelativeCWComplex {X Y : TopCat.{u}} (f : X ⟶ Y) := RelativeCellComplex.{u} basicCell f /-- A CW-complex is a topological space such that `⊥_ _ ⟶ X` is a relative CW-complex. -/ abbrev CWComplex (X : TopCat.{u}) := RelativeCWComplex (initial.to X) end TopCat
.lake/packages/mathlib/Mathlib/Topology/CWComplex/Classical/Finite.lean	import Mathlib.Topology.CWComplex.Classical.Basic /-! # Finiteness notions on CW complexes In this file we define what it means for a CW complex to be finite dimensional, of finite type or finite. We define constructors with relaxed conditions for CW complexes of finite type and finite CW complexes. ## Main definitions * `RelCWComplex.FiniteDimensional`: a CW complex is finite dimensional if it has only finitely many nonempty indexing types for the cells. * `RelCWComplex.FiniteType`: a CW complex is of finite type if it has only finitely many cells in each dimension. * `RelCWComplex.Finite`: a CW complex is finite if it is finite dimensional and of finite type. ## Main statements * `RelCWComplex.mkFiniteType`: if we want to construct a CW complex of finite type, we can relax the condition `mapsTo`. * `RelCWComplex.mkFinite`: if we want to construct a finite CW complex, we can relax the condition `mapsTo` and can leave out the condition `closed'`. * `RelCWComplex.finite_iff_finite_cells`: a CW complex is finite iff the total number of its cells is finite. -/ open Metric Set namespace Topology /-- A CW complex is finite dimensional if `cell C n` is empty for all but finitely many `n`. -/ class RelCWComplex.FiniteDimensional.{u} {X : Type u} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] : Prop where /-- For some natural number `n`, the type `cell C m` is empty for all `m ≥ n`. -/ eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell C n) /-- A CW complex is of finite type if `cell C n` is finite for every `n`. -/ class RelCWComplex.FiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] : Prop where /-- `cell C n` is finite for every `n`. -/ finite_cell (n : ℕ) : Finite (cell C n) /-- A CW complex is finite if it is finite dimensional and of finite type. -/ class RelCWComplex.Finite {X : Type} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] extends FiniteDimensional C, FiniteType C variable {X : Type} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] lemma RelCWComplex.finite_of_finiteDimensional_finiteType [FiniteDimensional C] [FiniteType C] : Finite C where eventually_isEmpty_cell := FiniteDimensional.eventually_isEmpty_cell finite_cell n := FiniteType.finite_cell n namespace CWComplex export RelCWComplex (FiniteDimensional FiniteType Finite FiniteDimensional.eventually_isEmpty_cell FiniteType.finite_cell finite_of_finiteDimensional_finiteType) end CWComplex /-- If we want to construct a relative CW complex of finite type, we can add the condition `finite_cell` and relax the condition `mapsTo`. -/ @[simps -isSimp] def RelCWComplex.mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (map n i '' ball 0 1) D) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (D ∪ ⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (closed' : ∀ (A : Set X) (_ : A ⊆ C), ((∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) ∧ IsClosed (A ∩ D)) → IsClosed A) (isClosedBase : IsClosed D) (union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : RelCWComplex C D where cell := cell map := map source_eq := source_eq continuousOn := continuousOn continuousOn_symm := continuousOn_symm pairwiseDisjoint' := pairwiseDisjoint' disjointBase' := disjointBase' mapsTo n i := by use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m))) simp only [Finite.mem_toFinset, mem_univ, iUnion_true] exact mapsTo n i closed' := closed' isClosedBase := isClosedBase union' := union' /-- A CW complex that was constructed using `RelCWComplex.mkFiniteType` is of finite type. -/ lemma RelCWComplex.finiteType_mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (map n i '' ball 0 1) D) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (D ∪ ⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (closed' : ∀ (A : Set X) (_ : A ⊆ C), ((∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) ∧ IsClosed (A ∩ D)) → IsClosed A) (isClosedBase : IsClosed D) (union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : letI := mkFiniteType C D cell map finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' disjointBase' mapsTo closed' isClosedBase union' FiniteType C := letI := mkFiniteType C D cell map finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' disjointBase' mapsTo closed' isClosedBase union' { finite_cell := finite_cell } /-- If we want to construct a CW complex of finite type, we can add the condition `finite_cell` and relax the condition `mapsTo`. -/ @[simps -isSimp] def CWComplex.mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (closed' : ∀ (A : Set X) (_ : A ⊆ C), (∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) → IsClosed A) (union' : ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : CWComplex C where cell := cell map := map source_eq := source_eq continuousOn := continuousOn continuousOn_symm := continuousOn_symm pairwiseDisjoint' := pairwiseDisjoint' mapsTo' n i := by use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m))) simp only [Finite.mem_toFinset, mem_univ, iUnion_true] exact mapsTo n i closed' := closed' union' := union' /-- A CW complex that was constructed using `CWComplex.mkFiniteType` is of finite type. -/ lemma CWComplex.finiteType_mkFiniteType.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (closed' : ∀ (A : Set X) (_ : A ⊆ C), (∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) → IsClosed A) (union' : ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : letI := mkFiniteType C cell map finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' mapsTo closed' union' FiniteType C := letI := mkFiniteType C cell map finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' mapsTo closed' union' { finite_cell := finite_cell } /-- If we want to construct a finite relative CW complex we can add the conditions `eventually_isEmpty_cell` and `finite_cell`, relax the condition `mapsTo` and remove the condition `closed'`. -/ @[simps -isSimp] def RelCWComplex.mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n)) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (map n i '' ball 0 1) D) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (D ∪ ⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (isClosedBase : IsClosed D) (union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : RelCWComplex C D where cell := cell map := map source_eq := source_eq continuousOn := continuousOn continuousOn_symm := continuousOn_symm pairwiseDisjoint' := pairwiseDisjoint' disjointBase' := disjointBase' mapsTo n i := by use fun m ↦ finite_univ.toFinset (s := (univ : Set (cell m))) simp only [Finite.mem_toFinset, mem_univ, iUnion_true] exact mapsTo n i closed' A asubc := by intro h -- `A = A ∩ C = A ∩ (D ∪ ⋃ n, ⋃ j, closedCell n j)` is closed by assumption since `C` has only -- finitely many cells. rw [← inter_eq_left.2 asubc] simp_rw [Filter.eventually_atTop, ge_iff_le] at eventually_isEmpty_cell obtain ⟨N, hN⟩ := eventually_isEmpty_cell suffices IsClosed (A ∩ (D ∪ ⋃ (n : {n : ℕ // n < N}), ⋃ j, ↑(map n j) '' closedBall 0 1)) by convert this using 2 rw [← union', iUnion_subtype] congrm D ∪ ⋃ n, ?_ refine subset_antisymm ?_ (iUnion_subset (fun i ↦ by rfl)) apply iUnion_subset intro i have : n < N := Decidable.byContradiction fun h ↦ (hN n (Nat.ge_of_not_lt h)).false i exact subset_iUnion₂ (s := fun _ i ↦ (map n i) '' closedBall 0 1) this i simp_rw [inter_union_distrib_left, inter_iUnion] exact h.2.union (isClosed_iUnion_of_finite (fun n ↦ isClosed_iUnion_of_finite (h.1 n.1))) isClosedBase := isClosedBase union' := union' /-- A CW complex that was constructed using `RelCWComplex.mkFinite` is finite. -/ lemma RelCWComplex.finite_mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n)) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (disjointBase' : ∀ (n : ℕ) (i : cell n), Disjoint (map n i '' ball 0 1) D) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (D ∪ ⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (isClosedBase : IsClosed D) (union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : letI := mkFinite C D cell map eventually_isEmpty_cell finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' disjointBase' mapsTo isClosedBase union' Finite C := letI := mkFinite C D cell map eventually_isEmpty_cell finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' disjointBase' mapsTo isClosedBase union' { eventually_isEmpty_cell := eventually_isEmpty_cell finite_cell := finite_cell } /-- If we want to construct a finite CW complex we can add the conditions `eventually_isEmpty_cell` and `finite_cell`, relax the condition `mapsTo` and remove the condition `closed'`. -/ @[simps! -isSimp] def CWComplex.mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n)) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (mapsTo_iff_image_subset : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (union' : ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : CWComplex C := (RelCWComplex.mkFinite C ∅ (cell := cell) (map := map) (eventually_isEmpty_cell := eventually_isEmpty_cell) (finite_cell := finite_cell) (source_eq := source_eq) (continuousOn := continuousOn) (continuousOn_symm := continuousOn_symm) (pairwiseDisjoint' := pairwiseDisjoint') (disjointBase' := by simp only [disjoint_empty, implies_true]) (mapsTo := by simpa only [empty_union]) (isClosedBase := isClosed_empty) (union' := by simpa only [empty_union])).toCWComplex /-- A CW complex that was constructed using `CWComplex.mkFinite` is finite. -/ lemma CWComplex.finite_mkFinite.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (cell : (n : ℕ) → Type u) (map : (n : ℕ) → (i : cell n) → PartialEquiv (Fin n → ℝ) X) (eventually_isEmpty_cell : ∀ᶠ n in Filter.atTop, IsEmpty (cell n)) (finite_cell : ∀ (n : ℕ), _root_.Finite (cell n)) (source_eq : ∀ (n : ℕ) (i : cell n), (map n i).source = ball 0 1) (continuousOn : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i) (closedBall 0 1)) (continuousOn_symm : ∀ (n : ℕ) (i : cell n), ContinuousOn (map n i).symm (map n i).target) (pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1)) (mapsTo : ∀ (n : ℕ) (i : cell n), MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j : cell m), map m j '' closedBall 0 1)) (union' : ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C) : letI := mkFinite C cell map eventually_isEmpty_cell finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' mapsTo union' Finite C := letI := mkFinite C cell map eventually_isEmpty_cell finite_cell source_eq continuousOn continuousOn_symm pairwiseDisjoint' mapsTo union' { eventually_isEmpty_cell := eventually_isEmpty_cell finite_cell := finite_cell } variable {X : Type*} [TopologicalSpace X] {C D : Set X} [RelCWComplex C D] /-- If the collection of all cells (of any dimension) of a relative CW complex `C` is finite, then `C` is finite as a CW complex. -/ lemma RelCWComplex.finite_of_finite_cells (finite : _root_.Finite (Σ n, cell C n)) : Finite C where eventually_isEmpty_cell := by simp only [Filter.eventually_atTop, ge_iff_le] cases isEmpty_or_nonempty (Σ n, cell C n) · exact ⟨0, by simp_all⟩ -- We take the greatest `n` such that there is a `j : cell C n` and show that this fulfills -- the necessary conditions. have _ := Fintype.ofFinite (Σ n, cell C n) classical let A := (Finset.univ : Finset (Σ n, cell C n)).image Sigma.fst use A.max' (Finset.image_nonempty.2 Finset.univ_nonempty) + 1 intro m _ by_contra! h' have hmA : m ∈ A := by simp only [Finset.mem_image, Finset.mem_univ, true_and, A] simp only [← exists_true_iff_nonempty] at h' obtain ⟨j, _⟩ := h' use ⟨m, j⟩ linarith [A.le_max' m hmA] finite_cell _ := Finite.of_injective (Sigma.mk _) sigma_mk_injective /-- If `C` is finite as a CW complex then the collection of all cells (of any dimension) is finite. -/ lemma RelCWComplex.finite_cells_of_finite [finite : Finite C] : _root_.Finite (Σ n, cell C n) := by -- We show that there is a bijection between `Σ n, cell C n` and -- `Σ (m : {m : ℕ // m < n}), cell C m`. have h := finite.eventually_isEmpty_cell have _ := finite.finite_cell simp only [Filter.eventually_atTop, ge_iff_le] at h rcases h with ⟨n, hn⟩ have (m) (j : cell C m) : m < n := by by_contra h exact (hn m (not_lt.1 h)).false j let f : (Σ (m : {m : ℕ // m < n}), cell C m) ≃ Σ m, cell C m := { toFun := fun ⟨m, j⟩ ↦ ⟨m, j⟩ invFun := fun ⟨m, j⟩ ↦ ⟨⟨m, this m j⟩, j⟩ left_inv := by simp [Function.LeftInverse] right_inv := by simp [Function.RightInverse, Function.LeftInverse]} rw [← Equiv.finite_iff f] exact Finite.instSigma /-- A CW complex is finite iff the total number of its cells is finite. -/ lemma RelCWComplex.finite_iff_finite_cells : Finite C ↔ _root_.Finite (Σ n, cell C n) := ⟨fun h ↦ finite_cells_of_finite (finite := h), finite_of_finite_cells⟩ namespace CWComplex export RelCWComplex (finite_of_finite_cells finite_cells_of_finite finite_iff_finite_cells) end CWComplex end Topology
.lake/packages/mathlib/Mathlib/Topology/CWComplex/Classical/Subcomplex.lean	import Mathlib.Topology.CWComplex.Classical.Finite import Mathlib.Analysis.Normed.Module.RCLike.Real /-! # Subcomplexes In this file we discuss subcomplexes of CW complexes. The definintion of subcomplexes is in the file `Topology.CWComplex.Classical.Basic`. ## Main results * `RelCWComplex.Subcomplex.instRelCWComplex`: a subcomplex of a (relative) CW complex is again a (relative) CW complex. ## References * [K. Jänich, Topology][Janich1984] -/ noncomputable section open Metric Set namespace Topology variable {X : Type*} [t : TopologicalSpace X] {C D : Set X} lemma RelCWComplex.Subcomplex.closedCell_subset_of_mem [T2Space X] [RelCWComplex C D] (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) : closedCell n i ⊆ E := by rw [← closure_openCell_eq_closedCell, E.closed.closure_subset_iff, ← E.union] apply subset_union_of_subset_right exact subset_iUnion_of_subset n (subset_iUnion (fun (j : ↑(E.I n)) ↦ openCell (C := C) n j) ⟨i, hi⟩) lemma RelCWComplex.Subcomplex.openCell_subset_of_mem [T2Space X] [RelCWComplex C D] (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) : openCell n i ⊆ E := (openCell_subset_closedCell n i).trans (closedCell_subset_of_mem E hi) lemma RelCWComplex.Subcomplex.cellFrontier_subset_of_mem [T2Space X] [RelCWComplex C D] (E : Subcomplex C) {n : ℕ} {i : cell C n} (hi : i ∈ E.I n) : cellFrontier n i ⊆ E := (cellFrontier_subset_closedCell n i).trans (closedCell_subset_of_mem E hi) /-- A subcomplex is the union of its closed cells and its base. -/ lemma RelCWComplex.Subcomplex.union_closedCell [T2Space X] [RelCWComplex C D] (E : Subcomplex C) : D ∪ ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E := by apply subset_antisymm · apply union_subset E.base_subset exact iUnion₂_subset fun n i ↦ closedCell_subset_of_mem E i.2 · rw [← E.union] apply union_subset_union_right apply iUnion₂_mono fun n i ↦ ?_ exact openCell_subset_closedCell (C := C) n i /-- A subcomplex is the union of its closed cells. -/ lemma CWComplex.Subcomplex.union_closedCell [T2Space X] [CWComplex C] (E : Subcomplex C) : ⋃ (n : ℕ) (j : E.I n), closedCell (C := C) n j = E := (empty_union _).symm.trans (RelCWComplex.Subcomplex.union_closedCell E) lemma RelCWComplex.Subcomplex.disjoint_openCell_subcomplex_of_not_mem [RelCWComplex C D] (E : Subcomplex C) {n : ℕ} {i : cell C n} (h : i ∉ E.I n) : Disjoint (openCell n i) E := by simp_rw [← union, disjoint_union_right, disjoint_iUnion_right] exact ⟨disjointBase n i , fun _ _ ↦ disjoint_openCell_of_ne (by aesop)⟩ open Classical in /-- A subcomplex is again a CW complex. -/ @[simps] instance RelCWComplex.Subcomplex.instRelCWComplex [T2Space X] [RelCWComplex C D] (E : Subcomplex C) : RelCWComplex E D where cell n := E.I n map n i := map (C := C) n i source_eq n i := source_eq (C := C) n i continuousOn n i := continuousOn (C := C) n i continuousOn_symm n i := continuousOn_symm (C := C) n i pairwiseDisjoint' := by intro ⟨n, i⟩ _ ⟨m, j⟩ _ hne refine @pairwiseDisjoint' _ _ C D _ ⟨n, i⟩ trivial ⟨m, j⟩ trivial ?_ exact Function.injective_id.sigma_map (fun _ ↦ Subtype.val_injective) \|>.ne hne disjointBase' n i := disjointBase' (C := C) n i mapsTo := by intro n i rcases cellFrontier_subset_finite_openCell (C := C) n i with ⟨J, hJ⟩ use fun m ↦ Finset.preimage (J m) Subtype.val Subtype.val_injective.injOn rw [mapsTo_iff_image_subset] intro x hx specialize hJ hx simp_rw [iUnion_coe_set, mem_union, mem_iUnion, Finset.mem_preimage, exists_prop, Decidable.or_iff_not_imp_left] at hJ ⊢ intro h specialize hJ h obtain ⟨m, hmn, j, hj, hxj⟩ := hJ suffices j ∈ E.I m from ⟨m, hmn, j, this, hj, openCell_subset_closedCell _ _ hxj⟩ have : x ∈ (E : Set X) := E.cellFrontier_subset_of_mem i.2 hx by_contra hj' exact E.disjoint_openCell_subcomplex_of_not_mem hj' \|>.notMem_of_mem_left hxj this closed' A hA h := by apply isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell (subset_trans hA (subset_complex (C := C) E)) h.2 intro n _ j by_cases hj : j ∈ E.I n · exact Or.intro_right _ (h.1 n ⟨j, hj⟩) · exact Or.intro_left _ ((disjoint_openCell_subcomplex_of_not_mem E hj).symm.mono_left hA) isClosedBase := isClosedBase (C := C) union' := union_closedCell E /-- A subcomplex is again a CW complex. -/ instance CWComplex.Subcomplex.instCWComplex [T2Space X] [CWComplex C] (E : Subcomplex C) : CWComplex (E : Set X) := RelCWComplex.toCWComplex (E : Set X) @[simp] lemma CWComplex.Subcomplex.cell_def [T2Space X] [CWComplex C] (E : Subcomplex C) (n : ℕ) : cell (E : Set X) n = E.I (C := C) n := rfl @[simp] lemma CWComplex.Subcomplex.map_def [T2Space X] [CWComplex C] (E : Subcomplex C) (n : ℕ) (i : E.I n) : map (C := E) n i = map (C := C) n i := rfl @[simp] lemma RelCWComplex.Subcomplex.openCell_eq [T2Space X] [RelCWComplex C D] (E : Subcomplex C) (n : ℕ) (i : E.I n) : openCell (C := E) n i = openCell n (i : cell C n) := by rfl @[simp] lemma RelCWComplex.Subcomplex.closedCell_eq [T2Space X] [RelCWComplex C D] (E : Subcomplex C) (n : ℕ) (i : E.I n) : closedCell (C := E) n i = closedCell n (i : cell C n) := by rfl @[simp] lemma RelCWComplex.Subcomplex.cellFrontier_eq [T2Space X] [RelCWComplex C D] (E : Subcomplex C) (n : ℕ) (i : E.I n) : cellFrontier (C := E) n i = cellFrontier n (i : cell C n) := by rfl instance RelCWComplex.Subcomplex.finiteType_subcomplex_of_finiteType [T2Space X] [RelCWComplex C D] [FiniteType C] (E : Subcomplex C) : FiniteType (E : Set X) where finite_cell n := let _ := FiniteType.finite_cell (C := C) (D := D) n Subtype.finite instance RelCWComplex.Subcomplex.finiteDimensional_subcomplex_of_finiteDimensional [T2Space X] [RelCWComplex C D] [FiniteDimensional C] (E : Subcomplex C) : FiniteDimensional (E : Set X) where eventually_isEmpty_cell := by filter_upwards [FiniteDimensional.eventually_isEmpty_cell (C := C) (D := D)] with n hn simp [isEmpty_subtype] /-- A subcomplex of a finite CW complex is again finite. -/ instance RelCWComplex.Subcomplex.finite_subcomplex_of_finite [T2Space X] [RelCWComplex C D] [Finite C] (E : Subcomplex C) : Finite (E : Set X) := finite_of_finiteDimensional_finiteType _ namespace CWComplex.Subcomplex export RelCWComplex.Subcomplex (closedCell_subset_of_mem openCell_subset_of_mem cellFrontier_subset_of_mem disjoint_openCell_subcomplex_of_not_mem subset_complex finiteType_subcomplex_of_finiteType finiteDimensional_subcomplex_of_finiteDimensional finite_subcomplex_of_finite) end CWComplex.Subcomplex end Topology
.lake/packages/mathlib/Mathlib/Topology/CWComplex/Classical/Basic.lean	import Mathlib.Analysis.Normed.Module.RCLike.Real import Mathlib.Data.ENat.Basic import Mathlib.Logic.Equiv.PartialEquiv import Mathlib.Topology.MetricSpace.ProperSpace.Real /-! # CW complexes This file defines (relative) CW complexes and proves basic properties about them using the classical approach of Whitehead. A CW complex is a topological space that is made by gluing closed disks of different dimensions together. ## Main definitions * `RelCWComplex C D`: the class of CW structures on a subspace `C` relative to a base set `D` of a topological space `X`. * `CWComplex C`: an abbreviation for `RelCWComplex C ∅`. The class of CW structures on a subspace `C` of the topological space `X`. * `openCell n`: indexed family of all open cells of dimension `n`. * `closedCell n`: indexed family of all closed cells of dimension `n`. * `cellFrontier n`: indexed family of the boundaries of cells of dimension `n`. * `skeleton C n`: the `n`-skeleton of the (relative) CW complex `C`. ## Main statements * `iUnion_openCell_eq_skeleton`: the skeletons can also be seen as a union of open cells. * `cellFrontier_subset_finite_openCell`: the edge of a cell is contained in a finite union of open cells of a lower dimension. ## Implementation notes * We use the historical definition of CW complexes, due to Whitehead: a CW complex is a collection of cells with attaching maps - all cells are subspaces of one ambient topological space. This way, we avoid having to work with a lot of different topological spaces. On the other hand, it requires the union of all cells to be closed. If that is not the case, you need to consider that union as a subspace of itself. * For a categorical approach that defines CW complexes via colimits and transfinite compositions, see `Mathlib/Topology/CWComplex/Abstract/Basic.lean`. The two approaches are equivalent but serve different purposes: * This approach is more convenient for concrete geometric arguments * The categorical approach is more suitable for abstract arguments and generalizations * The definition `RelCWComplex` does not require `X` to be a Hausdorff space. A lot of the lemmas will however require this property. * This definition is a class to ease working with different constructions and their properties. Overall this means that being a CW complex is treated more like a property than data. * The natural number is explicit in `openCell`, `closedCell` and `cellFrontier` because `cell n` and `cell m` might be the same type in an explicit CW complex even when `n` and `m` are different. * `CWComplex` is a separate class from `RelCWComplex`. This not only gives absolute CW complexes a better constructor but also aids typeclass inference: a construction on relative CW complexes may yield a base that for the special case of CW complexes is provably equal to the empty set but not definitionally so. In that case we define an instance specifically for absolute CW complexes and want this to be inferred over the relative version. Since the base is an `outParam` this is especially necessary since you cannot provide typeclass inference with a specified base. But having the type `CWComplex` be separate from `RelCWComplex` makes this specification possible. * For a similar reason to the previous bullet point we make the instance `CWComplex.instRelCWComplex` have high priority. For example, when talking about the type of cells `cell C` of an absolute CW complex `C`, this actually refers to `RelCWComplex.cell C` through this instance. Again, we want typeclass inference to first consider absolute CW structures. * For statements, the auxiliary construction `skeletonLT` is preferred over `skeleton` as it makes the base case of inductions easier. The statement about `skeleton` should then be derived from the one about `skeletonLT`. ## References * [A. Hatcher, Algebraic Topology][hatcher02] -/ noncomputable section open Metric Set namespace Topology /-- A CW complex of a topological space `X` relative to another subspace `D` is the data of its `n`-cells `cell n i` for each `n : ℕ` along with attaching maps that satisfy a number of properties with the most important being closure-finiteness (`mapsTo`) and weak topology (`closed'`). Note that this definition requires `C` and `D` to be closed subspaces. If `C` is not closed choose `X` to be `C`. -/ class RelCWComplex.{u} {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) where /-- The indexing type of the cells of dimension `n`. -/ cell (n : ℕ) : Type u /-- The characteristic map of the `n`-cell given by the index `i`. This map is a bijection when restricting to `ball 0 1`, where we consider `(Fin n → ℝ)` endowed with the maximum metric. -/ map (n : ℕ) (i : cell n) : PartialEquiv (Fin n → ℝ) X /-- The source of every characteristic map of dimension `n` is `(ball 0 1 : Set (Fin n → ℝ))`. -/ source_eq (n : ℕ) (i : cell n) : (map n i).source = ball 0 1 /-- The characteristic maps are continuous when restricting to `closedBall 0 1`. -/ continuousOn (n : ℕ) (i : cell n) : ContinuousOn (map n i) (closedBall 0 1) /-- The inverse of the restriction to `ball 0 1` is continuous on the image. -/ continuousOn_symm (n : ℕ) (i : cell n) : ContinuousOn (map n i).symm (map n i).target /-- The open cells are pairwise disjoint. Use `RelCWComplex.pairwiseDisjoint` or `RelCWComplex.disjoint_openCell_of_ne` instead. -/ pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1) /-- All open cells are disjoint with the base. Use `RelCWComplex.disjointBase` instead. -/ disjointBase' (n : ℕ) (i : cell n) : Disjoint (map n i '' ball 0 1) D /-- The boundary of a cell is contained in the union of the base with a finite union of closed cells of a lower dimension. Use `RelCWComplex.cellFrontier_subset_base_union_finite_closedCell` instead. -/ mapsTo (n : ℕ) (i : cell n) : ∃ I : Π m, Finset (cell m), MapsTo (map n i) (sphere 0 1) (D ∪ ⋃ (m < n) (j ∈ I m), map m j '' closedBall 0 1) /-- A CW complex has weak topology, i.e. a set `A` in `X` is closed iff its intersection with every closed cell and `D` is closed. Use `RelCWComplex.closed` instead. -/ closed' (A : Set X) (hAC : A ⊆ C) : ((∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) ∧ IsClosed (A ∩ D)) → IsClosed A /-- The base `D` is closed. -/ isClosedBase : IsClosed D /-- The union of all closed cells equals `C`. Use `RelCWComplex.union` instead. -/ union' : D ∪ ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C /-- Characterizing when a subspace `C` of a topological space `X` is a CW complex. Note that this requires `C` to be closed. If `C` is not closed choose `X` to be `C`. -/ class CWComplex.{u} {X : Type u} [TopologicalSpace X] (C : Set X) where /-- The indexing type of the cells of dimension `n`. -/ protected cell (n : ℕ) : Type u /-- The characteristic map of the `n`-cell given by the index `i`. This map is a bijection when restricting to `ball 0 1`, where we consider `(Fin n → ℝ)` endowed with the maximum metric. -/ protected map (n : ℕ) (i : cell n) : PartialEquiv (Fin n → ℝ) X /-- The source of every characteristic map of dimension `n` is `(ball 0 1 : Set (Fin n → ℝ))`. -/ protected source_eq (n : ℕ) (i : cell n) : (map n i).source = ball 0 1 /-- The characteristic maps are continuous when restricting to `closedBall 0 1`. -/ protected continuousOn (n : ℕ) (i : cell n) : ContinuousOn (map n i) (closedBall 0 1) /-- The inverse of the restriction to `ball 0 1` is continuous on the image. -/ protected continuousOn_symm (n : ℕ) (i : cell n) : ContinuousOn (map n i).symm (map n i).target /-- The open cells are pairwise disjoint. Use `CWComplex.pairwiseDisjoint` or `CWComplex.disjoint_openCell_of_ne` instead. -/ protected pairwiseDisjoint' : (univ : Set (Σ n, cell n)).PairwiseDisjoint (fun ni ↦ map ni.1 ni.2 '' ball 0 1) /-- The boundary of a cell is contained in a finite union of closed cells of a lower dimension. Use `CWComplex.mapsTo` or `CWComplex.cellFrontier_subset_finite_closedCell` instead. -/ protected mapsTo' (n : ℕ) (i : cell n) : ∃ I : Π m, Finset (cell m), MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j ∈ I m), map m j '' closedBall 0 1) /-- A CW complex has weak topology, i.e. a set `A` in `X` is closed iff its intersection with every closed cell is closed. Use `CWComplex.closed` instead. -/ protected closed' (A : Set X) (hAC : A ⊆ C) : (∀ n j, IsClosed (A ∩ map n j '' closedBall 0 1)) → IsClosed A /-- The union of all closed cells equals `C`. Use `CWComplex.union` instead. -/ protected union' : ⋃ (n : ℕ) (j : cell n), map n j '' closedBall 0 1 = C @[simps -isSimp] instance (priority := high) CWComplex.instRelCWComplex {X : Type} [TopologicalSpace X] (C : Set X) [CWComplex C] : RelCWComplex C ∅ where cell := CWComplex.cell C map := CWComplex.map source_eq := CWComplex.source_eq continuousOn := CWComplex.continuousOn continuousOn_symm := CWComplex.continuousOn_symm pairwiseDisjoint' := CWComplex.pairwiseDisjoint' disjointBase' := by simp [disjoint_empty] mapsTo := by simpa only [empty_union] using CWComplex.mapsTo' closed' := by simpa only [inter_empty, isClosed_empty, and_true] using CWComplex.closed' isClosedBase := isClosed_empty union' := by simpa only [empty_union] using CWComplex.union' /-- A relative CW complex with an empty base is an absolute CW complex. -/ @[simps -isSimp] def RelCWComplex.toCWComplex {X : Type} [TopologicalSpace X] (C : Set X) [RelCWComplex C ∅] : CWComplex C where cell := cell C map := map source_eq := source_eq continuousOn := continuousOn continuousOn_symm := continuousOn_symm pairwiseDisjoint' := pairwiseDisjoint' mapsTo' := by simpa using mapsTo (C := C) closed' := by simpa using closed' (C := C) union' := by simpa using union' (C := C) lemma RelCWComplex.toCWComplex_eq {X : Type} [TopologicalSpace X] (C : Set X) [h : RelCWComplex C ∅] : (toCWComplex C).instRelCWComplex = h := rfl variable {X : Type} [t : TopologicalSpace X] {C D : Set X} /-- The open `n`-cell given by the index `i`. Use this instead of `map n i '' ball 0 1` whenever possible. -/ def RelCWComplex.openCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : Set X := map n i '' ball 0 1 /-- The closed `n`-cell given by the index `i`. Use this instead of `map n i '' closedBall 0 1` whenever possible. -/ def RelCWComplex.closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : Set X := map n i '' closedBall 0 1 /-- The boundary of the `n`-cell given by the index `i`. Use this instead of `map n i '' sphere 0 1` whenever possible. -/ def RelCWComplex.cellFrontier [RelCWComplex C D] (n : ℕ) (i : cell C n) : Set X := map n i '' sphere 0 1 namespace CWComplex export RelCWComplex (cell map source_eq continuousOn continuousOn_symm mapsTo isClosedBase openCell closedCell cellFrontier) end CWComplex lemma CWComplex.mapsTo [CWComplex C] (n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m), MapsTo (map n i) (sphere 0 1) (⋃ (m < n) (j ∈ I m), map m j '' closedBall 0 1) := by have := RelCWComplex.mapsTo n i simp_rw [empty_union] at this exact this lemma RelCWComplex.pairwiseDisjoint [RelCWComplex C D] : (univ : Set (Σ n, cell C n)).PairwiseDisjoint (fun ni ↦ openCell ni.1 ni.2) := RelCWComplex.pairwiseDisjoint' lemma RelCWComplex.disjointBase [RelCWComplex C D] (n : ℕ) (i : cell C n) : Disjoint (openCell n i) D := RelCWComplex.disjointBase' n i lemma RelCWComplex.disjoint_openCell_of_ne [RelCWComplex C D] {n m : ℕ} {i : cell C n} {j : cell C m} (ne : (⟨n, i⟩ : Σ n, cell C n) ≠ ⟨m, j⟩) : Disjoint (openCell n i) (openCell m j) := pairwiseDisjoint (mem_univ _) (mem_univ _) ne lemma RelCWComplex.cellFrontier_subset_base_union_finite_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆ D ∪ ⋃ (m < n) (j ∈ I m), closedCell m j := by rcases mapsTo n i with ⟨I, hI⟩ use I rw [mapsTo_iff_image_subset] at hI exact hI lemma CWComplex.cellFrontier_subset_finite_closedCell [CWComplex C] (n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆ ⋃ (m < n) (j ∈ I m), closedCell m j := by rcases RelCWComplex.mapsTo n i with ⟨I, hI⟩ use I rw [mapsTo_iff_image_subset, empty_union] at hI exact hI lemma RelCWComplex.union [RelCWComplex C D] : D ∪ ⋃ (n : ℕ) (j : cell C n), closedCell n j = C := RelCWComplex.union' lemma CWComplex.union [CWComplex C] : ⋃ (n : ℕ) (j : cell C n), closedCell n j = C := by have := RelCWComplex.union' (C := C) rw [empty_union] at this exact this lemma RelCWComplex.openCell_subset_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : openCell n i ⊆ closedCell n i := image_mono Metric.ball_subset_closedBall lemma RelCWComplex.cellFrontier_subset_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : cellFrontier n i ⊆ closedCell n i := image_mono Metric.sphere_subset_closedBall lemma RelCWComplex.cellFrontier_union_openCell_eq_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : cellFrontier n i ∪ openCell n i = closedCell n i := by rw [cellFrontier, openCell, closedCell, ← image_union] congrm map n i '' ?_ exact sphere_union_ball lemma RelCWComplex.map_zero_mem_openCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : map n i 0 ∈ openCell n i := by apply mem_image_of_mem simp only [mem_ball, dist_self, zero_lt_one] lemma RelCWComplex.map_zero_mem_closedCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : map n i 0 ∈ closedCell n i := openCell_subset_closedCell _ _ (map_zero_mem_openCell _ _) /-- This is an auxiliary lemma used to prove `RelCWComplex.eq_of_eq_union_iUnion`. -/ private lemma RelCWComplex.subset_of_eq_union_iUnion [RelCWComplex C D] (I J : Π n, Set (cell C n)) (hIJ : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = D ∪ ⋃ (n : ℕ) (j : J n), openCell (C := C) n j) (n : ℕ) : I n ⊆ J n := by intro i hi by_contra hJ have h : openCell n i ⊆ D ∪ ⋃ n, ⋃ (j : J n), openCell (C := C) n j := hIJ.symm ▸ subset_union_of_subset_right (subset_iUnion_of_subset n (subset_iUnion_of_subset ⟨i, hi⟩ (subset_refl (openCell n i)))) D have h' : Disjoint (openCell n i) (D ∪ ⋃ n, ⋃ (j : J n), openCell (C := C) n j) := by simp_rw [disjoint_union_right, disjoint_iUnion_right] exact ⟨disjointBase n i, fun m j ↦ disjoint_openCell_of_ne (by aesop)⟩ rw [disjoint_of_subset_iff_left_eq_empty h] at h' exact notMem_empty _ (h' ▸ map_zero_mem_openCell n i) lemma RelCWComplex.eq_of_eq_union_iUnion [RelCWComplex C D] (I J : Π n, Set (cell C n)) (hIJ : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = D ∪ ⋃ (n : ℕ) (j : J n), openCell (C := C) n j) : I = J := by ext n x exact ⟨fun h ↦ subset_of_eq_union_iUnion I J hIJ n h, fun h ↦ subset_of_eq_union_iUnion J I hIJ.symm n h⟩ lemma CWComplex.eq_of_eq_union_iUnion [CWComplex C] (I J : Π n, Set (cell C n)) (hIJ : ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = ⋃ (n : ℕ) (j : J n), openCell (C := C) n j) : I = J := by apply RelCWComplex.eq_of_eq_union_iUnion simp_rw [empty_union, hIJ] lemma RelCWComplex.isCompact_closedCell [RelCWComplex C D] {n : ℕ} {i : cell C n} : IsCompact (closedCell n i) := (isCompact_closedBall _ _).image_of_continuousOn (continuousOn n i) lemma RelCWComplex.isClosed_closedCell [RelCWComplex C D] [T2Space X] {n : ℕ} {i : cell C n} : IsClosed (closedCell n i) := isCompact_closedCell.isClosed lemma RelCWComplex.isCompact_cellFrontier [RelCWComplex C D] {n : ℕ} {i : cell C n} : IsCompact (cellFrontier n i) := (isCompact_sphere _ _).image_of_continuousOn ((continuousOn n i).mono sphere_subset_closedBall) lemma RelCWComplex.isClosed_cellFrontier [RelCWComplex C D] [T2Space X] {n : ℕ} {i : cell C n} : IsClosed (cellFrontier n i) := isCompact_cellFrontier.isClosed lemma RelCWComplex.closure_openCell_eq_closedCell [RelCWComplex C D] [T2Space X] {n : ℕ} {j : cell C n} : closure (openCell n j) = closedCell n j := by apply subset_antisymm (isClosed_closedCell.closure_subset_iff.2 (openCell_subset_closedCell n j)) rw [closedCell, ← closure_ball 0 (by exact one_ne_zero)] apply ContinuousOn.image_closure rw [closure_ball 0 (by exact one_ne_zero)] exact continuousOn n j lemma RelCWComplex.closed (C : Set X) {D : Set X} [RelCWComplex C D] [T2Space X] (A : Set X) (asubc : A ⊆ C) : IsClosed A ↔ (∀ n (j : cell C n), IsClosed (A ∩ closedCell n j)) ∧ IsClosed (A ∩ D) := by refine ⟨?_, closed' A asubc⟩ exact fun closedA ↦ ⟨fun _ _ ↦ closedA.inter isClosed_closedCell, closedA.inter (isClosedBase C)⟩ lemma CWComplex.closed (C : Set X) [CWComplex C] [T2Space X] (A : Set X) (asubc : A ⊆ C) : IsClosed A ↔ ∀ n (j : cell C n), IsClosed (A ∩ closedCell n j) := by have := RelCWComplex.closed C A asubc simp_all lemma RelCWComplex.closedCell_subset_complex [RelCWComplex C D] (n : ℕ) (j : cell C n) : closedCell n j ⊆ C := by simp_rw [← union] exact subset_union_of_subset_right (subset_iUnion₂ _ _) _ lemma RelCWComplex.openCell_subset_complex [RelCWComplex C D] (n : ℕ) (j : cell C n) : openCell n j ⊆ C := (openCell_subset_closedCell _ _).trans (closedCell_subset_complex _ _) lemma RelCWComplex.cellFrontier_subset_complex [RelCWComplex C D] (n : ℕ) (j : cell C n) : cellFrontier n j ⊆ C := (cellFrontier_subset_closedCell n j).trans (closedCell_subset_complex n j) lemma RelCWComplex.closedCell_zero_eq_singleton [RelCWComplex C D] {j : cell C 0} : closedCell 0 j = {map 0 j ![]} := by simp [closedCell, Matrix.empty_eq] lemma RelCWComplex.openCell_zero_eq_singleton [RelCWComplex C D] {j : cell C 0} : openCell 0 j = {map 0 j ![]} := by simp [openCell, Matrix.empty_eq] lemma RelCWComplex.cellFrontier_zero_eq_empty [RelCWComplex C D] {j : cell C 0} : cellFrontier 0 j = ∅ := by simp [cellFrontier, sphere_eq_empty_of_subsingleton] lemma RelCWComplex.base_subset_complex [RelCWComplex C D] : D ⊆ C := by simp_rw [← union] exact subset_union_left lemma RelCWComplex.isClosed [T2Space X] [RelCWComplex C D] : IsClosed C := by rw [closed C C (by rfl)] constructor · intros rw [inter_eq_right.2 (closedCell_subset_complex _ _)] exact isClosed_closedCell · rw [inter_eq_right.2 base_subset_complex] exact isClosedBase C /-- A helper lemma that is essentially the same as `RelCWComplex.iUnion_openCell_eq_skeletonLT`. Use that lemma instead. -/ private lemma RelCWComplex.iUnion_openCell_eq_iUnion_closedCell [RelCWComplex C D] (n : ℕ∞) : D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), openCell m j = D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), closedCell m j := by apply subset_antisymm · apply union_subset · exact subset_union_left · apply iUnion₂_subset fun m hm ↦ iUnion_subset fun j ↦ ?_ apply subset_union_of_subset_right apply subset_iUnion₂_of_subset m hm apply subset_iUnion_of_subset j exact openCell_subset_closedCell m j · apply union_subset subset_union_left refine iUnion₂_subset fun m hm ↦ iUnion_subset fun j ↦ ?_ rw [← cellFrontier_union_openCell_eq_closedCell] apply union_subset · induction m using Nat.case_strong_induction_on with \| hz => simp [cellFrontier_zero_eq_empty] \| hi m hm' => obtain ⟨I, hI⟩ := cellFrontier_subset_base_union_finite_closedCell (m + 1) j apply hI.trans apply union_subset subset_union_left apply iUnion₂_subset fun l hl ↦ iUnion₂_subset fun i _ ↦ ?_ rw [← cellFrontier_union_openCell_eq_closedCell] apply union_subset · exact (hm' l (Nat.le_of_lt_succ hl) ((ENat.coe_lt_coe.2 hl).trans hm) i) · apply subset_union_of_subset_right exact subset_iUnion₂_of_subset l ((ENat.coe_lt_coe.2 hl).trans hm) <\| subset_iUnion _ i · exact subset_union_of_subset_right (subset_iUnion₂_of_subset m hm (subset_iUnion _ j)) _ lemma RelCWComplex.union_iUnion_openCell_eq_complex [RelCWComplex C D] : D ∪ ⋃ (n : ℕ) (j : cell C n), openCell n j = C := by suffices D ∪ ⋃ n, ⋃ (j : cell C n), openCell n j = D ∪ ⋃ (m : ℕ) (_ : m < (⊤ : ℕ∞)) (j : cell C m), closedCell m j by simpa [union] using this simp_rw [← RelCWComplex.iUnion_openCell_eq_iUnion_closedCell, ENat.coe_lt_top, iUnion_true] lemma CWComplex.iUnion_openCell_eq_complex [CWComplex C] : ⋃ (n : ℕ) (j : cell C n), openCell n j = C := by simpa using RelCWComplex.union_iUnion_openCell_eq_complex (C := C) /-- The contrapositive of `disjoint_openCell_of_ne`. -/ lemma RelCWComplex.eq_of_not_disjoint_openCell [RelCWComplex C D] {n : ℕ} {j : cell C n} {m : ℕ} {i : cell C m} (h : ¬ Disjoint (openCell n j) (openCell m i)) : (⟨n, j⟩ : (Σ n, cell C n)) = ⟨m, i⟩ := by contrapose! h exact disjoint_openCell_of_ne h lemma RelCWComplex.disjoint_base_iUnion_openCell [RelCWComplex C D] : Disjoint D (⋃ (n : ℕ) (j : cell C n), openCell n j) := by simp_rw [disjoint_iff_inter_eq_empty, inter_iUnion, iUnion_eq_empty] intro n i rw [inter_comm, (disjointBase n i).inter_eq] /-- If for all `m ≤ n` and every `i : cell C m` the intersection `A ∩ closedCell m j` is closed and `A ∩ D` is closed then `A ∩ cellFrontier (n + 1) j` is closed for every `j : cell C (n + 1)`. -/ lemma RelCWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell [RelCWComplex C D] [T2Space X] {A : Set X} {n : ℕ} (hn : ∀ m ≤ n, ∀ (j : cell C m), IsClosed (A ∩ closedCell m j)) (j : cell C (n + 1)) (hD : IsClosed (A ∩ D)) : IsClosed (A ∩ cellFrontier (n + 1) j) := by -- this is a consequence of `cellFrontier_subset_base_union_finite_closedCell` obtain ⟨I, hI⟩ := cellFrontier_subset_base_union_finite_closedCell (n + 1) j rw [← inter_eq_right.2 hI, ← inter_assoc] refine IsClosed.inter ?_ isClosed_cellFrontier simp_rw [inter_union_distrib_left, inter_iUnion, ← iUnion_subtype (fun m ↦ m < n + 1) (fun m ↦ ⋃ i ∈ I m, A ∩ closedCell m i)] apply hD.union apply isClosed_iUnion_of_finite intro ⟨m, mlt⟩ rw [← iUnion_subtype (fun i ↦ i ∈ I m) (fun i ↦ A ∩ closedCell m i.1)] exact isClosed_iUnion_of_finite (fun ⟨j, _⟩ ↦ hn m (Nat.le_of_lt_succ mlt) j) lemma CWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell [CWComplex C] [T2Space X] {A : Set X} {n : ℕ} (hn : ∀ m ≤ n, ∀ (j : cell C m), IsClosed (A ∩ closedCell m j)) (j : cell C (n + 1)) : IsClosed (A ∩ cellFrontier (n + 1) j) := RelCWComplex.isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell hn j (by simp only [inter_empty, isClosed_empty]) /-- If for every cell either `A ∩ openCell n j` or `A ∩ closedCell n j` is closed then `A` is closed. -/ lemma RelCWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell [RelCWComplex C D] [T2Space X] {A : Set X} (hAC : A ⊆ C) (hDA : IsClosed (A ∩ D)) (h : ∀ n (_ : 0 < n), ∀ (j : cell C n), IsClosed (A ∩ openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A := by rw [closed C A hAC] refine ⟨?_, hDA⟩ intro n j induction n using Nat.case_strong_induction_on with \| hz => rw [closedCell_zero_eq_singleton] exact isClosed_inter_singleton \| hi n hn => specialize h n.succ n.zero_lt_succ j rcases h with h1 \| h2 · rw [← cellFrontier_union_openCell_eq_closedCell, inter_union_distrib_left] exact (isClosed_inter_cellFrontier_succ_of_le_isClosed_inter_closedCell hn j hDA).union h1 · exact h2 /-- If for every cell either `A ∩ openCell n j` is empty or `A ∩ closedCell n j` is closed then `A` is closed. -/ lemma RelCWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell [RelCWComplex C D] [T2Space X] {A : Set X} (hAC : A ⊆ C) (hDA : IsClosed (A ∩ D)) (h : ∀ n (_ : 0 < n), ∀ (j : cell C n), Disjoint A (openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A := by apply isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell hAC hDA intro n hn j rcases h n hn j with h \| h · left rw [disjoint_iff_inter_eq_empty.1 h] exact isClosed_empty · exact Or.inr h /-- If for every cell either `A ∩ openCell n j` or `A ∩ closedCell n j` is closed then `A` is closed. -/ lemma CWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell [CWComplex C] [T2Space X] {A : Set X} (hAC : A ⊆ C) (h : ∀ n (_ : 0 < n), ∀ (j : cell C n), IsClosed (A ∩ openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A := RelCWComplex.isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell hAC (by simp) h /-- If for every cell either `A ∩ openCell n j` is empty or `A ∩ closedCell n j` is closed then `A` is closed. -/ lemma CWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell [CWComplex C] [T2Space X] {A : Set X} (hAC : A ⊆ C) (h : ∀ n (_ : 0 < n), ∀ (j : cell C n), Disjoint A (openCell n j) ∨ IsClosed (A ∩ closedCell n j)) : IsClosed A := RelCWComplex.isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell hAC (by simp) h /-- A version of `cellFrontier_subset_base_union_finite_closedCell` using open cells: The boundary of a cell is contained in a finite union of open cells of a lower dimension. -/ lemma RelCWComplex.cellFrontier_subset_finite_openCell [RelCWComplex C D] (n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆ D ∪ (⋃ (m < n) (j ∈ I m), openCell m j) := by induction n using Nat.case_strong_induction_on with \| hz => simp [cellFrontier_zero_eq_empty] \| hi n hn => -- We apply `cellFrontier_subset_base_union_finite_closedCell` once and then apply -- the induction hypothesis to the finitely many cells that -- `cellFrontier_subset_base_union_finite_closedCell` gives us. classical obtain ⟨J, hJ⟩ := cellFrontier_subset_base_union_finite_closedCell n.succ i choose p hp using hn let I m := J m ∪ ((Finset.range n.succ).biUnion (fun l ↦ (J l).biUnion (fun y ↦ if h : l ≤ n then p l h y m else ∅))) use I intro x hx specialize hJ hx simp only [mem_union, mem_iUnion, exists_prop] at hJ ⊢ rcases hJ with hJ \| hJ · exact .inl hJ obtain ⟨l, hln, j, hj, hxj⟩ := hJ rw [← cellFrontier_union_openCell_eq_closedCell] at hxj rcases hxj with hxj \| hxj · specialize hp l (Nat.le_of_lt_succ hln) j hxj simp_rw [mem_union, mem_iUnion, exists_prop] at hp refine .imp_right (fun ⟨k, hkl, i, hi, hxi⟩ ↦ ⟨k, lt_trans hkl hln, i, ?_, hxi⟩) hp simp only [Nat.succ_eq_add_one, Finset.mem_union, Finset.mem_biUnion, Finset.mem_range, I] exact .inr ⟨l, hln, j, hj, by simp [Nat.le_of_lt_succ hln, hi]⟩ · right use l, hln, j simp only [Nat.succ_eq_add_one, Finset.mem_union, I] exact ⟨Or.intro_left _ hj, hxj⟩ /-- A version of `cellFrontier_subset_finite_closedCell` using open cells: The boundary of a cell is contained in a finite union of open cells of a lower dimension. -/ lemma CWComplex.cellFrontier_subset_finite_openCell [CWComplex C] (n : ℕ) (i : cell C n) : ∃ I : Π m, Finset (cell C m), cellFrontier n i ⊆ ⋃ (m < n) (j ∈ I m), openCell m j := by simpa using RelCWComplex.cellFrontier_subset_finite_openCell n i section Subcomplex namespace RelCWComplex /-- A subcomplex is a closed subspace of a CW complex that is the union of open cells of the CW complex. -/ structure Subcomplex (C : Set X) {D : Set X} [RelCWComplex C D] where /-- The underlying set of the subcomplex. -/ carrier : Set X /-- The indexing set of cells of the subcomplex. -/ I : Π n, Set (cell C n) /-- A subcomplex is closed. -/ closed' : IsClosed carrier /-- The union of all open cells of the subcomplex equals the subcomplex. -/ union' : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = carrier namespace Subcomplex variable [RelCWComplex C D] instance : SetLike (Subcomplex C) X where coe E := E.carrier coe_injective' E F h := by obtain ⟨E, _, _, hE⟩ := E obtain ⟨F, _, _, hF⟩ := F congr apply eq_of_eq_union_iUnion rw [hE, hF] simpa using h initialize_simps_projections Subcomplex (carrier → coe, as_prefix coe) lemma mem_carrier {E : Subcomplex C} {x : X} : x ∈ E.carrier ↔ x ∈ (E : Set X) := Iff.rfl lemma coe_eq_carrier {E : Subcomplex C} : (E : Set X) = E.carrier := rfl @[ext] lemma ext {E F : Subcomplex C} (h : ∀ x, x ∈ E ↔ x ∈ F) : E = F := SetLike.ext h lemma eq_iff (E F : Subcomplex C) : E = F ↔ (E : Set X) = F := SetLike.coe_injective.eq_iff.symm /-- Copy of a `Subcomplex` with a new `carrier` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (E : Subcomplex C) (F : Set X) (hF : F = E) (J : (n : ℕ) → Set (cell C n)) (hJ : J = E.I) : Subcomplex C := { carrier := F I := J closed' := hF.symm ▸ E.closed' union' := hF.symm ▸ hJ ▸ E.union' } @[simp] lemma coe_copy (E : Subcomplex C) (F : Set X) (hF : F = E) (J : (n : ℕ) → Set (cell C n)) (hJ : J = E.I) : (E.copy F hF J hJ : Set X) = F := rfl lemma copy_eq (E : Subcomplex C) (F : Set X) (hF : F = E) (J : (n : ℕ) → Set (cell C n)) (hJ : J = E.I) : E.copy F hF J hJ = E := SetLike.coe_injective hF lemma union (E : Subcomplex C) : D ∪ ⋃ (n : ℕ) (j : E.I n), openCell (C := C) n j.1 = E := by rw [E.union'] rfl lemma closed (E : Subcomplex C) : IsClosed (E : Set X) := E.closed' end Subcomplex end RelCWComplex namespace CWComplex export RelCWComplex (Subcomplex) namespace Subcomplex export RelCWComplex.Subcomplex (I closed union mem_carrier coe_eq_carrier ext copy coe_copy copy_eq) end CWComplex.Subcomplex lemma CWComplex.Subcomplex.union {C : Set X} [CWComplex C] {E : Subcomplex C} : ⋃ (n : ℕ) (j : E.I n), openCell (C := C) n j = E := by have := RelCWComplex.Subcomplex.union E (C := C) rw [empty_union] at this exact this /-- An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires that for every cell of the subcomplex the corresponding closed cell is a subset of `E`. -/ @[simps -isSimp] def RelCWComplex.Subcomplex.mk' [T2Space X] (C : Set X) {D : Set X} [RelCWComplex C D] (E : Set X) (I : Π n, Set (cell C n)) (closedCell_subset : ∀ (n : ℕ) (i : I n), closedCell (C := C) n i ⊆ E) (union : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C where carrier := E I := I closed' := by have hEC : (E : Set X) ⊆ C := by simp_rw [← union, ← union_iUnion_openCell_eq_complex (C := C)] exact union_subset_union_right D (iUnion_mono fun n ↦ iUnion_subset fun i ↦ subset_iUnion _ (i : cell C n)) apply isClosed_of_disjoint_openCell_or_isClosed_inter_closedCell hEC · have : D ⊆ E := by rw [← union] exact subset_union_left rw [inter_eq_right.2 this] exact isClosedBase C intro n _ j by_cases h : j ∈ I n · right suffices closedCell n j ⊆ E by rw [inter_eq_right.2 this] exact isClosed_closedCell exact closedCell_subset n ⟨j, h⟩ · left simp_rw [← union, disjoint_union_left, disjoint_iUnion_left] exact ⟨disjointBase n j \|>.symm, fun _ _ ↦ disjoint_openCell_of_ne (by aesop)⟩ union' := union /-- An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires that for every cell of the subcomplex the corresponding closed cell is a subset of `E`. -/ @[simps! -isSimp] def CWComplex.Subcomplex.mk' [T2Space X] (C : Set X) [CWComplex C] (E : Set X) (I : Π n, Set (cell C n)) (closedCell_subset : ∀ (n : ℕ) (i : I n), closedCell (C := C) n i ⊆ E) (union : ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C := RelCWComplex.Subcomplex.mk' C E I closedCell_subset (by rw [empty_union]; exact union) /-- An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires that `E` is a CW-complex. -/ @[simps -isSimp] def RelCWComplex.Subcomplex.mk'' [T2Space X] (C : Set X) {D : Set X} [RelCWComplex C D] (E : Set X) (I : Π n, Set (cell C n)) [RelCWComplex E D] (union : D ∪ ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C where carrier := E I := I closed' := isClosed union' := union /-- An alternative version of `Subcomplex.mk`: Instead of requiring that `E` is closed it requires that `E` is a CW-complex. -/ @[simps -isSimp] def CWComplex.Subcomplex.mk'' [T2Space X] (C : Set X) [h : CWComplex C] (E : Set X) (I : Π n, Set (cell C n)) [CWComplex E] (union : ⋃ (n : ℕ) (j : I n), openCell (C := C) n j = E) : Subcomplex C where carrier := E I := I closed' := RelCWComplex.isClosed union' := by rw [empty_union] exact union lemma RelCWComplex.Subcomplex.subset_complex {C D : Set X} [RelCWComplex C D] (E : Subcomplex C) : ↑E ⊆ C := by simp_rw [← union, ← RelCWComplex.union_iUnion_openCell_eq_complex] exact union_subset_union_right _ (iUnion_mono fun _ ↦ iUnion_mono' fun j ↦ ⟨j, subset_rfl⟩) lemma RelCWComplex.Subcomplex.base_subset {C D : Set X} [RelCWComplex C D] (E : Subcomplex C) : D ⊆ E := by simp_rw [← union] exact subset_union_left namespace CWComplex.Subcomplex export RelCWComplex.Subcomplex (subset_complex base_subset) end CWComplex.Subcomplex end Subcomplex section skeleton variable [T2Space X] namespace RelCWComplex /-- A non-standard definition of the `n`-skeleton of a CW complex for `n ∈ ℕ ∪ {∞}`. This allows the base case of induction to be about the base instead of being about the union of the base and some points. The standard `skeleton` is defined in terms of `skeletonLT`. `skeletonLT` is preferred in statements. You should then derive the statement about `skeleton`. -/ @[simps! -isSimp, irreducible] def skeletonLT (C : Set X) {D : Set X} [RelCWComplex C D] (n : ℕ∞) : Subcomplex C := Subcomplex.mk' _ (D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), closedCell m j) (fun l ↦ {x : cell C l \| l < n}) (by intro l ⟨i, hi⟩ apply subset_union_of_subset_right apply subset_iUnion₂_of_subset l hi exact subset_iUnion _ _) (by rw [← RelCWComplex.iUnion_openCell_eq_iUnion_closedCell] congrm D ∪ ?_ apply iUnion_congr fun m ↦ ?_ rw [iUnion_subtype, iUnion_comm] rfl) /-- The `n`-skeleton of a CW complex, for `n ∈ ℕ ∪ {∞}`. For statements use `skeletonLT` instead and then derive the statement about `skeleton`. -/ abbrev skeleton (C : Set X) {D : Set X} [RelCWComplex C D] (n : ℕ∞) : Subcomplex C := skeletonLT C (n + 1) end RelCWComplex namespace CWComplex export RelCWComplex (skeletonLT coe_skeletonLT skeletonLT_I skeleton) end CWComplex lemma RelCWComplex.skeletonLT_zero_eq_base [RelCWComplex C D] : skeletonLT C 0 = D := by simp [coe_skeletonLT] lemma CWComplex.skeletonLT_zero_eq_empty [CWComplex C] : (skeletonLT C 0 : Set X) = ∅ := RelCWComplex.skeletonLT_zero_eq_base @[simp] lemma RelCWComplex.skeletonLT_top [RelCWComplex C D] : skeletonLT C ⊤ = C := by simp [coe_skeletonLT, union] @[simp] lemma RelCWComplex.skeleton_top [RelCWComplex C D] : skeleton C ⊤ = C := skeletonLT_top lemma RelCWComplex.skeletonLT_mono [RelCWComplex C D] {n m : ℕ∞} (h : m ≤ n) : (skeletonLT C m : Set X) ⊆ skeletonLT C n := by simp_rw [coe_skeletonLT] apply union_subset_union_right intro x xmem simp_rw [mem_iUnion, exists_prop] at xmem ⊢ obtain ⟨l, lltm, xmeml⟩ := xmem exact ⟨l, lt_of_lt_of_le lltm h, xmeml⟩ lemma RelCWComplex.skeletonLT_monotone [RelCWComplex C D] : Monotone (skeletonLT C) := fun _ _ h ↦ skeletonLT_mono h lemma RelCWComplex.skeleton_mono [RelCWComplex C D] {n m : ℕ∞} (h : m ≤ n) : (skeleton C m : Set X) ⊆ skeleton C n := skeletonLT_mono (by gcongr) lemma RelCWComplex.skeleton_monotone [RelCWComplex C D] : Monotone (skeleton C) := fun _ _ h ↦ skeleton_mono h lemma RelCWComplex.closedCell_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) : closedCell n j ⊆ skeletonLT C (n + 1) := by intro x xmem rw [coe_skeletonLT] right simp_rw [mem_iUnion, exists_prop] refine ⟨n, (by norm_cast; exact lt_add_one n), ⟨j,xmem⟩⟩ lemma RelCWComplex.closedCell_subset_skeleton [RelCWComplex C D] (n : ℕ) (j : cell C n) : closedCell n j ⊆ skeleton C n := closedCell_subset_skeletonLT n j lemma RelCWComplex.openCell_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) : openCell n j ⊆ skeletonLT C (n + 1) := (openCell_subset_closedCell _ _).trans (closedCell_subset_skeletonLT _ _) lemma RelCWComplex.openCell_subset_skeleton [RelCWComplex C D] (n : ℕ) (j : cell C n) : openCell n j ⊆ skeleton C n := (openCell_subset_closedCell _ _).trans (closedCell_subset_skeleton _ _) lemma RelCWComplex.cellFrontier_subset_skeletonLT [RelCWComplex C D] (n : ℕ) (j : cell C n) : cellFrontier n j ⊆ skeletonLT C n := by obtain ⟨I, hI⟩ := cellFrontier_subset_base_union_finite_closedCell n j apply subset_trans hI rw [coe_skeletonLT] apply union_subset_union_right intro x xmem simp only [mem_iUnion, exists_prop] at xmem ⊢ obtain ⟨i, iltn, j, _, xmem⟩ := xmem exact ⟨i, by norm_cast, j, xmem⟩ lemma RelCWComplex.cellFrontier_subset_skeleton [RelCWComplex C D] (n : ℕ) (j : cell C (n + 1)) : cellFrontier (n + 1) j ⊆ skeleton C n := cellFrontier_subset_skeletonLT _ _ lemma RelCWComplex.iUnion_cellFrontier_subset_skeletonLT [RelCWComplex C D] (l : ℕ) : ⋃ (j : cell C l), cellFrontier l j ⊆ skeletonLT C l := iUnion_subset (fun _ ↦ cellFrontier_subset_skeletonLT _ _) lemma RelCWComplex.iUnion_cellFrontier_subset_skeleton [RelCWComplex C D] (l : ℕ) : ⋃ (j : cell C l), cellFrontier l j ⊆ skeleton C l := (iUnion_cellFrontier_subset_skeletonLT l).trans (skeletonLT_mono le_self_add) lemma RelCWComplex.skeletonLT_union_iUnion_closedCell_eq_skeletonLT_succ [RelCWComplex C D] (n : ℕ) : (skeletonLT C n : Set X) ∪ ⋃ (j : cell C n), closedCell n j = skeletonLT C (n + 1) := by rw [coe_skeletonLT, coe_skeletonLT, union_assoc] congr norm_cast exact (biUnion_lt_succ _ _).symm lemma RelCWComplex.skeleton_union_iUnion_closedCell_eq_skeleton_succ [RelCWComplex C D] (n : ℕ) : (skeleton C n : Set X) ∪ ⋃ (j : cell C (n + 1)), closedCell (n + 1) j = skeleton C (n + 1) := skeletonLT_union_iUnion_closedCell_eq_skeletonLT_succ _ /-- A version of the definition of `skeletonLT` with open cells. -/ lemma RelCWComplex.iUnion_openCell_eq_skeletonLT [RelCWComplex C D] (n : ℕ∞) : D ∪ ⋃ (m : ℕ) (_ : m < n) (j : cell C m), openCell m j = skeletonLT C n := (coe_skeletonLT C _).symm ▸ RelCWComplex.iUnion_openCell_eq_iUnion_closedCell n lemma CWComplex.iUnion_openCell_eq_skeletonLT [CWComplex C] (n : ℕ∞) : ⋃ (m : ℕ) (_ : m < n) (j : cell C m), openCell m j = skeletonLT C n := by rw [← RelCWComplex.iUnion_openCell_eq_skeletonLT, empty_union] lemma RelCWComplex.iUnion_openCell_eq_skeleton [RelCWComplex C D] (n : ℕ∞) : D ∪ ⋃ (m : ℕ) (_ : m < n + 1) (j : cell C m), openCell m j = skeleton C n := iUnion_openCell_eq_skeletonLT _ lemma CWComplex.iUnion_openCell_eq_skeleton [CWComplex C] (n : ℕ∞) : ⋃ (m : ℕ) (_ : m < n + 1) (j : cell C m), openCell m j = skeleton C n := iUnion_openCell_eq_skeletonLT _ lemma RelCWComplex.iUnion_skeletonLT_eq_complex [RelCWComplex C D] : ⋃ (n : ℕ), skeletonLT C n = C := by apply subset_antisymm (iUnion_subset_iff.2 fun _ ↦ (skeletonLT C _).subset_complex) simp_rw [← union_iUnion_openCell_eq_complex, union_subset_iff, iUnion₂_subset_iff] exact ⟨subset_iUnion_of_subset 0 (skeletonLT C 0).base_subset, fun n i ↦ subset_iUnion_of_subset _ (openCell_subset_skeletonLT n i)⟩ lemma RelCWComplex.iUnion_skeleton_eq_complex [RelCWComplex C D] : ⋃ (n : ℕ), skeleton C n = C := by apply subset_antisymm (iUnion_subset_iff.2 fun _ ↦ (skeleton C _).subset_complex) simp_rw [← union_iUnion_openCell_eq_complex, union_subset_iff, iUnion₂_subset_iff] exact ⟨subset_iUnion_of_subset 0 (skeleton C 0).base_subset, fun n i ↦ subset_iUnion_of_subset _ (openCell_subset_skeleton n i)⟩ lemma RelCWComplex.mem_skeletonLT_iff [RelCWComplex C D] {n : ℕ∞} {x : X} : x ∈ skeletonLT C n ↔ x ∈ D ∨ ∃ (m : ℕ) (_ : m < n) (j : cell C m), x ∈ openCell m j := by simp [← SetLike.mem_coe, ← iUnion_openCell_eq_skeletonLT] lemma CWComplex.mem_skeletonLT_iff [CWComplex C] {n : ℕ∞} {x : X} : x ∈ skeletonLT C n ↔ ∃ (m : ℕ) (_ : m < n) (j : cell C m), x ∈ openCell m j := by simp [← SetLike.mem_coe, ← iUnion_openCell_eq_skeletonLT] lemma RelCWComplex.mem_skeleton_iff [RelCWComplex C D] {n : ℕ∞} {x : X} : x ∈ skeleton C n ↔ x ∈ D ∨ ∃ (m : ℕ) (_ : m ≤ n) (j : cell C m), x ∈ openCell m j := by rw [skeleton, mem_skeletonLT_iff] suffices ∀ (m : ℕ), m < n + 1 ↔ m ≤ n by simp_rw [this] intro m cases n · simp · rw [← Nat.cast_one, ← Nat.cast_add, Nat.cast_lt, Nat.cast_le, Order.lt_add_one_iff] lemma CWComplex.exists_mem_openCell_of_mem_skeleton [CWComplex C] {n : ℕ∞} {x : X} : x ∈ skeleton C n ↔ ∃ (m : ℕ) (_ : m ≤ n) (j : cell C m), x ∈ openCell m j := by rw [RelCWComplex.mem_skeleton_iff, mem_empty_iff_false, false_or] /-- A skeleton and an open cell of a higher dimension are disjoint. -/ lemma RelCWComplex.disjoint_skeletonLT_openCell [RelCWComplex C D] {n : ℕ∞} {m : ℕ} {j : cell C m} (hnm : n ≤ m) : Disjoint (skeletonLT C n : Set X) (openCell m j) := by -- This is a consequence of `iUnion_openCell_eq_skeletonLT` and `disjoint_openCell_of_ne` simp_rw [← iUnion_openCell_eq_skeletonLT, disjoint_union_left, disjoint_iUnion_left] refine ⟨(disjointBase m j).symm, ?_⟩ intro l hln i apply disjoint_openCell_of_ne intro simp_all only [Sigma.mk.inj_iff] exact (lt_self_iff_false m).mp (ENat.coe_lt_coe.1 (hln.trans_le hnm)) /-- A skeleton and an open cell of a higher dimension are disjoint. -/ lemma RelCWComplex.disjoint_skeleton_openCell [RelCWComplex C D] {n : ℕ∞} {m : ℕ} {j : cell C m} (nlem : n < m) : Disjoint (skeleton C n : Set X) (openCell m j) := disjoint_skeletonLT_openCell (Order.add_one_le_of_lt nlem) /-- A skeleton intersected with a closed cell of a higher dimension is the skeleton intersected with the boundary of the cell. -/ lemma RelCWComplex.skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier [RelCWComplex C D] {n : ℕ∞} {m : ℕ} {j : cell C m} (hnm : n ≤ m) : (skeletonLT C n : Set X) ∩ closedCell m j = (skeletonLT C n : Set X) ∩ cellFrontier m j := by refine subset_antisymm ?_ (inter_subset_inter_right _ (cellFrontier_subset_closedCell _ _)) rw [← cellFrontier_union_openCell_eq_closedCell, inter_union_distrib_left] apply union_subset (by rfl) rw [(disjoint_skeletonLT_openCell hnm).inter_eq] exact empty_subset _ /-- Version of `skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier` using `skeleton`. -/ lemma RelCWComplex.skeleton_inter_closedCell_eq_skeleton_inter_cellFrontier [RelCWComplex C D] {n : ℕ∞} {m : ℕ} {j : cell C m} (hnm : n < m) : (skeleton C n : Set X) ∩ closedCell m j = (skeleton C n : Set X) ∩ cellFrontier m j := skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier (Order.add_one_le_of_lt hnm) end skeleton lemma RelCWComplex.disjoint_interior_base_closedCell [T2Space X] [RelCWComplex C D] {n : ℕ} {j : cell C n} : Disjoint (interior D) (closedCell n j) := by rw [disjoint_iff_inter_eq_empty] by_contra! h rw [← closure_openCell_eq_closedCell, inter_comm, closure_inter_open_nonempty_iff isOpen_interior] at h rcases h with ⟨x, xmemcell, xmemD⟩ suffices x ∈ (skeletonLT C 0 : Set X) ∩ openCell n j by rwa [(disjoint_skeletonLT_openCell n.cast_nonneg').inter_eq] at this exact ⟨(skeletonLT C 0).base_subset (interior_subset xmemD), xmemcell⟩ lemma RelCWComplex.disjoint_interior_base_iUnion_closedCell [T2Space X] [RelCWComplex C D] : Disjoint (interior D) (⋃ (n : ℕ) (j : cell C n), closedCell n j) := by simp_rw [disjoint_iff_inter_eq_empty, inter_iUnion, disjoint_interior_base_closedCell.inter_eq, iUnion_empty] namespace CWComplex export RelCWComplex (pairwiseDisjoint disjoint_openCell_of_ne openCell_subset_closedCell cellFrontier_subset_closedCell cellFrontier_union_openCell_eq_closedCell map_zero_mem_openCell map_zero_mem_closedCell isCompact_closedCell isClosed_closedCell isCompact_cellFrontier isClosed_cellFrontier closure_openCell_eq_closedCell skeletonLT_top skeleton_top skeletonLT_mono skeleton_mono skeletonLT_monotone skeleton_monotone closedCell_subset_skeletonLT closedCell_subset_skeleton closedCell_subset_complex openCell_subset_skeletonLT openCell_subset_skeleton openCell_subset_complex cellFrontier_subset_skeletonLT cellFrontier_subset_skeleton cellFrontier_subset_complex iUnion_cellFrontier_subset_skeletonLT iUnion_cellFrontier_subset_skeleton closedCell_zero_eq_singleton openCell_zero_eq_singleton cellFrontier_zero_eq_empty isClosed skeletonLT_union_iUnion_closedCell_eq_skeletonLT_succ skeleton_union_iUnion_closedCell_eq_skeleton_succ iUnion_skeletonLT_eq_complex iUnion_skeleton_eq_complex eq_of_not_disjoint_openCell disjoint_skeletonLT_openCell disjoint_skeleton_openCell skeletonLT_inter_closedCell_eq_skeletonLT_inter_cellFrontier skeleton_inter_closedCell_eq_skeleton_inter_cellFrontier) end CWComplex end Topology
.lake/packages/mathlib/Mathlib/Topology/LocallyConstant/Basic.lean	import Mathlib.Algebra.Notation.Indicator import Mathlib.Tactic.FinCases import Mathlib.Topology.Connected.LocallyConnected import Mathlib.Topology.Sets.Closeds /-! # Locally constant functions This file sets up the theory of locally constant function from a topological space to a type. ## Main definitions and constructions * `IsLocallyConstant f` : a map `f : X → Y` where `X` is a topological space is locally constant if every set in `Y` has an open preimage. * `LocallyConstant X Y` : the type of locally constant maps from `X` to `Y` * `LocallyConstant.map` : push-forward of locally constant maps * `LocallyConstant.comap` : pull-back of locally constant maps -/ variable {X Y Z α : Type} [TopologicalSpace X] open Set Filter open scoped Topology /-- A function between topological spaces is locally constant if the preimage of any set is open. -/ def IsLocallyConstant (f : X → Y) : Prop := ∀ s : Set Y, IsOpen (f ⁻¹' s) namespace IsLocallyConstant open List in protected theorem tfae (f : X → Y) : TFAE [IsLocallyConstant f, ∀ x, ∀ᶠ x' in 𝓝 x, f x' = f x, ∀ x, IsOpen { x' \| f x' = f x }, ∀ y, IsOpen (f ⁻¹' {y}), ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by tfae_have 1 → 4 := fun h y => h {y} tfae_have 4 → 3 := fun h x => h (f x) tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl tfae_have 2 → 5 \| h, x => by rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩ exact ⟨U, hU, hx, eq⟩ tfae_have 5 → 1 \| h, s => by refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_ rcases h x with ⟨U, hU, hxU, eq⟩ exact ⟨U, fun x' hx' => mem_preimage.2 <\| (eq x' hx').symm ▸ hx, hU, hxU⟩ tfae_finish @[nontriviality] theorem of_discrete [DiscreteTopology X] (f : X → Y) : IsLocallyConstant f := fun _ => isOpen_discrete _ theorem isOpen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsOpen { x \| f x = y } := hf {y} theorem isClosed_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClosed { x \| f x = y } := ⟨hf {y}ᶜ⟩ theorem isClopen_fiber {f : X → Y} (hf : IsLocallyConstant f) (y : Y) : IsClopen { x \| f x = y } := ⟨isClosed_fiber hf _, isOpen_fiber hf _⟩ theorem iff_exists_open (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x := (IsLocallyConstant.tfae f).out 0 4 theorem iff_eventually_eq (f : X → Y) : IsLocallyConstant f ↔ ∀ x, ∀ᶠ y in 𝓝 x, f y = f x := (IsLocallyConstant.tfae f).out 0 1 theorem exists_open {f : X → Y} (hf : IsLocallyConstant f) (x : X) : ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x := (iff_exists_open f).1 hf x protected theorem eventually_eq {f : X → Y} (hf : IsLocallyConstant f) (x : X) : ∀ᶠ y in 𝓝 x, f y = f x := (iff_eventually_eq f).1 hf x theorem iff_isOpen_fiber_apply {f : X → Y} : IsLocallyConstant f ↔ ∀ x, IsOpen (f ⁻¹' {f x}) := (IsLocallyConstant.tfae f).out 0 2 theorem iff_isOpen_fiber {f : X → Y} : IsLocallyConstant f ↔ ∀ y, IsOpen (f ⁻¹' {y}) := (IsLocallyConstant.tfae f).out 0 3 protected theorem continuous [TopologicalSpace Y] {f : X → Y} (hf : IsLocallyConstant f) : Continuous f := ⟨fun _ _ => hf _⟩ theorem iff_continuous {_ : TopologicalSpace Y} [DiscreteTopology Y] (f : X → Y) : IsLocallyConstant f ↔ Continuous f := ⟨IsLocallyConstant.continuous, fun h s => h.isOpen_preimage s (isOpen_discrete _)⟩ theorem of_constant (f : X → Y) (h : ∀ x y, f x = f y) : IsLocallyConstant f := (iff_eventually_eq f).2 fun _ => Eventually.of_forall fun _ => h _ _ protected theorem const (y : Y) : IsLocallyConstant (Function.const X y) := of_constant _ fun _ _ => rfl protected theorem comp {f : X → Y} (hf : IsLocallyConstant f) (g : Y → Z) : IsLocallyConstant (g ∘ f) := fun s => by rw [Set.preimage_comp] exact hf _ theorem prodMk {Y'} {f : X → Y} {f' : X → Y'} (hf : IsLocallyConstant f) (hf' : IsLocallyConstant f') : IsLocallyConstant fun x => (f x, f' x) := (iff_eventually_eq _).2 fun x => (hf.eventually_eq x).mp <\| (hf'.eventually_eq x).mono fun _ hf' hf => Prod.ext hf hf' theorem comp₂ {Y₁ Y₂ Z : Type} {f : X → Y₁} {g : X → Y₂} (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) (h : Y₁ → Y₂ → Z) : IsLocallyConstant fun x => h (f x) (g x) := (hf.prodMk hg).comp fun x : Y₁ × Y₂ => h x.1 x.2 theorem comp_continuous [TopologicalSpace Y] {g : Y → Z} {f : X → Y} (hg : IsLocallyConstant g) (hf : Continuous f) : IsLocallyConstant (g ∘ f) := fun s => by rw [Set.preimage_comp] exact hf.isOpen_preimage _ (hg _) /-- A locally constant function is constant on any preconnected set. -/ theorem apply_eq_of_isPreconnected {f : X → Y} (hf : IsLocallyConstant f) {s : Set X} (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by let U := f ⁻¹' {f y} suffices x ∉ Uᶜ from Classical.not_not.1 this intro hxV specialize hs U Uᶜ (hf {f y}) (hf {f y}ᶜ) _ ⟨y, ⟨hy, rfl⟩⟩ ⟨x, ⟨hx, hxV⟩⟩ · simp only [union_compl_self, subset_univ] · simp only [inter_empty, Set.not_nonempty_empty, inter_compl_self] at hs theorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x y : X) : f x = f y := hf.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial theorem eq_const [PreconnectedSpace X] {f : X → Y} (hf : IsLocallyConstant f) (x : X) : f = Function.const X (f x) := funext fun y => hf.apply_eq_of_preconnectedSpace y x theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] {f : X → Y} (hf : IsLocallyConstant f) : ∃ y, f = Function.const X y := by rcases isEmpty_or_nonempty X with h \| h · exact ⟨Classical.arbitrary Y, funext <\| h.elim⟩ · exact ⟨f (Classical.arbitrary X), hf.eq_const _⟩ theorem iff_is_const [PreconnectedSpace X] {f : X → Y} : IsLocallyConstant f ↔ ∀ x y, f x = f y := ⟨fun h _ _ => h.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial, of_constant _⟩ theorem range_finite [CompactSpace X] {f : X → Y} (hf : IsLocallyConstant f) : (Set.range f).Finite := by letI : TopologicalSpace Y := ⊥; haveI := discreteTopology_bot Y exact (isCompact_range hf.continuous).finite_of_discrete @[to_additive] theorem one [One Y] : IsLocallyConstant (1 : X → Y) := IsLocallyConstant.const 1 @[to_additive] theorem inv [Inv Y] ⦃f : X → Y⦄ (hf : IsLocallyConstant f) : IsLocallyConstant f⁻¹ := hf.comp fun x => x⁻¹ @[to_additive] theorem mul [Mul Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) : IsLocallyConstant (f * g) := hf.comp₂ hg (· * ·) @[to_additive] theorem div [Div Y] ⦃f g : X → Y⦄ (hf : IsLocallyConstant f) (hg : IsLocallyConstant g) : IsLocallyConstant (f / g) := hf.comp₂ hg (· / ·) /-- If a composition of a function `f` followed by an injection `g` is locally constant, then the locally constant property descends to `f`. -/ theorem desc {α β : Type} (f : X → α) (g : α → β) (h : IsLocallyConstant (g ∘ f)) (inj : Function.Injective g) : IsLocallyConstant f := fun s => by rw [← preimage_image_eq s inj, preimage_preimage] exact h (g '' s) theorem of_constant_on_connected_components [LocallyConnectedSpace X] {f : X → Y} (h : ∀ x, ∀ y ∈ connectedComponent x, f y = f x) : IsLocallyConstant f := (iff_exists_open _).2 fun x => ⟨connectedComponent x, isOpen_connectedComponent, mem_connectedComponent, h x⟩ theorem of_constant_on_connected_clopens [LocallyConnectedSpace X] {f : X → Y} (h : ∀ U : Set X, IsConnected U → IsClopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x) : IsLocallyConstant f := of_constant_on_connected_components fun x => h (connectedComponent x) isConnected_connectedComponent isClopen_connectedComponent x mem_connectedComponent theorem of_constant_on_preconnected_clopens [LocallyConnectedSpace X] {f : X → Y} (h : ∀ U : Set X, IsPreconnected U → IsClopen U → ∀ x ∈ U, ∀ y ∈ U, f y = f x) : IsLocallyConstant f := of_constant_on_connected_clopens fun U hU ↦ h U hU.isPreconnected end IsLocallyConstant /-- A (bundled) locally constant function from a topological space `X` to a type `Y`. -/ structure LocallyConstant (X Y : Type) [TopologicalSpace X] where /-- The underlying function. -/ protected toFun : X → Y /-- The map is locally constant. -/ protected isLocallyConstant : IsLocallyConstant toFun namespace LocallyConstant instance [Inhabited Y] : Inhabited (LocallyConstant X Y) := ⟨⟨_, IsLocallyConstant.const default⟩⟩ instance : FunLike (LocallyConstant X Y) X Y where coe := LocallyConstant.toFun coe_injective' := by rintro ⟨_, _⟩ ⟨_, _⟩ _; congr /-- See Note [custom simps projections]. -/ def Simps.apply (f : LocallyConstant X Y) : X → Y := f initialize_simps_projections LocallyConstant (toFun → apply) @[simp] theorem toFun_eq_coe (f : LocallyConstant X Y) : f.toFun = f := rfl @[simp] theorem coe_mk (f : X → Y) (h) : ⇑(⟨f, h⟩ : LocallyConstant X Y) = f := rfl protected theorem congr_fun {f g : LocallyConstant X Y} (h : f = g) (x : X) : f x = g x := DFunLike.congr_fun h x protected theorem congr_arg (f : LocallyConstant X Y) {x y : X} (h : x = y) : f x = f y := DFunLike.congr_arg f h theorem coe_injective : @Function.Injective (LocallyConstant X Y) (X → Y) (↑) := fun _ _ => DFunLike.ext' @[norm_cast] theorem coe_inj {f g : LocallyConstant X Y} : (f : X → Y) = g ↔ f = g := coe_injective.eq_iff @[ext] theorem ext ⦃f g : LocallyConstant X Y⦄ (h : ∀ x, f x = g x) : f = g := DFunLike.ext _ _ h section CodomainTopologicalSpace variable [TopologicalSpace Y] (f : LocallyConstant X Y) protected theorem continuous : Continuous f := f.isLocallyConstant.continuous /-- We can turn a locally-constant function into a bundled `ContinuousMap`. -/ @[coe] def toContinuousMap : C(X, Y) := ⟨f, f.continuous⟩ /-- As a shorthand, `LocallyConstant.toContinuousMap` is available as a coercion -/ instance : Coe (LocallyConstant X Y) C(X, Y) := ⟨toContinuousMap⟩ @[simp] theorem coe_continuousMap : ((f : C(X, Y)) : X → Y) = (f : X → Y) := rfl theorem toContinuousMap_injective : Function.Injective (toContinuousMap : LocallyConstant X Y → C(X, Y)) := fun _ _ h => ext (ContinuousMap.congr_fun h) end CodomainTopologicalSpace /-- The constant locally constant function on `X` with value `y : Y`. -/ def const (X : Type) {Y : Type} [TopologicalSpace X] (y : Y) : LocallyConstant X Y := ⟨Function.const X y, IsLocallyConstant.const _⟩ @[simp] theorem coe_const (y : Y) : (const X y : X → Y) = Function.const X y := rfl /-- Evaluation/projection as a locally constant function. -/ @[simps] def eval {ι : Type} {X : ι → Type} [∀ i, TopologicalSpace (X i)] (i : ι) [DiscreteTopology (X i)] : LocallyConstant (Π i, X i) (X i) where toFun := fun f ↦ f i isLocallyConstant := (IsLocallyConstant.iff_continuous _).mpr <\| continuous_apply i /-- The locally constant function to `Fin 2` associated to a clopen set. -/ def ofIsClopen {X : Type} [TopologicalSpace X] {U : Set X} [∀ x, Decidable (x ∈ U)] (hU : IsClopen U) : LocallyConstant X (Fin 2) where toFun x := if x ∈ U then 0 else 1 isLocallyConstant := by refine IsLocallyConstant.iff_isOpen_fiber.2 <\| Fin.forall_fin_two.2 ⟨?_, ?_⟩ · convert hU.2 using 1 ext simp only [mem_singleton_iff, Fin.one_eq_zero_iff, mem_preimage, ite_eq_left_iff, Nat.succ_succ_ne_one] tauto · rw [← isClosed_compl_iff] convert hU.1 ext simp @[simp] theorem ofIsClopen_fiber_zero {X : Type} [TopologicalSpace X] {U : Set X} [∀ x, Decidable (x ∈ U)] (hU : IsClopen U) : ofIsClopen hU ⁻¹' ({0} : Set (Fin 2)) = U := by ext simp only [ofIsClopen, mem_singleton_iff, Fin.one_eq_zero_iff, coe_mk, mem_preimage, ite_eq_left_iff, Nat.succ_succ_ne_one] tauto @[simp] theorem ofIsClopen_fiber_one {X : Type} [TopologicalSpace X] {U : Set X} [∀ x, Decidable (x ∈ U)] (hU : IsClopen U) : ofIsClopen hU ⁻¹' ({1} : Set (Fin 2)) = Uᶜ := by ext simp only [ofIsClopen, mem_singleton_iff, coe_mk, Fin.zero_eq_one_iff, mem_preimage, ite_eq_right_iff, mem_compl_iff, Nat.succ_succ_ne_one] theorem locallyConstant_eq_of_fiber_zero_eq {X : Type} [TopologicalSpace X] (f g : LocallyConstant X (Fin 2)) (h : f ⁻¹' ({0} : Set (Fin 2)) = g ⁻¹' {0}) : f = g := by simp only [Set.ext_iff, mem_singleton_iff, mem_preimage] at h ext1 x exact Fin.fin_two_eq_of_eq_zero_iff (h x) theorem range_finite [CompactSpace X] (f : LocallyConstant X Y) : (Set.range f).Finite := f.isLocallyConstant.range_finite theorem apply_eq_of_isPreconnected (f : LocallyConstant X Y) {s : Set X} (hs : IsPreconnected s) {x y : X} (hx : x ∈ s) (hy : y ∈ s) : f x = f y := f.isLocallyConstant.apply_eq_of_isPreconnected hs hx hy theorem apply_eq_of_preconnectedSpace [PreconnectedSpace X] (f : LocallyConstant X Y) (x y : X) : f x = f y := f.isLocallyConstant.apply_eq_of_isPreconnected isPreconnected_univ trivial trivial theorem eq_const [PreconnectedSpace X] (f : LocallyConstant X Y) (x : X) : f = const X (f x) := ext fun _ => apply_eq_of_preconnectedSpace f _ _ theorem exists_eq_const [PreconnectedSpace X] [Nonempty Y] (f : LocallyConstant X Y) : ∃ y, f = const X y := by rcases Classical.em (Nonempty X) with (⟨⟨x⟩⟩ \| hX) · exact ⟨f x, f.eq_const x⟩ · exact ⟨Classical.arbitrary Y, ext fun x => (hX ⟨x⟩).elim⟩ /-- Push forward of locally constant maps under any map, by post-composition. -/ def map (f : Y → Z) (g : LocallyConstant X Y) : LocallyConstant X Z := ⟨f ∘ g, g.isLocallyConstant.comp f⟩ @[simp] theorem map_apply (f : Y → Z) (g : LocallyConstant X Y) : ⇑(map f g) = f ∘ g := rfl @[simp] theorem map_id : @map X Y Y _ id = id := rfl @[simp] theorem map_comp {Y₁ Y₂ Y₃ : Type} (g : Y₂ → Y₃) (f : Y₁ → Y₂) : @map X _ _ _ g ∘ map f = map (g ∘ f) := rfl /-- Given a locally constant function to `α → β`, construct a family of locally constant functions with values in β indexed by α. -/ def flip {X α β : Type} [TopologicalSpace X] (f : LocallyConstant X (α → β)) (a : α) : LocallyConstant X β := f.map fun f => f a /-- If α is finite, this constructs a locally constant function to `α → β` given a family of locally constant functions with values in β indexed by α. -/ def unflip {X α β : Type} [Finite α] [TopologicalSpace X] (f : α → LocallyConstant X β) : LocallyConstant X (α → β) where toFun x a := f a x isLocallyConstant := IsLocallyConstant.iff_isOpen_fiber.2 fun g => by have : (fun (x : X) (a : α) => f a x) ⁻¹' {g} = ⋂ a : α, f a ⁻¹' {g a} := by ext; simp [funext_iff] rw [this] exact isOpen_iInter_of_finite fun a => (f a).isLocallyConstant _ @[simp] theorem unflip_flip {X α β : Type} [Finite α] [TopologicalSpace X] (f : LocallyConstant X (α → β)) : unflip f.flip = f := rfl @[simp] theorem flip_unflip {X α β : Type} [Finite α] [TopologicalSpace X] (f : α → LocallyConstant X β) : (unflip f).flip = f := rfl section Comap variable [TopologicalSpace Y] /-- Pull back of locally constant maps under a continuous map, by pre-composition. -/ def comap (f : C(X, Y)) (g : LocallyConstant Y Z) : LocallyConstant X Z := ⟨g ∘ f, g.isLocallyConstant.comp_continuous f.continuous⟩ @[simp] theorem coe_comap (f : C(X, Y)) (g : LocallyConstant Y Z) : (comap f g) = g ∘ f := rfl theorem coe_comap_apply (f : C(X, Y)) (g : LocallyConstant Y Z) (x : X) : comap f g x = g (f x) := rfl @[simp] theorem comap_id : comap (@ContinuousMap.id X _) = @id (LocallyConstant X Z) := rfl theorem comap_comp {W : Type} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) : comap (Z := Z) (g.comp f) = comap f ∘ comap g := rfl theorem comap_comap {W : Type} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) (x : LocallyConstant Y Z) : comap f (comap g x) = comap (g.comp f) x := rfl theorem comap_const (f : C(X, Y)) (y : Y) (h : ∀ x, f x = y) : (comap f : LocallyConstant Y Z → LocallyConstant X Z) = fun g => const X (g y) := by ext; simp [h] lemma comap_injective (f : C(X, Y)) (hfs : f.1.Surjective) : (comap (Z := Z) f).Injective := by intro a b h ext y obtain ⟨x, hx⟩ := hfs y simpa [← hx] using LocallyConstant.congr_fun h x end Comap section Desc /-- If a locally constant function factors through an injection, then it factors through a locally constant function. -/ def desc {X α β : Type} [TopologicalSpace X] {g : α → β} (f : X → α) (h : LocallyConstant X β) (cond : g ∘ f = h) (inj : Function.Injective g) : LocallyConstant X α where toFun := f isLocallyConstant := IsLocallyConstant.desc _ g (cond.symm ▸ h.isLocallyConstant) inj @[simp] theorem coe_desc {X α β : Type} [TopologicalSpace X] (f : X → α) (g : α → β) (h : LocallyConstant X β) (cond : g ∘ f = h) (inj : Function.Injective g) : ⇑(desc f h cond inj) = f := rfl end Desc section Indicator variable {R : Type} [One R] {U : Set X} (f : LocallyConstant X R) /-- Given a clopen set `U` and a locally constant function `f`, `LocallyConstant.mulIndicator` returns the locally constant function that is `f` on `U` and `1` otherwise. -/ @[to_additive (attr := simps) /-- Given a clopen set `U` and a locally constant function `f`, `LocallyConstant.indicator` returns the locally constant function that is `f` on `U` and `0` otherwise. -/] noncomputable def mulIndicator (hU : IsClopen U) : LocallyConstant X R where toFun := Set.mulIndicator U f isLocallyConstant := fun s => by rw [mulIndicator_preimage, Set.ite, Set.diff_eq] exact ((f.2 s).inter hU.isOpen).union ((IsLocallyConstant.const 1 s).inter hU.compl.isOpen) variable (a : X) open Classical in @[to_additive] theorem mulIndicator_apply_eq_if (hU : IsClopen U) : mulIndicator f hU a = if a ∈ U then f a else 1 := Set.mulIndicator_apply U f a variable {a} @[to_additive] theorem mulIndicator_of_mem (hU : IsClopen U) (h : a ∈ U) : f.mulIndicator hU a = f a := Set.mulIndicator_of_mem h _ @[to_additive] theorem mulIndicator_of_notMem (hU : IsClopen U) (h : a ∉ U) : f.mulIndicator hU a = 1 := Set.mulIndicator_of_notMem h _ @[deprecated (since := "2025-05-23")] alias indicator_of_not_mem := indicator_of_notMem @[to_additive existing, deprecated (since := "2025-05-23")] alias mulIndicator_of_not_mem := mulIndicator_of_notMem end Indicator section Equiv /-- The equivalence between `LocallyConstant X Z` and `LocallyConstant Y Z` given a homeomorphism `X ≃ₜ Y` -/ @[simps] def congrLeft [TopologicalSpace Y] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ LocallyConstant Y Z where toFun := comap e.symm invFun := comap e left_inv := by intro simp [comap_comap] right_inv := by intro simp [comap_comap] /-- The equivalence between `LocallyConstant X Y` and `LocallyConstant X Z` given an equivalence `Y ≃ Z` -/ @[simps] def congrRight (e : Y ≃ Z) : LocallyConstant X Y ≃ LocallyConstant X Z where toFun := map e invFun := map e.symm left_inv := by intro; ext; simp right_inv := by intro; ext; simp variable (X) in /-- The set of clopen subsets of a topological space is equivalent to the locally constant maps to a two-element set -/ def equivClopens [∀ (s : Set X) x, Decidable (x ∈ s)] : LocallyConstant X (Fin 2) ≃ TopologicalSpace.Clopens X where toFun f := ⟨f ⁻¹' {0}, f.2.isClopen_fiber _⟩ invFun s := ofIsClopen s.2 left_inv _ := locallyConstant_eq_of_fiber_zero_eq _ _ (by simp) right_inv _ := by simp end Equiv section Piecewise /-- Given two closed sets covering a topological space, and locally constant maps on these two sets, then if these two locally constant maps agree on the intersection, we get a piecewise defined locally constant map on the whole space. TODO: Generalise this construction to `ContinuousMap`. -/ def piecewise {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z) (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩) [DecidablePred (· ∈ C₁)] : LocallyConstant X Z where toFun i := if hi : i ∈ C₁ then f ⟨i, hi⟩ else g ⟨i, (Set.compl_subset_iff_union.mpr h) hi⟩ isLocallyConstant := by let dZ : TopologicalSpace Z := ⊥ haveI : DiscreteTopology Z := discreteTopology_bot Z obtain ⟨f, hf⟩ := f obtain ⟨g, hg⟩ := g rw [IsLocallyConstant.iff_continuous] at hf hg ⊢ dsimp only [coe_mk] rw [Set.union_eq_iUnion] at h refine (locallyFinite_of_finite _).continuous h (fun i ↦ ?_) (fun i ↦ ?_) · cases i <;> [exact h₂; exact h₁] · cases i <;> rw [continuousOn_iff_continuous_restrict] · convert hg ext x simp only [cond_false, restrict_apply, Subtype.coe_eta, dite_eq_right_iff] exact fun hx ↦ hfg x ⟨hx, x.prop⟩ · simp only [cond_true, restrict_dite, Subtype.coe_eta] exact hf @[simp] lemma piecewise_apply_left {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z) (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩) [DecidablePred (· ∈ C₁)] (x : X) (hx : x ∈ C₁) : piecewise h₁ h₂ h f g hfg x = f ⟨x, hx⟩ := by simp only [piecewise, coe_mk] rw [dif_pos hx] @[simp] lemma piecewise_apply_right {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z) (hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩) [DecidablePred (· ∈ C₁)] (x : X) (hx : x ∈ C₂) : piecewise h₁ h₂ h f g hfg x = g ⟨x, hx⟩ := by simp only [piecewise, coe_mk] split_ifs with h · exact hfg x ⟨h, hx⟩ · rfl /-- A variant of `LocallyConstant.piecewise` where the two closed sets cover a subset. TODO: Generalise this construction to `ContinuousMap`. -/ def piecewise' {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z) [DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩) : LocallyConstant C₀ Z := letI : ∀ j : C₀, Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j ↦ decidable_of_iff (↑j ∈ C₁) Iff.rfl piecewise (h₁.preimage continuous_subtype_val) (h₂.preimage continuous_subtype_val) (by simpa [eq_univ_iff_forall] using h₀) (f₁.comap ⟨(restrictPreimage C₁ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩) (f₂.comap ⟨(restrictPreimage C₂ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩) <\| by rintro ⟨x, hx₀⟩ ⟨hx₁ : x ∈ C₁, hx₂ : x ∈ C₂⟩ simpa using hf x ⟨hx₁, hx₂⟩ @[simp] lemma piecewise'_apply_left {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z) [DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩) (x : C₀) (hx : x.val ∈ C₁) : piecewise' h₀ h₁ h₂ f₁ f₂ hf x = f₁ ⟨x.val, hx⟩ := by letI : ∀ j : C₀, Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j ↦ decidable_of_iff (↑j ∈ C₁) Iff.rfl rw [piecewise', piecewise_apply_left (f := (f₁.comap ⟨(restrictPreimage C₁ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩)) (hx := hx)] rfl @[simp] lemma piecewise'_apply_right {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z) [DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩) (x : C₀) (hx : x.val ∈ C₂) : piecewise' h₀ h₁ h₂ f₁ f₂ hf x = f₂ ⟨x.val, hx⟩ := by letI : ∀ j : C₀, Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j ↦ decidable_of_iff (↑j ∈ C₁) Iff.rfl rw [piecewise', piecewise_apply_right (f := (f₁.comap ⟨(restrictPreimage C₁ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩)) (hx := hx)] rfl end Piecewise end LocallyConstant
.lake/packages/mathlib/Mathlib/Topology/LocallyConstant/Algebra.lean	import Mathlib.Algebra.Algebra.Pi import Mathlib.Algebra.GroupWithZero.Indicator import Mathlib.LinearAlgebra.Pi import Mathlib.Topology.LocallyConstant.Basic /-! # Algebraic structure on locally constant functions This file puts algebraic structure (`Group`, `AddGroup`, etc) on the type of locally constant functions. -/ namespace LocallyConstant variable {X Y : Type} [TopologicalSpace X] @[to_additive] instance [One Y] : One (LocallyConstant X Y) where one := const X 1 @[to_additive (attr := simp)] theorem coe_one [One Y] : ⇑(1 : LocallyConstant X Y) = (1 : X → Y) := rfl @[to_additive] theorem one_apply [One Y] (x : X) : (1 : LocallyConstant X Y) x = 1 := rfl @[to_additive] instance [Inv Y] : Inv (LocallyConstant X Y) where inv f := ⟨f⁻¹, f.isLocallyConstant.inv⟩ @[to_additive (attr := simp)] theorem coe_inv [Inv Y] (f : LocallyConstant X Y) : ⇑(f⁻¹ : LocallyConstant X Y) = (f : X → Y)⁻¹ := rfl @[to_additive] theorem inv_apply [Inv Y] (f : LocallyConstant X Y) (x : X) : f⁻¹ x = (f x)⁻¹ := rfl @[to_additive] instance [Mul Y] : Mul (LocallyConstant X Y) where mul f g := ⟨f g, f.isLocallyConstant.mul g.isLocallyConstant⟩ @[to_additive (attr := simp)] theorem coe_mul [Mul Y] (f g : LocallyConstant X Y) : ⇑(f * g) = f * g := rfl @[to_additive] theorem mul_apply [Mul Y] (f g : LocallyConstant X Y) (x : X) : (f * g) x = f x * g x := rfl @[to_additive] instance [MulOneClass Y] : MulOneClass (LocallyConstant X Y) := Function.Injective.mulOneClass DFunLike.coe DFunLike.coe_injective' rfl fun _ _ => rfl /-- `DFunLike.coe` as a `MonoidHom`. -/ @[to_additive (attr := simps) /-- `DFunLike.coe` as an `AddMonoidHom`. -/] def coeFnMonoidHom [MulOneClass Y] : LocallyConstant X Y →* X → Y where toFun := DFunLike.coe map_one' := rfl map_mul' _ _ := rfl /-- The constant-function embedding, as a multiplicative monoid hom. -/ @[to_additive (attr := simps) /-- The constant-function embedding, as an additive monoid hom. -/] def constMonoidHom [MulOneClass Y] : Y →* LocallyConstant X Y where toFun := const X map_one' := rfl map_mul' _ _ := rfl instance [MulZeroClass Y] : MulZeroClass (LocallyConstant X Y) := Function.Injective.mulZeroClass DFunLike.coe DFunLike.coe_injective' rfl fun _ _ => rfl instance [MulZeroOneClass Y] : MulZeroOneClass (LocallyConstant X Y) := Function.Injective.mulZeroOneClass DFunLike.coe DFunLike.coe_injective' rfl rfl fun _ _ => rfl section CharFn variable (Y) [MulZeroOneClass Y] {U V : Set X} /-- Characteristic functions are locally constant functions taking `x : X` to `1` if `x ∈ U`, where `U` is a clopen set, and `0` otherwise. -/ noncomputable def charFn (hU : IsClopen U) : LocallyConstant X Y := indicator 1 hU theorem coe_charFn (hU : IsClopen U) : (charFn Y hU : X → Y) = Set.indicator U 1 := rfl theorem charFn_eq_one [Nontrivial Y] (x : X) (hU : IsClopen U) : charFn Y hU x = (1 : Y) ↔ x ∈ U := Set.indicator_eq_one_iff_mem _ theorem charFn_eq_zero [Nontrivial Y] (x : X) (hU : IsClopen U) : charFn Y hU x = (0 : Y) ↔ x ∉ U := Set.indicator_eq_zero_iff_notMem _ theorem charFn_inj [Nontrivial Y] (hU : IsClopen U) (hV : IsClopen V) (h : charFn Y hU = charFn Y hV) : U = V := Set.indicator_one_inj Y <\| coe_inj.mpr h end CharFn @[to_additive] instance [Div Y] : Div (LocallyConstant X Y) where div f g := ⟨f / g, f.isLocallyConstant.div g.isLocallyConstant⟩ @[to_additive] theorem coe_div [Div Y] (f g : LocallyConstant X Y) : ⇑(f / g) = f / g := rfl @[to_additive] theorem div_apply [Div Y] (f g : LocallyConstant X Y) (x : X) : (f / g) x = f x / g x := rfl @[to_additive] instance [Semigroup Y] : Semigroup (LocallyConstant X Y) := Function.Injective.semigroup DFunLike.coe DFunLike.coe_injective' fun _ _ => rfl instance [SemigroupWithZero Y] : SemigroupWithZero (LocallyConstant X Y) := Function.Injective.semigroupWithZero DFunLike.coe DFunLike.coe_injective' rfl fun _ _ => rfl @[to_additive] instance [CommSemigroup Y] : CommSemigroup (LocallyConstant X Y) := Function.Injective.commSemigroup DFunLike.coe DFunLike.coe_injective' fun _ _ => rfl variable {α R : Type} @[to_additive] instance smul [SMul α Y] : SMul α (LocallyConstant X Y) where smul n f := f.map (n • ·) @[to_additive (attr := simp)] theorem coe_smul [SMul R Y] (r : R) (f : LocallyConstant X Y) : ⇑(r • f) = r • (f : X → Y) := rfl @[to_additive] theorem smul_apply [SMul R Y] (r : R) (f : LocallyConstant X Y) (x : X) : (r • f) x = r • f x := rfl @[to_additive existing LocallyConstant.smul] instance [Pow Y α] : Pow (LocallyConstant X Y) α where pow f n := f.map (· ^ n) @[to_additive] instance [Monoid Y] : Monoid (LocallyConstant X Y) := Function.Injective.monoid DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) fun _ _ => rfl instance [NatCast Y] : NatCast (LocallyConstant X Y) where natCast n := const X n instance [IntCast Y] : IntCast (LocallyConstant X Y) where intCast n := const X n instance [AddMonoidWithOne Y] : AddMonoidWithOne (LocallyConstant X Y) := Function.Injective.addMonoidWithOne DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) fun _ => rfl @[to_additive] instance [CommMonoid Y] : CommMonoid (LocallyConstant X Y) := Function.Injective.commMonoid DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) fun _ _ => rfl @[to_additive] instance [Group Y] : Group (LocallyConstant X Y) := Function.Injective.group DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) @[to_additive] instance [CommGroup Y] : CommGroup (LocallyConstant X Y) := Function.Injective.commGroup DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [Distrib Y] : Distrib (LocallyConstant X Y) := Function.Injective.distrib DFunLike.coe DFunLike.coe_injective' (fun _ _ => rfl) fun _ _ => rfl instance [NonUnitalNonAssocSemiring Y] : NonUnitalNonAssocSemiring (LocallyConstant X Y) := Function.Injective.nonUnitalNonAssocSemiring DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [NonUnitalSemiring Y] : NonUnitalSemiring (LocallyConstant X Y) := Function.Injective.nonUnitalSemiring DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [NonAssocSemiring Y] : NonAssocSemiring (LocallyConstant X Y) := Function.Injective.nonAssocSemiring DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) fun _ => rfl /-- The constant-function embedding, as a ring hom. -/ @[simps] def constRingHom [NonAssocSemiring Y] : Y →+ LocallyConstant X Y := { constMonoidHom, constAddMonoidHom with toFun := const X } instance [Semiring Y] : Semiring (LocallyConstant X Y) := Function.Injective.semiring DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) fun _ => rfl instance [NonUnitalCommSemiring Y] : NonUnitalCommSemiring (LocallyConstant X Y) := Function.Injective.nonUnitalCommSemiring DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [CommSemiring Y] : CommSemiring (LocallyConstant X Y) := Function.Injective.commSemiring DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) fun _ => rfl instance [NonUnitalNonAssocRing Y] : NonUnitalNonAssocRing (LocallyConstant X Y) := Function.Injective.nonUnitalNonAssocRing DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [NonUnitalRing Y] : NonUnitalRing (LocallyConstant X Y) := Function.Injective.nonUnitalRing DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [NonAssocRing Y] : NonAssocRing (LocallyConstant X Y) := Function.Injective.nonAssocRing DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ => rfl) instance [Ring Y] : Ring (LocallyConstant X Y) := Function.Injective.ring DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) fun _ => rfl instance [NonUnitalCommRing Y] : NonUnitalCommRing (LocallyConstant X Y) := Function.Injective.nonUnitalCommRing DFunLike.coe DFunLike.coe_injective' rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) instance [CommRing Y] : CommRing (LocallyConstant X Y) := Function.Injective.commRing DFunLike.coe DFunLike.coe_injective' rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) fun _ => rfl variable {R : Type} instance [Monoid R] [MulAction R Y] : MulAction R (LocallyConstant X Y) := Function.Injective.mulAction _ coe_injective fun _ _ => rfl instance [Monoid R] [AddMonoid Y] [DistribMulAction R Y] : DistribMulAction R (LocallyConstant X Y) := Function.Injective.distribMulAction coeFnAddMonoidHom coe_injective fun _ _ => rfl instance [Semiring R] [AddCommMonoid Y] [Module R Y] : Module R (LocallyConstant X Y) := Function.Injective.module R coeFnAddMonoidHom coe_injective fun _ _ => rfl section Algebra variable [CommSemiring R] [Semiring Y] [Algebra R Y] instance : Algebra R (LocallyConstant X Y) where algebraMap := constRingHom.comp <\| algebraMap R Y commutes' := by intros ext exact Algebra.commutes' _ _ smul_def' := by intros ext exact Algebra.smul_def' _ _ @[simp] theorem coe_algebraMap (r : R) : ⇑(algebraMap R (LocallyConstant X Y) r) = algebraMap R (X → Y) r := rfl end Algebra section coeFn /-- `DFunLike.coe` as a `RingHom`. -/ @[simps!] def coeFnRingHom [Semiring Y] : LocallyConstant X Y →+ X → Y where toMonoidHom := coeFnMonoidHom __ := coeFnAddMonoidHom /-- `DFunLike.coe` as a linear map. -/ @[simps!] def coeFnₗ (R : Type) [Semiring R] [AddCommMonoid Y] [Module R Y] : LocallyConstant X Y →ₗ[R] X → Y where toAddHom := coeFnAddMonoidHom.toAddHom map_smul' _ _ := rfl /-- `DFunLike.coe` as an `AlgHom`. -/ @[simps!] def coeFnAlgHom (R : Type) [CommSemiring R] [Semiring Y] [Algebra R Y] : LocallyConstant X Y →ₐ[R] X → Y where toRingHom := coeFnRingHom commutes' _ := rfl end coeFn section Eval /-- Evaluation as a `MonoidHom` -/ @[to_additive (attr := simps!) /-- Evaluation as an `AddMonoidHom` -/] def evalMonoidHom [MulOneClass Y] (x : X) : LocallyConstant X Y →* Y := (Pi.evalMonoidHom _ x).comp coeFnMonoidHom /-- Evaluation as a linear map -/ @[simps!] def evalₗ (R : Type) [Semiring R] [AddCommMonoid Y] [Module R Y] (x : X) : LocallyConstant X Y →ₗ[R] Y := (LinearMap.proj x).comp (coeFnₗ R) /-- Evaluation as a `RingHom` -/ @[simps!] def evalRingHom [Semiring Y] (x : X) : LocallyConstant X Y →+ Y := (Pi.evalRingHom _ x).comp coeFnRingHom /-- Evaluation as an `AlgHom` -/ @[simps!] def evalₐ (R : Type) [CommSemiring R] [Semiring Y] [Algebra R Y] (x : X) : LocallyConstant X Y →ₐ[R] Y := (Pi.evalAlgHom _ _ x).comp (coeFnAlgHom R) end Eval section Comap variable [TopologicalSpace Y] {Z : Type} /-- `LocallyConstant.comap` as a `MonoidHom`. -/ @[to_additive (attr := simps) /-- `LocallyConstant.comap` as an `AddMonoidHom`. -/] def comapMonoidHom [MulOneClass Z] (f : C(X, Y)) : LocallyConstant Y Z →* LocallyConstant X Z where toFun := comap f map_one' := rfl map_mul' _ _ := rfl /-- `LocallyConstant.comap` as a linear map. -/ @[simps!] def comapₗ (R : Type) [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) : LocallyConstant Y Z →ₗ[R] LocallyConstant X Z where toFun := comap f map_add' := map_add (comapAddMonoidHom f) map_smul' _ _ := rfl /-- `LocallyConstant.comap` as a `RingHom`. -/ @[simps!] def comapRingHom [Semiring Z] (f : C(X, Y)) : LocallyConstant Y Z →+ LocallyConstant X Z where toMonoidHom := comapMonoidHom f __ := (comapAddMonoidHom f) /-- `LocallyConstant.comap` as an `AlgHom` -/ @[simps!] def comapₐ (R : Type) [CommSemiring R] [Semiring Z] [Algebra R Z] (f : C(X, Y)) : LocallyConstant Y Z →ₐ[R] LocallyConstant X Z where toRingHom := comapRingHom f commutes' _ := rfl lemma ker_comapₗ [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) (hfs : Function.Surjective f) : LinearMap.ker (comapₗ R f : LocallyConstant Y Z →ₗ[R] LocallyConstant X Z) = ⊥ := LinearMap.ker_eq_bot_of_injective <\| comap_injective _ hfs /-- `LocallyConstant.congrLeft` as a linear equivalence. -/ @[simps!] def congrLeftₗ (R : Type) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ₗ[R] LocallyConstant Y Z where toLinearMap := comapₗ R ⟨_, e.symm.continuous⟩ __ := congrLeft e /-- `LocallyConstant.congrLeft` as a `RingEquiv`. -/ @[simps!] def congrLeftRingEquiv [Semiring Z] (e : X ≃ₜ Y) : LocallyConstant X Z ≃+* LocallyConstant Y Z where toEquiv := congrLeft e __ := comapMonoidHom ⟨_, e.symm.continuous⟩ __ := comapAddMonoidHom ⟨_, e.symm.continuous⟩ /-- `LocallyConstant.congrLeft` as an `AlgEquiv`. -/ @[simps!] def congrLeftₐ (R : Type) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ₐ[R] LocallyConstant Y Z where toEquiv := congrLeft e __ := comapₐ R ⟨_, e.symm.continuous⟩ end Comap section Map variable {Z : Type} /-- `LocallyConstant.map` as a `MonoidHom`. -/ @[to_additive (attr := simps) /-- `LocallyConstant.map` as an `AddMonoidHom`. -/] def mapMonoidHom [MulOneClass Y] [MulOneClass Z] (f : Y →* Z) : LocallyConstant X Y →* LocallyConstant X Z where toFun := map f map_one' := by aesop map_mul' := by aesop /-- `LocallyConstant.map` as a linear map. -/ @[simps!] def mapₗ (R : Type) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (f : Y →ₗ[R] Z) : LocallyConstant X Y →ₗ[R] LocallyConstant X Z where toFun := map f map_add' := by aesop map_smul' := by aesop /-- `LocallyConstant.map` as a `RingHom`. -/ @[simps!] def mapRingHom [Semiring Y] [Semiring Z] (f : Y →+ Z) : LocallyConstant X Y →+* LocallyConstant X Z where toMonoidHom := mapMonoidHom f __ := (mapAddMonoidHom f.toAddMonoidHom) /-- `LocallyConstant.map` as an `AlgHom` -/ @[simps!] def mapₐ (R : Type) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (f : Y →ₐ[R] Z) : LocallyConstant X Y →ₐ[R] LocallyConstant X Z where toRingHom := mapRingHom f commutes' _ := by aesop /-- `LocallyConstant.congrRight` as a linear equivalence. -/ @[simps!] def congrRightₗ (R : Type) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) : LocallyConstant X Y ≃ₗ[R] LocallyConstant X Z where toLinearMap := mapₗ R e __ := congrRight e.toEquiv /-- `LocallyConstant.congrRight` as a `RingEquiv`. -/ @[simps!] def congrRightRingEquiv [Semiring Y] [Semiring Z] (e : Y ≃+* Z) : LocallyConstant X Y ≃+* LocallyConstant X Z where toEquiv := congrRight e __ := mapMonoidHom e.toMonoidHom __ := mapAddMonoidHom e.toAddMonoidHom /-- `LocallyConstant.congrRight` as an `AlgEquiv`. -/ @[simps!] def congrRightₐ (R : Type) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (e : Y ≃ₐ[R] Z) : LocallyConstant X Y ≃ₐ[R] LocallyConstant X Z where toEquiv := congrRight e __ := mapₐ R e.toAlgHom end Map section Const /-- `LocallyConstant.const` as a linear map. -/ @[simps!] def constₗ (R : Type) [Semiring R] [AddCommMonoid Y] [Module R Y] : Y →ₗ[R] LocallyConstant X Y where toFun := const X map_add' _ _ := rfl map_smul' _ _ := rfl /-- `LocallyConstant.const` as an `AlgHom` -/ @[simps!] def constₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] : Y →ₐ[R] LocallyConstant X Y where toRingHom := constRingHom commutes' _ := rfl end Const end LocallyConstant
.lake/packages/mathlib/Mathlib/Topology/Category/Sequential.lean	import Mathlib.CategoryTheory.Elementwise import Mathlib.Topology.Sequences import Mathlib.Topology.Instances.Discrete import Mathlib.Topology.Category.TopCat.Basic /-! # The category of sequential topological spaces We define the category `Sequential` of sequential topological spaces. We follow the usual template for defining categories of topological spaces, by giving it the induced category structure from `TopCat`. -/ open CategoryTheory universe u /-- The type sequential topological spaces. -/ structure Sequential where /-- The underlying topological space of an object of `Sequential`. -/ toTop : TopCat.{u} -- TODO: turn this into `extends` /-- The underlying topological space is sequential. -/ [is_sequential : SequentialSpace toTop] namespace Sequential instance : Inhabited Sequential.{u} := ⟨{ toTop := TopCat.of (ULift (Fin 37)) }⟩ instance : CoeSort Sequential Type* := ⟨fun X => X.toTop⟩ attribute [instance] is_sequential instance : Category.{u, u+1} Sequential.{u} := InducedCategory.category toTop instance : ConcreteCategory.{u} Sequential.{u} (C(·, ·)) := InducedCategory.concreteCategory toTop variable (X : Type u) [TopologicalSpace X] [SequentialSpace X] /-- Constructor for objects of the category `Sequential`. -/ abbrev of : Sequential.{u} where toTop := TopCat.of X is_sequential := ‹_› /-- The fully faithful embedding of `Sequential` in `TopCat`. -/ @[simps!] def sequentialToTop : Sequential.{u} ⥤ TopCat.{u} := inducedFunctor _ /-- The functor to `TopCat` is indeed fully faithful. -/ def fullyFaithfulSequentialToTop : sequentialToTop.FullyFaithful := fullyFaithfulInducedFunctor _ instance : sequentialToTop.{u}.Full := inferInstanceAs (inducedFunctor _).Full instance : sequentialToTop.{u}.Faithful := inferInstanceAs (inducedFunctor _).Faithful /-- Construct an isomorphism from a homeomorphism. -/ @[simps hom inv] def isoOfHomeo {X Y : Sequential.{u}} (f : X ≃ₜ Y) : X ≅ Y where hom := TopCat.ofHom ⟨f, f.continuous⟩ inv := TopCat.ofHom ⟨f.symm, f.symm.continuous⟩ hom_inv_id := by ext x exact f.symm_apply_apply x inv_hom_id := by ext x exact f.apply_symm_apply x /-- Construct a homeomorphism from an isomorphism. -/ @[simps] def homeoOfIso {X Y : Sequential.{u}} (f : X ≅ Y) : X ≃ₜ Y where toFun := f.hom invFun := f.inv left_inv := f.hom_inv_id_apply right_inv := f.inv_hom_id_apply continuous_toFun := f.hom.hom.continuous continuous_invFun := f.inv.hom.continuous /-- The equivalence between isomorphisms in `Sequential` and homeomorphisms of topological spaces. -/ @[simps] def isoEquivHomeo {X Y : Sequential.{u}} : (X ≅ Y) ≃ (X ≃ₜ Y) where toFun := homeoOfIso invFun := isoOfHomeo end Sequential
.lake/packages/mathlib/Mathlib/Topology/Category/CompactlyGenerated.lean	import Mathlib.Topology.Compactness.CompactlyGeneratedSpace import Mathlib.CategoryTheory.Elementwise /-! # Compactly generated topological spaces This file defines the category of compactly generated topological spaces. These are spaces `X` such that a map `f : X → Y` is continuous whenever the composition `S → X → Y` is continuous for all compact Hausdorff spaces `S` mapping continuously to `X`. ## TODO * `CompactlyGenerated` is a reflective subcategory of `TopCat`. * `CompactlyGenerated` is Cartesian closed. * Every first-countable space is `u`-compactly generated for every universe `u`. -/ universe u w open CategoryTheory Topology TopologicalSpace /-- `CompactlyGenerated.{u, w}` is the type of `u`-compactly generated `w`-small topological spaces. This should always be used with explicit universe parameters. -/ structure CompactlyGenerated where /-- The underlying topological space of an object of `CompactlyGenerated`. -/ toTop : TopCat.{w} /-- The underlying topological space is compactly generated. -/ [is_compactly_generated : UCompactlyGeneratedSpace.{u} toTop] namespace CompactlyGenerated instance : Inhabited CompactlyGenerated.{u, w} := ⟨{ toTop := TopCat.of (ULift (Fin 37)) }⟩ instance : CoeSort CompactlyGenerated Type* := ⟨fun X => X.toTop⟩ attribute [instance] is_compactly_generated instance : Category.{w, w+1} CompactlyGenerated.{u, w} := InducedCategory.category toTop instance : ConcreteCategory.{w} CompactlyGenerated.{u, w} (C(·, ·)) := InducedCategory.concreteCategory toTop variable (X : Type w) [TopologicalSpace X] [UCompactlyGeneratedSpace.{u} X] /-- Constructor for objects of the category `CompactlyGenerated`. -/ abbrev of : CompactlyGenerated.{u, w} where toTop := TopCat.of X is_compactly_generated := ‹_› section variable {X} {Y : Type w} [TopologicalSpace Y] [UCompactlyGeneratedSpace.{u} Y] /-- Typecheck a `ContinuousMap` as a morphism in `CompactlyGenerated`. -/ abbrev ofHom (f : C(X, Y)) : of X ⟶ of Y := ConcreteCategory.ofHom f end /-- The fully faithful embedding of `CompactlyGenerated` in `TopCat`. -/ @[simps!] def compactlyGeneratedToTop : CompactlyGenerated.{u, w} ⥤ TopCat.{w} := inducedFunctor _ /-- `compactlyGeneratedToTop` is fully faithful. -/ def fullyFaithfulCompactlyGeneratedToTop : compactlyGeneratedToTop.{u, w}.FullyFaithful := fullyFaithfulInducedFunctor _ instance : compactlyGeneratedToTop.{u, w}.Full := fullyFaithfulCompactlyGeneratedToTop.full instance : compactlyGeneratedToTop.{u, w}.Faithful := fullyFaithfulCompactlyGeneratedToTop.faithful /-- Construct an isomorphism from a homeomorphism. -/ @[simps hom inv] def isoOfHomeo {X Y : CompactlyGenerated.{u, w}} (f : X ≃ₜ Y) : X ≅ Y where hom := ofHom ⟨f, f.continuous⟩ inv := ofHom ⟨f.symm, f.symm.continuous⟩ hom_inv_id := by ext x exact f.symm_apply_apply x inv_hom_id := by ext x exact f.apply_symm_apply x /-- Construct a homeomorphism from an isomorphism. -/ @[simps] def homeoOfIso {X Y : CompactlyGenerated.{u, w}} (f : X ≅ Y) : X ≃ₜ Y where toFun := f.hom invFun := f.inv left_inv := f.hom_inv_id_apply right_inv := f.inv_hom_id_apply continuous_toFun := f.hom.hom.continuous continuous_invFun := f.inv.hom.continuous /-- The equivalence between isomorphisms in `CompactlyGenerated` and homeomorphisms of topological spaces. -/ @[simps] def isoEquivHomeo {X Y : CompactlyGenerated.{u, w}} : (X ≅ Y) ≃ (X ≃ₜ Y) where toFun := homeoOfIso invFun := isoOfHomeo end CompactlyGenerated
.lake/packages/mathlib/Mathlib/Topology/Category/Born.lean	import Mathlib.CategoryTheory.ConcreteCategory.Basic import Mathlib.Topology.Bornology.Hom /-! # The category of bornologies This defines `Born`, the category of bornologies. -/ universe u open CategoryTheory /-- The category of bornologies. -/ structure Born where /-- Construct a bundled `Born` from a `Bornology`. -/ of :: /-- The underlying bornology. -/ carrier : Type* [str : Bornology carrier] attribute [instance] Born.str namespace Born instance : CoeSort Born Type* := ⟨carrier⟩ instance : Inhabited Born := ⟨of PUnit⟩ instance : LargeCategory.{u} Born where Hom X Y := LocallyBoundedMap X Y id X := LocallyBoundedMap.id X comp f g := g.comp f instance : ConcreteCategory Born (LocallyBoundedMap · ·) where hom f := f ofHom f := f end Born
.lake/packages/mathlib/Mathlib/Topology/Category/TopCommRingCat.lean	import Mathlib.Algebra.Category.Ring.Basic import Mathlib.Topology.Category.TopCat.Basic import Mathlib.Topology.Algebra.Ring.Basic /-! # Category of topological commutative rings We introduce the category `TopCommRingCat` of topological commutative rings together with the relevant forgetful functors to topological spaces and commutative rings. -/ universe u open CategoryTheory /-- A bundled topological commutative ring. -/ structure TopCommRingCat where /-- Construct a bundled `TopCommRingCat` from the underlying type and the appropriate typeclasses. -/ of :: /-- carrier of a topological commutative ring. -/ α : Type u [isCommRing : CommRing α] [isTopologicalSpace : TopologicalSpace α] [isTopologicalRing : IsTopologicalRing α] namespace TopCommRingCat instance : Inhabited TopCommRingCat := ⟨⟨PUnit⟩⟩ instance : CoeSort TopCommRingCat (Type u) := ⟨TopCommRingCat.α⟩ attribute [instance] isCommRing isTopologicalSpace isTopologicalRing instance : Category TopCommRingCat.{u} where Hom R S := { f : R →+* S // Continuous f } id R := ⟨RingHom.id R, by rw [RingHom.id]; continuity⟩ comp f g := ⟨g.val.comp f.val, by -- TODO automate cases f cases g continuity⟩ instance (R S : TopCommRingCat.{u}) : FunLike { f : R →+* S // Continuous f } R S where coe f := f.val coe_injective' _ _ h := Subtype.ext (DFunLike.coe_injective h) instance : ConcreteCategory TopCommRingCat.{u} fun R S => { f : R →+* S // Continuous f } where hom f := f ofHom f := f theorem coe_of (X : Type u) [CommRing X] [TopologicalSpace X] [IsTopologicalRing X] : (of X : Type u) = X := rfl instance hasForgetToCommRingCat : HasForget₂ TopCommRingCat CommRingCat := HasForget₂.mk' (fun R => CommRingCat.of R) (fun _ => rfl) (fun f => CommRingCat.ofHom f.val) HEq.rfl instance forgetToCommRingCatTopologicalSpace (R : TopCommRingCat) : TopologicalSpace ((forget₂ TopCommRingCat CommRingCat).obj R) := R.isTopologicalSpace /-- The forgetful functor to `TopCat`. -/ instance hasForgetToTopCat : HasForget₂ TopCommRingCat TopCat := HasForget₂.mk' (fun R => TopCat.of R) (fun _ => rfl) (fun f => TopCat.ofHom ⟨⇑f.1, f.2⟩) HEq.rfl instance forgetToTopCatCommRing (R : TopCommRingCat) : CommRing ((forget₂ TopCommRingCat TopCat).obj R) := R.isCommRing instance forgetToTopCatTopologicalRing (R : TopCommRingCat) : IsTopologicalRing ((forget₂ TopCommRingCat TopCat).obj R) := R.isTopologicalRing /-- The forgetful functors to `Type` do not reflect isomorphisms, but the forgetful functor from `TopCommRingCat` to `TopCat` does. -/ instance : (forget₂ TopCommRingCat.{u} TopCat.{u}).ReflectsIsomorphisms where reflects {X Y} f _ := by -- We have an isomorphism in `TopCat`, let i_Top := asIso ((forget₂ TopCommRingCat TopCat).map f) -- and a `RingEquiv`. let e_Ring : X ≃+* Y := { f.1, ((forget TopCat).mapIso i_Top).toEquiv with } -- Putting these together we obtain the isomorphism we're after: exact ⟨⟨⟨e_Ring.symm, i_Top.inv.hom.2⟩, ⟨by ext x exact e_Ring.left_inv x, by ext x exact e_Ring.right_inv x⟩⟩⟩ end TopCommRingCat
.lake/packages/mathlib/Mathlib/Topology/Category/FinTopCat.lean	import Mathlib.CategoryTheory.FintypeCat import Mathlib.Topology.Category.TopCat.Basic /-! # Category of finite topological spaces Definition of the category of finite topological spaces with the canonical forgetful functors. -/ universe u open CategoryTheory /-- A bundled finite topological space. -/ structure FinTopCat where /-- carrier of a finite topological space. -/ toTop : TopCat.{u} -- TODO: turn this into an `extends`? [fintype : Fintype toTop] namespace FinTopCat instance : Inhabited FinTopCat := ⟨{ toTop := TopCat.of PEmpty }⟩ instance : CoeSort FinTopCat (Type u) := ⟨fun X => X.toTop⟩ attribute [instance] fintype instance : Category FinTopCat := InducedCategory.category toTop instance : ConcreteCategory FinTopCat (C(·, ·)) := InducedCategory.concreteCategory toTop /-- Construct a bundled `FinTopCat` from the underlying type and the appropriate typeclasses. -/ def of (X : Type u) [Fintype X] [TopologicalSpace X] : FinTopCat where toTop := TopCat.of X fintype := ‹_› @[simp] theorem coe_of (X : Type u) [Fintype X] [TopologicalSpace X] : (of X : Type u) = X := rfl /-- The forgetful functor to `FintypeCat`. -/ instance : HasForget₂ FinTopCat FintypeCat := HasForget₂.mk' (fun X ↦ FintypeCat.of X) (fun _ ↦ rfl) (fun f ↦ f.hom.toFun) HEq.rfl instance (X : FinTopCat) : TopologicalSpace ((forget₂ FinTopCat FintypeCat).obj X) := inferInstanceAs <\| TopologicalSpace X /-- The forgetful functor to `TopCat`. -/ instance : HasForget₂ FinTopCat TopCat := InducedCategory.hasForget₂ _ instance (X : FinTopCat) : Fintype ((forget₂ FinTopCat TopCat).obj X) := X.fintype end FinTopCat namespace FintypeCatDiscrete /-- Scoped topological space instance on objects of the category of finite types, assigning the discrete topology. -/ scoped instance (X : FintypeCat) : TopologicalSpace X := ⊥ scoped instance (X : FintypeCat) : DiscreteTopology X := ⟨rfl⟩ /-- The forgetful functor from finite types to topological spaces, forgetting discreteness. This is a scoped instance. -/ scoped instance : HasForget₂ FintypeCat TopCat where forget₂.obj X := TopCat.of X forget₂.map f := TopCat.ofHom ⟨f, continuous_of_discreteTopology⟩ end FintypeCatDiscrete
.lake/packages/mathlib/Mathlib/Topology/Category/UniformSpace.lean	import Mathlib.CategoryTheory.Adjunction.Reflective import Mathlib.CategoryTheory.Monad.Limits import Mathlib.Topology.Category.TopCat.Basic import Mathlib.Topology.UniformSpace.Completion /-! # The category of uniform spaces We construct the category of uniform spaces, show that the complete separated uniform spaces form a reflective subcategory, and hence possess all limits that uniform spaces do. TODO: show that uniform spaces actually have all limits! -/ universe u open CategoryTheory /-- An object in the category of uniform spaces. -/ structure UniformSpaceCat : Type (u + 1) where /-- Construct a bundled `UniformSpace` from the underlying type and the typeclass. -/ of :: /-- The underlying uniform space. -/ carrier : Type u [str : UniformSpace carrier] attribute [instance] UniformSpaceCat.str namespace UniformSpaceCat instance : CoeSort UniformSpaceCat Type* := ⟨carrier⟩ /-- A bundled uniform continuous map. -/ @[ext] structure Hom (X Y : UniformSpaceCat) where /-- The underlying `UniformContinuous` function. -/ hom' : { f : X → Y // UniformContinuous f } instance : LargeCategory.{u} UniformSpaceCat.{u} where Hom := Hom id X := ⟨id, uniformContinuous_id⟩ comp f g := ⟨⟨g.hom'.val ∘ f.hom'.val, g.hom'.property.comp f.hom'.property⟩⟩ id_comp := by intros; apply Hom.ext; simp comp_id := by intros; apply Hom.ext; simp assoc := by intros; apply Hom.ext; ext; simp instance instFunLike (X Y : UniformSpaceCat) : FunLike { f : X → Y // UniformContinuous f } X Y where coe := Subtype.val coe_injective' _ _ h := Subtype.ext h instance : ConcreteCategory UniformSpaceCat ({ f : · → · // UniformContinuous f }) where hom f := f.hom' ofHom f := ⟨f⟩ /-- Turn a morphism in `UniformSpaceCat` back into a function which is `UniformContinuous`. -/ abbrev Hom.hom {X Y : UniformSpaceCat} (f : Hom X Y) := ConcreteCategory.hom (C := UniformSpaceCat) f /-- Typecheck a function which is `UniformContinuous` as a morphism in `UniformSpaceCat`. -/ abbrev ofHom {X Y : Type u} [UniformSpace X] [UniformSpace Y] (f : { f : X → Y // UniformContinuous f }) : of X ⟶ of Y := ConcreteCategory.ofHom f instance : Inhabited UniformSpaceCat := ⟨UniformSpaceCat.of Empty⟩ theorem coe_of (X : Type u) [UniformSpace X] : (of X : Type u) = X := rfl @[simp] theorem hom_comp {X Y Z : UniformSpaceCat} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).hom = ⟨g ∘ f, g.hom.prop.comp f.hom.prop⟩ := rfl @[simp] theorem hom_id (X : UniformSpaceCat) : (𝟙 X : X ⟶ X).hom = ⟨id, uniformContinuous_id⟩ := rfl @[simp] theorem hom_ofHom {X Y : Type u} [UniformSpace X] [UniformSpace Y] (f : { f : X → Y // UniformContinuous f }) : (ofHom f).hom = f := rfl theorem coe_comp {X Y Z : UniformSpaceCat} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f := rfl theorem coe_id (X : UniformSpaceCat) : (𝟙 X : X → X) = id := rfl theorem coe_mk {X Y : UniformSpaceCat} (f : X → Y) (hf : UniformContinuous f) : (⟨f, hf⟩ : X ⟶ Y).hom = f := rfl @[ext] theorem hom_ext {X Y : UniformSpaceCat} {f g : X ⟶ Y} (h : (f : X → Y) = g) : f = g := Hom.ext (Subtype.ext h) /-- The forgetful functor from uniform spaces to topological spaces. -/ instance hasForgetToTop : HasForget₂ UniformSpaceCat.{u} TopCat.{u} where forget₂ := { obj := fun X => TopCat.of X map := fun f => TopCat.ofHom { toFun := f continuous_toFun := f.hom.property.continuous } } end UniformSpaceCat /-- A (bundled) complete separated uniform space. -/ structure CpltSepUniformSpace where /-- The underlying space -/ α : Type u [isUniformSpace : UniformSpace α] [isCompleteSpace : CompleteSpace α] [isT0 : T0Space α] namespace CpltSepUniformSpace instance : CoeSort CpltSepUniformSpace (Type u) := ⟨CpltSepUniformSpace.α⟩ attribute [instance] isUniformSpace isCompleteSpace isT0 /-- The function forgetting that a complete separated uniform spaces is complete and separated. -/ def toUniformSpace (X : CpltSepUniformSpace) : UniformSpaceCat := UniformSpaceCat.of X instance completeSpace (X : CpltSepUniformSpace) : CompleteSpace (toUniformSpace X).carrier := CpltSepUniformSpace.isCompleteSpace X instance t0Space (X : CpltSepUniformSpace) : T0Space (toUniformSpace X).carrier := CpltSepUniformSpace.isT0 X /-- Construct a bundled `UniformSpace` from the underlying type and the appropriate typeclasses. -/ def of (X : Type u) [UniformSpace X] [CompleteSpace X] [T0Space X] : CpltSepUniformSpace := ⟨X⟩ @[simp] theorem coe_of (X : Type u) [UniformSpace X] [CompleteSpace X] [T0Space X] : (of X : Type u) = X := rfl instance : Inhabited CpltSepUniformSpace := ⟨CpltSepUniformSpace.of Empty⟩ /-- The category instance on `CpltSepUniformSpace`. -/ instance category : LargeCategory CpltSepUniformSpace := InducedCategory.category toUniformSpace instance instFunLike (X Y : CpltSepUniformSpace) : FunLike { f : X → Y // UniformContinuous f } X Y where coe := Subtype.val coe_injective' _ _ h := Subtype.ext h /-- The concrete category instance on `CpltSepUniformSpace`. -/ instance concreteCategory : ConcreteCategory CpltSepUniformSpace ({ f : · → · // UniformContinuous f }) := InducedCategory.concreteCategory toUniformSpace instance hasForgetToUniformSpace : HasForget₂ CpltSepUniformSpace UniformSpaceCat := InducedCategory.hasForget₂ toUniformSpace @[simp] theorem hom_comp {X Y Z : CpltSepUniformSpace} (f : X ⟶ Y) (g : Y ⟶ Z) : ConcreteCategory.hom (f ≫ g) = ⟨g ∘ f, g.hom.prop.comp f.hom.prop⟩ := rfl @[simp] theorem hom_id (X : CpltSepUniformSpace) : ConcreteCategory.hom (𝟙 X : X ⟶ X) = ⟨id, uniformContinuous_id⟩ := rfl @[simp] theorem hom_ofHom {X Y : Type u} [UniformSpace X] [UniformSpace Y] (f : { f : X → Y // UniformContinuous f }) : (UniformSpaceCat.ofHom f).hom = f := rfl end CpltSepUniformSpace namespace UniformSpaceCat open UniformSpace open CpltSepUniformSpace /-- The functor turning uniform spaces into complete separated uniform spaces. -/ @[simps map] noncomputable def completionFunctor : UniformSpaceCat ⥤ CpltSepUniformSpace where obj X := CpltSepUniformSpace.of (Completion X) map f := ofHom ⟨Completion.map f.1, Completion.uniformContinuous_map⟩ map_id _ := hom_ext Completion.map_id map_comp f g := hom_ext (Completion.map_comp g.hom.property f.hom.property).symm /-- The inclusion of a uniform space into its completion. -/ def completionHom (X : UniformSpaceCat) : X ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj (completionFunctor.obj X) where hom'.val := ((↑) : X → Completion X) hom'.property := Completion.uniformContinuous_coe X @[simp] theorem completionHom_val (X : UniformSpaceCat) (x) : (completionHom X) x = (x : Completion X) := rfl /-- The mate of a morphism from a `UniformSpace` to a `CpltSepUniformSpace`. -/ noncomputable def extensionHom {X : UniformSpaceCat} {Y : CpltSepUniformSpace} (f : X ⟶ (forget₂ CpltSepUniformSpace UniformSpaceCat).obj Y) : completionFunctor.obj X ⟶ Y where hom'.val := Completion.extension f hom'.property := Completion.uniformContinuous_extension @[simp] theorem extensionHom_val {X : UniformSpaceCat} {Y : CpltSepUniformSpace} (f : X ⟶ (forget₂ _ _).obj Y) (x) : (extensionHom f) x = Completion.extension f x := rfl @[simp] theorem extension_comp_coe {X : UniformSpaceCat} {Y : CpltSepUniformSpace} (f : toUniformSpace (CpltSepUniformSpace.of (Completion X)) ⟶ toUniformSpace Y) : extensionHom (completionHom X ≫ f) = f := by ext x exact congr_fun (Completion.extension_comp_coe f.hom.property) x /-- The completion functor is left adjoint to the forgetful functor. -/ noncomputable def adj : completionFunctor ⊣ forget₂ CpltSepUniformSpace UniformSpaceCat := Adjunction.mkOfHomEquiv { homEquiv := fun X Y => { toFun := fun f => completionHom X ≫ f invFun := fun f => extensionHom f left_inv := fun f => by dsimp; rw [extension_comp_coe] right_inv := fun f => by ext x rcases f with ⟨⟨_, _⟩⟩ exact @Completion.extension_coe _ _ _ _ _ (CpltSepUniformSpace.t0Space _) ‹_› _ } homEquiv_naturality_left_symm := fun {X' X Y} f g => by ext x dsimp [-Function.comp_apply] erw [Completion.extension_map (γ := Y) g.hom.2 f.hom.2] rfl } noncomputable instance : Reflective (forget₂ CpltSepUniformSpace UniformSpaceCat) where L := completionFunctor adj := adj map_surjective f := ⟨f, rfl⟩ open CategoryTheory.Limits -- TODO Once someone defines `HasLimits UniformSpace`, turn this into an instance. example [HasLimits.{u} UniformSpaceCat.{u}] : HasLimits.{u} CpltSepUniformSpace.{u} := hasLimits_of_reflective <\| forget₂ CpltSepUniformSpace UniformSpaceCat.{u} end UniformSpaceCat
.lake/packages/mathlib/Mathlib/Topology/Category/DeltaGenerated.lean	import Mathlib.CategoryTheory.Monad.Limits import Mathlib.Topology.Category.TopCat.Limits.Basic import Mathlib.Topology.Compactness.DeltaGeneratedSpace /-! # Delta-generated topological spaces The category of delta-generated spaces. See https://ncatlab.org/nlab/show/Delta-generated+topological+space. Adapted from `Mathlib/Topology/Category/CompactlyGenerated.lean`. ## TODO * `DeltaGenerated` is Cartesian closed. -/ universe u open CategoryTheory /-- The type of delta-generated topological spaces. -/ structure DeltaGenerated where /-- the underlying topological space -/ toTop : TopCat.{u} /-- The underlying topological space is delta-generated. -/ deltaGenerated : DeltaGeneratedSpace toTop := by infer_instance namespace DeltaGenerated instance : CoeSort DeltaGenerated Type* := ⟨fun X ↦ X.toTop⟩ attribute [instance] deltaGenerated instance : LargeCategory.{u} DeltaGenerated.{u} := InducedCategory.category toTop instance : ConcreteCategory.{u} DeltaGenerated.{u} (C(·, ·)) := InducedCategory.concreteCategory toTop /-- Constructor for objects of the category `DeltaGenerated` -/ abbrev of (X : Type u) [TopologicalSpace X] [DeltaGeneratedSpace X] : DeltaGenerated.{u} where toTop := TopCat.of X deltaGenerated := ‹_› /-- The forgetful functor `DeltaGenerated ⥤ TopCat` -/ @[simps!] def deltaGeneratedToTop : DeltaGenerated.{u} ⥤ TopCat.{u} := inducedFunctor _ /-- `deltaGeneratedToTop` is fully faithful. -/ def fullyFaithfulDeltaGeneratedToTop : deltaGeneratedToTop.{u}.FullyFaithful := fullyFaithfulInducedFunctor _ instance : deltaGeneratedToTop.{u}.Full := fullyFaithfulDeltaGeneratedToTop.full instance : deltaGeneratedToTop.{u}.Faithful := fullyFaithfulDeltaGeneratedToTop.faithful /-- The faithful (but not full) functor taking each topological space to its delta-generated coreflection. -/ @[simps!] def topToDeltaGenerated : TopCat.{u} ⥤ DeltaGenerated.{u} where obj X := of (DeltaGeneratedSpace.of X) map {_ Y} f := TopCat.ofHom ⟨f, (continuous_to_deltaGenerated (Y := Y)).mpr <\| continuous_le_dom deltaGenerated_le f.hom.continuous⟩ instance : topToDeltaGenerated.{u}.Faithful := ⟨fun h ↦ by ext x; exact CategoryTheory.congr_fun h x⟩ /-- The adjunction between the forgetful functor `DeltaGenerated ⥤ TopCat` and its coreflector. -/ def coreflectorAdjunction : deltaGeneratedToTop ⊣ topToDeltaGenerated := Adjunction.mkOfUnitCounit { unit := { app X := TopCat.ofHom ⟨id, continuous_iff_coinduced_le.mpr (eq_deltaGenerated (X := X)).le⟩ } counit := { app X := TopCat.ofHom ⟨DeltaGeneratedSpace.counit, DeltaGeneratedSpace.continuous_counit⟩ }} /-- The category of delta-generated spaces is coreflective in the category of topological spaces. -/ instance deltaGeneratedToTop.coreflective : Coreflective deltaGeneratedToTop where R := topToDeltaGenerated adj := coreflectorAdjunction noncomputable instance deltaGeneratedToTop.createsColimits : CreatesColimits deltaGeneratedToTop := comonadicCreatesColimits deltaGeneratedToTop instance hasLimits : Limits.HasLimits DeltaGenerated := hasLimits_of_coreflective deltaGeneratedToTop instance hasColimits : Limits.HasColimits DeltaGenerated := hasColimits_of_hasColimits_createsColimits deltaGeneratedToTop end DeltaGenerated
.lake/packages/mathlib/Mathlib/Topology/Category/Locale.lean	import Mathlib.Order.Category.Frm import Mathlib.Topology.Category.CompHaus.Frm /-! # The category of locales This file defines `Locale`, the category of locales. This is the opposite of the category of frames. -/ universe u open CategoryTheory Opposite Order TopologicalSpace /-- The category of locales. -/ def Locale := Frmᵒᵖ deriving LargeCategory namespace Locale instance : CoeSort Locale Type* := ⟨fun X => X.unop⟩ instance (X : Locale) : Frame X := X.unop.str /-- Construct a bundled `Locale` from a `Frame`. -/ def of (α : Type) [Frame α] : Locale := op <\| Frm.of α @[simp] theorem coe_of (α : Type) [Frame α] : ↥(of α) = α := rfl instance : Inhabited Locale := ⟨of PUnit⟩ end Locale /-- The forgetful functor from `Top` to `Locale` which forgets that the space has "enough points". -/ @[simps!] def topToLocale : TopCat ⥤ Locale := topCatOpToFrm.rightOp -- Note, `CompHaus` is too strong. We only need `T0Space`. instance CompHausToLocale.faithful : (compHausToTop ⋙ topToLocale.{u}).Faithful := ⟨fun h => by dsimp at h exact ConcreteCategory.ext (Opens.comap_injective (congr_arg Frm.Hom.hom (Quiver.Hom.op_inj h)))⟩
.lake/packages/mathlib/Mathlib/Topology/Category/Compactum.lean	import Mathlib.CategoryTheory.Monad.Types import Mathlib.CategoryTheory.Monad.Limits import Mathlib.CategoryTheory.Equivalence import Mathlib.Topology.Category.CompHaus.Basic import Mathlib.Topology.Category.Profinite.Basic import Mathlib.Data.Set.Constructions /-! # Compacta and Compact Hausdorff Spaces Recall that, given a monad `M` on `Type`, an algebra* for `M` consists of the following data: - A type `X : Type` - A "structure" map `M X → X`. This data must also satisfy a distributivity and unit axiom, and algebras for `M` form a category in an evident way. See the file `CategoryTheory.Monad.Algebra` for a general version, as well as the following link. https://ncatlab.org/nlab/show/monad This file proves the equivalence between the category of compact Hausdorff topological spaces* and the category of algebras for the ultrafilter monad. ## Notation: Here are the main objects introduced in this file. - `Compactum` is the type of compacta, which we define as algebras for the ultrafilter monad. - `compactumToCompHaus` is the functor `Compactum ⥤ CompHaus`. Here `CompHaus` is the usual category of compact Hausdorff spaces. - `compactumToCompHaus.isEquivalence` is a term of type `IsEquivalence compactumToCompHaus`. The proof of this equivalence is a bit technical. But the idea is quite simply that the structure map `Ultrafilter X → X` for an algebra `X` of the ultrafilter monad should be considered as the map sending an ultrafilter to its limit in `X`. The topology on `X` is then defined by mimicking the characterization of open sets in terms of ultrafilters. Any `X : Compactum` is endowed with a coercion to `Type`, as well as the following instances: - `TopologicalSpace X`. - `CompactSpace X`. - `T2Space X`. Any morphism `f : X ⟶ Y` of is endowed with a coercion to a function `X → Y`, which is shown to be continuous in `continuous_of_hom`. The function `Compactum.ofTopologicalSpace` can be used to construct a `Compactum` from a topological space which satisfies `CompactSpace` and `T2Space`. We also add wrappers around structures which already exist. Here are the main ones, all in the `Compactum` namespace: - `forget : Compactum ⥤ Type` is the forgetful functor, which induces a `ConcreteCategory` instance for `Compactum`. - `free : Type* ⥤ Compactum` is the left adjoint to `forget`, and the adjunction is in `adj`. - `str : Ultrafilter X → X` is the structure map for `X : Compactum`. The notation `X.str` is preferred. - `join : Ultrafilter (Ultrafilter X) → Ultrafilter X` is the monadic join for `X : Compactum`. Again, the notation `X.join` is preferred. - `incl : X → Ultrafilter X` is the unit for `X : Compactum`. The notation `X.incl` is preferred. ## References - E. Manes, Algebraic Theories, Graduate Texts in Mathematics 26, Springer-Verlag, 1976. - https://ncatlab.org/nlab/show/ultrafilter -/ universe u open CategoryTheory Filter Ultrafilter TopologicalSpace CategoryTheory.Limits FiniteInter open scoped Topology local notation "β" => ofTypeMonad Ultrafilter /-- The type `Compactum` of Compacta, defined as algebras for the ultrafilter monad. -/ def Compactum := Monad.Algebra β deriving Category, Inhabited namespace Compactum /-- The forgetful functor to Type* -/ def forget : Compactum ⥤ Type* := Monad.forget _ instance : forget.Faithful := show (Monad.forget _).Faithful from inferInstance noncomputable instance : CreatesLimits forget := show CreatesLimits <\| Monad.forget _ from inferInstance /-- The "free" Compactum functor. -/ def free : Type* ⥤ Compactum := Monad.free _ /-- The adjunction between `free` and `forget`. -/ def adj : free ⊣ forget := Monad.adj _ instance : CoeSort Compactum Type* := ⟨fun X => X.A⟩ instance {X Y : Compactum} : FunLike (X ⟶ Y) X Y where coe f := f.f coe_injective' _ _ h := (Monad.forget_faithful β).map_injective h -- Basic instances instance : ConcreteCategory Compactum (· ⟶ ·) where hom f := f ofHom f := f instance : HasLimits Compactum := hasLimits_of_hasLimits_createsLimits forget /-- The structure map for a compactum, essentially sending an ultrafilter to its limit. -/ def str (X : Compactum) : Ultrafilter X → X := X.a /-- The monadic join. -/ def join (X : Compactum) : Ultrafilter (Ultrafilter X) → Ultrafilter X := (β).μ.app _ /-- The inclusion of `X` into `Ultrafilter X`. -/ def incl (X : Compactum) : X → Ultrafilter X := (β).η.app _ @[simp] theorem str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x := by change ((β).η.app _ ≫ X.a) _ = _ rw [Monad.Algebra.unit] rfl @[simp] theorem str_hom_commute (X Y : Compactum) (f : X ⟶ Y) (xs : Ultrafilter X) : f (X.str xs) = Y.str (map f xs) := by change (X.a ≫ f.f) _ = _ rw [← f.h] rfl @[simp] theorem join_distrib (X : Compactum) (uux : Ultrafilter (Ultrafilter X)) : X.str (X.join uux) = X.str (map X.str uux) := by change ((β).μ.app _ ≫ X.a) _ = _ rw [Monad.Algebra.assoc] rfl instance {X : Compactum} : TopologicalSpace X where IsOpen U := ∀ F : Ultrafilter X, X.str F ∈ U → U ∈ F isOpen_univ _ _ := Filter.univ_sets _ isOpen_inter _ _ h3 h4 _ h6 := Filter.inter_sets _ (h3 _ h6.1) (h4 _ h6.2) isOpen_sUnion := fun _ h1 _ ⟨T, hT, h2⟩ => mem_of_superset (h1 T hT _ h2) (Set.subset_sUnion_of_mem hT) theorem isClosed_iff {X : Compactum} (S : Set X) : IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by rw [← isOpen_compl_iff] constructor · intro cond F h by_contra c specialize cond F c rw [compl_mem_iff_notMem] at cond contradiction · intro h1 F h2 specialize h1 F rcases F.mem_or_compl_mem S with h \| h exacts [absurd (h1 h) h2, h] instance {X : Compactum} : CompactSpace X := by constructor rw [isCompact_iff_ultrafilter_le_nhds] intro F _ refine ⟨X.str F, by tauto, ?_⟩ rw [le_nhds_iff] intro S h1 h2 exact h2 F h1 /-- A local definition used only in the proofs. -/ private def basic {X : Compactum} (A : Set X) : Set (Ultrafilter X) := { F \| A ∈ F } /-- A local definition used only in the proofs. -/ private def cl {X : Compactum} (A : Set X) : Set X := X.str '' basic A private theorem basic_inter {X : Compactum} (A B : Set X) : basic (A ∩ B) = basic A ∩ basic B := by ext G constructor · intro hG constructor <;> filter_upwards [hG] with _ exacts [And.left, And.right] · rintro ⟨h1, h2⟩ exact inter_mem h1 h2 private theorem subset_cl {X : Compactum} (A : Set X) : A ⊆ cl A := fun a ha => ⟨X.incl a, ha, by simp⟩ private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by rintro _ ⟨F, hF, rfl⟩ -- Notation to be used in this proof. let fsu := Finset (Set (Ultrafilter X)) let ssu := Set (Set (Ultrafilter X)) let ι : fsu → ssu := fun x ↦ ↑x let C0 : ssu := { Z \| ∃ B ∈ F, X.str ⁻¹' B = Z } let AA := { G : Ultrafilter X \| A ∈ G } let C1 := insert AA C0 let C2 := finiteInterClosure C1 -- C0 is closed under intersections. have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩ use Q ∩ R simp only [and_true, Set.preimage_inter] exact inter_sets _ hQ hR -- All sets in C0 are nonempty. have claim2 : ∀ B ∈ C0, Set.Nonempty B := by rintro B ⟨Q, hQ, rfl⟩ obtain ⟨q⟩ := Filter.nonempty_of_mem hQ use X.incl q simpa -- The intersection of AA with every set in C0 is nonempty. have claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty := by rintro B ⟨Q, hQ, rfl⟩ have : (Q ∩ cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF) rcases this with ⟨q, hq1, P, hq2, hq3⟩ refine ⟨P, hq2, ?_⟩ rw [← hq3] at hq1 simpa -- Suffices to show that the intersection of any finite subcollection of C1 is nonempty. suffices ∀ T : fsu, ι T ⊆ C1 → (⋂₀ ι T).Nonempty by obtain ⟨G, h1⟩ := exists_ultrafilter_of_finite_inter_nonempty _ this use X.join G have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr ⟨S, hS, rfl⟩) rw [join_distrib, this] exact ⟨h1 (Or.inl rfl), rfl⟩ -- C2 is closed under finite intersections (by construction!). have claim4 := finiteInterClosure_finiteInter C1 -- C0 is closed under finite intersections by claim1. have claim5 : FiniteInter C0 := ⟨⟨_, univ_mem, Set.preimage_univ⟩, claim1⟩ -- Every element of C2 is nonempty. have claim6 : ∀ P ∈ C2, (P : Set (Ultrafilter X)).Nonempty := by suffices ∀ P ∈ C2, P ∈ C0 ∨ ∃ Q ∈ C0, P = AA ∩ Q by intro P hP rcases this P hP with h \| h · exact claim2 _ h · rcases h with ⟨Q, hQ, rfl⟩ exact claim3 _ hQ intro P hP exact claim5.finiteInterClosure_insert _ hP intro T hT -- Suffices to show that the intersection of the T's is contained in C2. suffices ⋂₀ ι T ∈ C2 by exact claim6 _ this -- Finish apply claim4.finiteInter_mem T intro t ht exact finiteInterClosure.basic (@hT t ht) theorem isClosed_cl {X : Compactum} (A : Set X) : IsClosed (cl A) := by rw [isClosed_iff] intro F hF exact cl_cl _ ⟨F, hF, rfl⟩ theorem str_eq_of_le_nhds {X : Compactum} (F : Ultrafilter X) (x : X) : ↑F ≤ 𝓝 x → X.str F = x := by -- Notation to be used in this proof. let fsu := Finset (Set (Ultrafilter X)) let ssu := Set (Set (Ultrafilter X)) let ι : fsu → ssu := fun x ↦ ↑x let T0 : ssu := { S \| ∃ A ∈ F, S = basic A } let AA := X.str ⁻¹' {x} let T1 := insert AA T0 let T2 := finiteInterClosure T1 intro cond -- If F contains a closed set A, then x is contained in A. have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by intro A hA h by_contra H rw [le_nhds_iff] at cond specialize cond Aᶜ H hA.isOpen_compl rw [Ultrafilter.mem_coe, Ultrafilter.compl_mem_iff_notMem] at cond contradiction -- If A ∈ F, then x ∈ cl A. have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by intro A hA exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A)) -- T0 is closed under intersections. have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by rintro S1 ⟨S1, hS1, rfl⟩ S2 ⟨S2, hS2, rfl⟩ exact ⟨S1 ∩ S2, inter_mem hS1 hS2, by simp [basic_inter]⟩ -- For every S ∈ T0, the intersection AA ∩ S is nonempty. have claim4 : ∀ S ∈ T0, (AA ∩ S).Nonempty := by rintro S ⟨S, hS, rfl⟩ rcases claim2 _ hS with ⟨G, hG, hG2⟩ exact ⟨G, hG2, hG⟩ -- Every element of T0 is nonempty. have claim5 : ∀ S ∈ T0, Set.Nonempty S := by rintro S ⟨S, hS, rfl⟩ exact ⟨F, hS⟩ -- Every element of T2 is nonempty. have claim6 : ∀ S ∈ T2, Set.Nonempty S := by suffices ∀ S ∈ T2, S ∈ T0 ∨ ∃ Q ∈ T0, S = AA ∩ Q by intro S hS rcases this _ hS with h \| h · exact claim5 S h · rcases h with ⟨Q, hQ, rfl⟩ exact claim4 Q hQ intro S hS apply finiteInterClosure_insert · constructor · use Set.univ refine ⟨Filter.univ_sets _, ?_⟩ ext refine ⟨?_, by tauto⟩ · intro apply Filter.univ_sets · exact claim3 · exact hS -- It suffices to show that the intersection of any finite subset of T1 is nonempty. suffices ∀ F : fsu, ↑F ⊆ T1 → (⋂₀ ι F).Nonempty by obtain ⟨G, h1⟩ := Ultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ this have c1 : X.join G = F := Ultrafilter.coe_le_coe.1 fun P hP => h1 (Or.inr ⟨P, hP, rfl⟩) have c2 : G.map X.str = X.incl x := by refine Ultrafilter.coe_le_coe.1 fun P hP => ?_ apply mem_of_superset (h1 (Or.inl rfl)) rintro x ⟨rfl⟩ exact hP simp [← c1, c2] -- Finish... intro T hT refine claim6 _ (finiteInter_mem (.finiteInterClosure_finiteInter _) _ ?_) intro t ht exact finiteInterClosure.basic (@hT t ht) theorem le_nhds_of_str_eq {X : Compactum} (F : Ultrafilter X) (x : X) : X.str F = x → ↑F ≤ 𝓝 x := fun h => le_nhds_iff.mpr fun s hx hs => hs _ <\| by rwa [h] -- All the hard work above boils down to this `T2Space` instance. instance {X : Compactum} : T2Space X := by rw [t2_iff_ultrafilter] intro _ _ F hx hy rw [← str_eq_of_le_nhds _ _ hx, ← str_eq_of_le_nhds _ _ hy] /-- The structure map of a compactum actually computes limits. -/ theorem lim_eq_str {X : Compactum} (F : Ultrafilter X) : F.lim = X.str F := by rw [Ultrafilter.lim_eq_iff_le_nhds, le_nhds_iff] tauto theorem cl_eq_closure {X : Compactum} (A : Set X) : cl A = closure A := by ext rw [mem_closure_iff_ultrafilter] constructor · rintro ⟨F, h1, h2⟩ exact ⟨F, h1, le_nhds_of_str_eq _ _ h2⟩ · rintro ⟨F, h1, h2⟩ exact ⟨F, h1, str_eq_of_le_nhds _ _ h2⟩ /-- Any morphism of compacta is continuous. -/ theorem continuous_of_hom {X Y : Compactum} (f : X ⟶ Y) : Continuous f := by rw [continuous_iff_ultrafilter] intro x g h rw [Tendsto, ← coe_map] apply le_nhds_of_str_eq rw [← str_hom_commute, str_eq_of_le_nhds _ x _] apply h /-- Given any compact Hausdorff space, we construct a Compactum. -/ noncomputable def ofTopologicalSpace (X : Type*) [TopologicalSpace X] [CompactSpace X] [T2Space X] : Compactum where A := X a := Ultrafilter.lim unit := by ext x exact lim_eq (pure_le_nhds _) assoc := by ext FF change Ultrafilter (Ultrafilter X) at FF set x := (Ultrafilter.map Ultrafilter.lim FF).lim with c1 have c2 : ∀ (U : Set X) (F : Ultrafilter X), F.lim ∈ U → IsOpen U → U ∈ F := by intro U F h1 hU exact isOpen_iff_ultrafilter.mp hU _ h1 _ (Ultrafilter.le_nhds_lim _) have c3 : ↑(Ultrafilter.map Ultrafilter.lim FF) ≤ 𝓝 x := by rw [le_nhds_iff] intro U hx hU exact mem_coe.2 (c2 _ _ (by rwa [← c1]) hU) have c4 : ∀ U : Set X, x ∈ U → IsOpen U → { G : Ultrafilter X \| U ∈ G } ∈ FF := by intro U hx hU suffices Ultrafilter.lim ⁻¹' U ∈ FF by apply mem_of_superset this intro P hP exact c2 U P hP hU exact @c3 U (IsOpen.mem_nhds hU hx) apply lim_eq rw [le_nhds_iff] exact c4 /-- Any continuous map between Compacta is a morphism of compacta. -/ def homOfContinuous {X Y : Compactum} (f : X → Y) (cont : Continuous f) : X ⟶ Y := { f h := by rw [continuous_iff_ultrafilter] at cont ext (F : Ultrafilter X) specialize cont (X.str F) F (le_nhds_of_str_eq F (X.str F) rfl) simp only [types_comp_apply] exact str_eq_of_le_nhds (Ultrafilter.map f F) _ cont } end Compactum /-- The functor from Compactum to CompHaus. -/ def compactumToCompHaus : Compactum ⥤ CompHaus where obj X := { toTop := TopCat.of X, prop := trivial } map := fun f => CompHausLike.ofHom _ { toFun := f continuous_toFun := Compactum.continuous_of_hom _ } namespace compactumToCompHaus /-- The functor `compactumToCompHaus` is full. -/ instance full : compactumToCompHaus.{u}.Full where map_surjective f := ⟨Compactum.homOfContinuous f.1 f.hom.2, rfl⟩ /-- The functor `compactumToCompHaus` is faithful. -/ instance faithful : compactumToCompHaus.Faithful where -- Porting note: this used to be obviously (though it consumed a bit of memory) map_injective := by intro _ _ _ _ h -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): `ext` gets confused by coercion using forget. apply Monad.Algebra.Hom.ext apply congrArg (fun f => f.hom.toFun) h /-- This definition is used to prove essential surjectivity of `compactumToCompHaus`. -/ noncomputable def isoOfTopologicalSpace {D : CompHaus} : compactumToCompHaus.obj (Compactum.ofTopologicalSpace D) ≅ D where hom := CompHausLike.ofHom _ { toFun := id continuous_toFun := continuous_def.2 fun _ h => by rw [isOpen_iff_ultrafilter'] at h exact h } inv := CompHausLike.ofHom _ { toFun := id continuous_toFun := continuous_def.2 fun _ h1 => by rw [isOpen_iff_ultrafilter'] intro _ h2 exact h1 _ h2 } /-- The functor `compactumToCompHaus` is essentially surjective. -/ instance essSurj : compactumToCompHaus.EssSurj := { mem_essImage := fun X => ⟨Compactum.ofTopologicalSpace X, ⟨isoOfTopologicalSpace⟩⟩ } /-- The functor `compactumToCompHaus` is an equivalence of categories. -/ instance isEquivalence : compactumToCompHaus.IsEquivalence where end compactumToCompHaus /-- The forgetful functors of `Compactum` and `CompHaus` are compatible via `compactumToCompHaus`. -/ def compactumToCompHausCompForget : compactumToCompHaus ⋙ CategoryTheory.forget CompHaus ≅ Compactum.forget := NatIso.ofComponents fun _ => eqToIso rfl /- TODO: `forget CompHaus` is monadic, as it is isomorphic to the composition of an equivalence with the monadic functor `forget Compactum`. Once we have the API to transfer monadicity of functors along such isomorphisms, the instance `CreatesLimits (forget CompHaus)` can be deduced from this monadicity. -/ noncomputable instance CompHaus.forgetCreatesLimits : CreatesLimits (forget CompHaus) := by let e : forget CompHaus ≅ compactumToCompHaus.inv ⋙ Compactum.forget := (((forget CompHaus).leftUnitor.symm ≪≫ Functor.isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso (forget CompHaus)) ≪≫ compactumToCompHaus.inv.associator compactumToCompHaus (forget CompHaus)) ≪≫ Functor.isoWhiskerLeft _ compactumToCompHausCompForget exact createsLimitsOfNatIso e.symm noncomputable instance Profinite.forgetCreatesLimits : CreatesLimits (forget Profinite) := by change CreatesLimits (profiniteToCompHaus ⋙ forget _) infer_instance
.lake/packages/mathlib/Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean	import Mathlib.Topology.Category.CompHausLike.EffectiveEpi import Mathlib.Topology.Category.LightProfinite.Limits /-! # Effective epimorphisms in `LightProfinite` This file proves that `EffectiveEpi` and `Surjective` are equivalent in `LightProfinite`. As a consequence we deduce from the material in `Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean` that `LightProfinite` is `Preregular` and `Precoherent`. -/ universe u open CategoryTheory Limits CompHausLike namespace LightProfinite theorem effectiveEpi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) : EffectiveEpi f ↔ Function.Surjective f := by refine ⟨fun h ↦ ?_, fun h ↦ ⟨⟨effectiveEpiStruct f h⟩⟩⟩ rw [← epi_iff_surjective] infer_instance instance : Preregular LightProfinite.{u} := by apply CompHausLike.preregular intro _ _ f exact (effectiveEpi_iff_surjective f).mp example : Precoherent LightProfinite.{u} := inferInstance end LightProfinite
.lake/packages/mathlib/Mathlib/Topology/Category/LightProfinite/Sequence.lean	import Mathlib.Topology.Compactification.OnePoint.Basic import Mathlib.Topology.Category.LightProfinite.Basic /-! # The light profinite set classifying convergent sequences This file defines the light profinite set `ℕ∪{∞}`, defined as the one point compactification of `ℕ`. -/ open CategoryTheory OnePoint TopologicalSpace Topology namespace LightProfinite /-- The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0`. -/ noncomputable def natUnionInftyEmbedding : C(OnePoint ℕ, ℝ) where toFun \| ∞ => 0 \| OnePoint.some n => 1 / (n + 1 : ℝ) continuous_toFun := OnePoint.continuous_iff_from_nat _ \|>.mpr tendsto_one_div_add_atTop_nhds_zero_nat /-- The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0` is a closed embedding. -/ lemma isClosedEmbedding_natUnionInftyEmbedding : IsClosedEmbedding natUnionInftyEmbedding := by refine .of_continuous_injective_isClosedMap natUnionInftyEmbedding.continuous ?_ ?_ · rintro (_ \| n) (_ \| m) h · rfl · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, zero_eq_inv] at h assumption_mod_cast · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, inv_eq_zero] at h assumption_mod_cast · simp only [natUnionInftyEmbedding, one_div, ContinuousMap.coe_mk, inv_inj, add_left_inj, Nat.cast_inj] at h rw [h] · exact fun _ hC => (hC.isCompact.image natUnionInftyEmbedding.continuous).isClosed instance : MetrizableSpace (OnePoint ℕ) := isClosedEmbedding_natUnionInftyEmbedding.metrizableSpace /-- The one point compactification of the natural numbers as a light profinite set. -/ abbrev NatUnionInfty : LightProfinite := of (OnePoint ℕ) @[inherit_doc] scoped notation "ℕ∪{∞}" => NatUnionInfty instance : Coe ℕ ℕ∪{∞} := optionCoe open Filter Topology lemma continuous_iff_convergent {Y : Type*} [TopologicalSpace Y] (f : ℕ∪{∞} → Y) : Continuous f ↔ Tendsto (fun x : ℕ ↦ f x) atTop (𝓝 (f ∞)) := continuous_iff_from_nat f end LightProfinite
.lake/packages/mathlib/Mathlib/Topology/Category/LightProfinite/Basic.lean	import Mathlib.CategoryTheory.Limits.Shapes.Countable import Mathlib.Topology.Category.Profinite.AsLimit import Mathlib.Topology.Category.Profinite.CofilteredLimit import Mathlib.Topology.ClopenBox /-! # Light profinite spaces We construct the category `LightProfinite` of light profinite topological spaces. These are implemented as totally disconnected second countable compact Hausdorff spaces. This file also defines the category `LightDiagram`, which consists of those spaces that can be written as a sequential limit (in `Profinite`) of finite sets. We define an equivalence of categories `LightProfinite ≌ LightDiagram` and prove that these are essentially small categories. ## Implementation The category `LightProfinite` is defined using the structure `CompHausLike`. See the file `CompHausLike.Basic` for more information. -/ /- The basic API for `LightProfinite` is largely copied from the API of `Profinite`; where possible, try to keep them in sync -/ universe v u open CategoryTheory Limits Opposite FintypeCat Topology TopologicalSpace CompHausLike /-- `LightProfinite` is the category of second countable profinite spaces. -/ abbrev LightProfinite := CompHausLike (fun X ↦ TotallyDisconnectedSpace X ∧ SecondCountableTopology X) namespace LightProfinite instance (X : Type) [TopologicalSpace X] [TotallyDisconnectedSpace X] [SecondCountableTopology X] : HasProp (fun Y ↦ TotallyDisconnectedSpace Y ∧ SecondCountableTopology Y) X := ⟨⟨(inferInstance : TotallyDisconnectedSpace X), (inferInstance : SecondCountableTopology X)⟩⟩ /-- Construct a term of `LightProfinite` from a type endowed with the structure of a compact, Hausdorff, totally disconnected and second countable topological space. -/ abbrev of (X : Type) [TopologicalSpace X] [CompactSpace X] [T2Space X] [TotallyDisconnectedSpace X] [SecondCountableTopology X] : LightProfinite := CompHausLike.of _ X instance : Inhabited LightProfinite := ⟨LightProfinite.of PEmpty⟩ instance {X : LightProfinite} : TotallyDisconnectedSpace X := X.prop.1 instance {X : LightProfinite} : SecondCountableTopology X := X.prop.2 end LightProfinite /-- The fully faithful embedding of `LightProfinite` in `Profinite`. -/ abbrev lightToProfinite : LightProfinite ⥤ Profinite := CompHausLike.toCompHausLike (fun _ ↦ inferInstance) /-- `lightToProfinite` is fully faithful. -/ abbrev lightToProfiniteFullyFaithful : lightToProfinite.FullyFaithful := fullyFaithfulToCompHausLike _ /-- The fully faithful embedding of `LightProfinite` in `CompHaus`. -/ abbrev lightProfiniteToCompHaus : LightProfinite ⥤ CompHaus := compHausLikeToCompHaus _ /-- The fully faithful embedding of `LightProfinite` in `TopCat`. This is definitionally the same as the obvious composite. -/ abbrev LightProfinite.toTopCat : LightProfinite ⥤ TopCat := CompHausLike.compHausLikeToTop _ section DiscreteTopology attribute [local instance] FintypeCat.botTopology attribute [local instance] FintypeCat.discreteTopology /-- The natural functor from `Fintype` to `LightProfinite`, endowing a finite type with the discrete topology. -/ @[simps! -isSimp map_hom_apply] def FintypeCat.toLightProfinite : FintypeCat ⥤ LightProfinite where obj A := LightProfinite.of A map f := CompHausLike.ofHom _ ⟨f, by continuity⟩ /-- `FintypeCat.toLightProfinite` is fully faithful. -/ def FintypeCat.toLightProfiniteFullyFaithful : toLightProfinite.FullyFaithful where preimage f := (f : _ → _) map_preimage _ := rfl preimage_map _ := rfl instance : FintypeCat.toLightProfinite.Faithful := FintypeCat.toLightProfiniteFullyFaithful.faithful instance : FintypeCat.toLightProfinite.Full := FintypeCat.toLightProfiniteFullyFaithful.full instance (X : FintypeCat.{u}) : Fintype (FintypeCat.toLightProfinite.obj X) := inferInstanceAs (Fintype X) instance (X : FintypeCat.{u}) : Fintype (LightProfinite.of X) := inferInstanceAs (Fintype X) end DiscreteTopology namespace LightProfinite instance {J : Type v} [SmallCategory J] (F : J ⥤ LightProfinite.{max u v}) : TotallyDisconnectedSpace (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).pt.toTop := by change TotallyDisconnectedSpace ({ u : ∀ j : J, F.obj j \| _ } : Type _) exact Subtype.totallyDisconnectedSpace /-- An explicit limit cone for a functor `F : J ⥤ LightProfinite`, for a countable category `J` defined in terms of `CompHaus.limitCone`, which is defined in terms of `TopCat.limitCone`. -/ def limitCone {J : Type v} [SmallCategory J] [CountableCategory J] (F : J ⥤ LightProfinite.{max u v}) : Limits.Cone F where pt := { toTop := (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).pt.toTop prop := by constructor · infer_instance · change SecondCountableTopology ({ u : ∀ j : J, F.obj j \| _ } : Type _) apply IsInducing.subtypeVal.secondCountableTopology } π := { app := (CompHaus.limitCone.{v, u} (F ⋙ lightProfiniteToCompHaus)).π.app naturality := by intro j k f ext ⟨g, p⟩ exact (p f).symm } /-- The limit cone `LightProfinite.limitCone F` is indeed a limit cone. -/ def limitConeIsLimit {J : Type v} [SmallCategory J] [CountableCategory J] (F : J ⥤ LightProfinite.{max u v}) : Limits.IsLimit (limitCone F) where lift S := (CompHaus.limitConeIsLimit.{v, u} (F ⋙ lightProfiniteToCompHaus)).lift (lightProfiniteToCompHaus.mapCone S) uniq S _ h := (CompHaus.limitConeIsLimit.{v, u} _).uniq (lightProfiniteToCompHaus.mapCone S) _ h noncomputable instance createsCountableLimits {J : Type v} [SmallCategory J] [CountableCategory J] : CreatesLimitsOfShape J lightToProfinite.{max v u} where CreatesLimit {F} := createsLimitOfFullyFaithfulOfIso (limitCone.{v, u} F).pt <\| (Profinite.limitConeIsLimit.{v, u} (F ⋙ lightToProfinite)).conePointUniqueUpToIso (limit.isLimit _) instance : HasCountableLimits LightProfinite where out _ := { has_limit := fun F ↦ ⟨limitCone F, limitConeIsLimit F⟩ } instance : PreservesLimitsOfShape ℕᵒᵖ (forget LightProfinite.{u}) := have : PreservesLimitsOfSize.{0, 0} (forget Profinite.{u}) := preservesLimitsOfSize_shrink _ inferInstanceAs (PreservesLimitsOfShape ℕᵒᵖ (lightToProfinite ⋙ forget Profinite)) variable {X Y : LightProfinite.{u}} (f : X ⟶ Y) /-- Any morphism of light profinite spaces is a closed map. -/ theorem isClosedMap : IsClosedMap f := CompHausLike.isClosedMap _ /-- Any continuous bijection of light profinite spaces induces an isomorphism. -/ theorem isIso_of_bijective (bij : Function.Bijective f) : IsIso f := haveI := CompHausLike.isIso_of_bijective (lightProfiniteToCompHaus.map f) bij isIso_of_fully_faithful lightProfiniteToCompHaus _ /-- Any continuous bijection of light profinite spaces induces an isomorphism. -/ noncomputable def isoOfBijective (bij : Function.Bijective f) : X ≅ Y := letI := LightProfinite.isIso_of_bijective f bij asIso f instance forget_reflectsIsomorphisms : (forget LightProfinite).ReflectsIsomorphisms := by constructor intro A B f hf rw [isIso_iff_bijective] at hf exact LightProfinite.isIso_of_bijective _ hf theorem epi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by constructor · -- Note: in mathlib3 `contrapose` saw through `Function.Surjective`. dsimp [Function.Surjective] contrapose! rintro ⟨y, hy⟩ hf let C := Set.range f have hC : IsClosed C := (isCompact_range f.hom.continuous).isClosed let U := Cᶜ have hyU : y ∈ U := by refine Set.mem_compl ?_ rintro ⟨y', hy'⟩ exact hy y' hy' have hUy : U ∈ 𝓝 y := hC.compl_mem_nhds hyU obtain ⟨V, hV, hyV, hVU⟩ := isTopologicalBasis_isClopen.mem_nhds_iff.mp hUy classical let Z := of (ULift.{u} <\| Fin 2) let g : Y ⟶ Z := CompHausLike.ofHom _ ⟨(LocallyConstant.ofIsClopen hV).map ULift.up, LocallyConstant.continuous _⟩ let h : Y ⟶ Z := CompHausLike.ofHom _ ⟨fun _ => ⟨1⟩, continuous_const⟩ have H : h = g := by rw [← cancel_epi f] ext x dsimp [g, LocallyConstant.ofIsClopen] rw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, hom_ofHom, ContinuousMap.coe_mk, Function.comp_apply, if_neg] refine mt (fun α => hVU α) ?_ simp [U, C] apply_fun fun e => (e y).down at H dsimp [g, LocallyConstant.ofIsClopen] at H rw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Function.comp_apply, if_pos hyV] at H exact top_ne_bot H · rw [← CategoryTheory.epi_iff_surjective] apply (forget LightProfinite).epi_of_epi_map instance : lightToProfinite.PreservesEpimorphisms where preserves f _ := (Profinite.epi_iff_surjective _).mpr ((epi_iff_surjective f).mp inferInstance) end LightProfinite /-- A structure containing the data of sequential limit in `Profinite` of finite sets. -/ structure LightDiagram : Type (u+1) where /-- The indexing diagram. -/ diagram : ℕᵒᵖ ⥤ FintypeCat /-- The limit cone. -/ cone : Cone (diagram ⋙ FintypeCat.toProfinite.{u}) /-- The limit cone is limiting. -/ isLimit : IsLimit cone namespace LightDiagram /-- The underlying `Profinite` of a `LightDiagram`. -/ def toProfinite (S : LightDiagram) : Profinite := S.cone.pt @[simps!] instance : Category LightDiagram := InducedCategory.category toProfinite instance hasForget : ConcreteCategory LightDiagram (fun X Y => C(X.toProfinite, Y.toProfinite)) := InducedCategory.concreteCategory toProfinite end LightDiagram /-- The fully faithful embedding `LightDiagram ⥤ Profinite` -/ @[simps!] def lightDiagramToProfinite : LightDiagram ⥤ Profinite := inducedFunctor _ instance : lightDiagramToProfinite.Faithful := show (inducedFunctor _).Faithful from inferInstance instance : lightDiagramToProfinite.Full := show (inducedFunctor _).Full from inferInstance namespace LightProfinite instance (S : LightProfinite) : Countable (Clopens S) := by rw [TopologicalSpace.Clopens.countable_iff_secondCountable] infer_instance instance instCountableDiscreteQuotient (S : LightProfinite) : Countable (DiscreteQuotient ((lightToProfinite.obj S))) := (DiscreteQuotient.finsetClopens_inj S).countable /-- A profinite space which is light gives rise to a light profinite space. -/ noncomputable def toLightDiagram (S : LightProfinite.{u}) : LightDiagram.{u} where diagram := IsCofiltered.sequentialFunctor _ ⋙ (lightToProfinite.obj S).fintypeDiagram cone := (Functor.Initial.limitConeComp (IsCofiltered.sequentialFunctor _) (lightToProfinite.obj S).lim).cone isLimit := (Functor.Initial.limitConeComp (IsCofiltered.sequentialFunctor _) (lightToProfinite.obj S).lim).isLimit end LightProfinite /-- The functor part of the equivalence `LightProfinite ≌ LightDiagram` -/ @[simps] noncomputable def lightProfiniteToLightDiagram : LightProfinite.{u} ⥤ LightDiagram.{u} where obj X := X.toLightDiagram map f := f open scoped Classical in instance (S : LightDiagram.{u}) : SecondCountableTopology S.cone.pt := by rw [← TopologicalSpace.Clopens.countable_iff_secondCountable] refine @Countable.of_equiv _ _ ?_ (LocallyConstant.equivClopens (X := S.cone.pt)) refine @Function.Surjective.countable (Σ (n : ℕ), LocallyConstant ((S.diagram ⋙ FintypeCat.toProfinite).obj ⟨n⟩) (Fin 2)) _ ?_ ?_ ?_ · apply @instCountableSigma _ _ _ ?_ intro n refine @Finite.to_countable _ ?_ refine @Finite.of_injective _ ((S.diagram ⋙ FintypeCat.toProfinite).obj ⟨n⟩ → (Fin 2)) ?_ _ LocallyConstant.coe_injective refine @Pi.finite _ _ ?_ _ simp only [Functor.comp_obj] exact show (Finite (S.diagram.obj _)) from inferInstance · exact fun a ↦ a.snd.comap (S.cone.π.app ⟨a.fst⟩).hom · intro a obtain ⟨n, g, h⟩ := Profinite.exists_locallyConstant S.cone S.isLimit a exact ⟨⟨unop n, g⟩, h.symm⟩ /-- The inverse part of the equivalence `LightProfinite ≌ LightDiagram` -/ @[simps obj map] def lightDiagramToLightProfinite : LightDiagram.{u} ⥤ LightProfinite.{u} where obj X := LightProfinite.of X.cone.pt map f := f /-- The equivalence of categories `LightProfinite ≌ LightDiagram` -/ noncomputable def LightProfinite.equivDiagram : LightProfinite.{u} ≌ LightDiagram.{u} where functor := lightProfiniteToLightDiagram inverse := lightDiagramToLightProfinite unitIso := Iso.refl _ counitIso := NatIso.ofComponents (fun _ ↦ lightDiagramToProfinite.preimageIso (Iso.refl _)) (by intro _ _ f simp only [Functor.comp_obj, lightDiagramToLightProfinite_obj, lightProfiniteToLightDiagram_obj, Functor.id_obj, Functor.comp_map, lightDiagramToLightProfinite_map, lightProfiniteToLightDiagram_map, lightDiagramToProfinite_obj, Functor.preimageIso_hom, Iso.refl_hom, Functor.id_map] erw [lightDiagramToProfinite.preimage_id, lightDiagramToProfinite.preimage_id, Category.comp_id f]) functor_unitIso_comp _ := by simpa using lightDiagramToProfinite.preimage_id instance : lightProfiniteToLightDiagram.IsEquivalence := show LightProfinite.equivDiagram.functor.IsEquivalence from inferInstance instance : lightDiagramToLightProfinite.IsEquivalence := show LightProfinite.equivDiagram.inverse.IsEquivalence from inferInstance noncomputable section EssentiallySmall open LightDiagram /-- This is an auxiliary definition used to show that `LightDiagram` is essentially small. Note that below we put a category instance on this structure which is completely different from the category instance on `ℕᵒᵖ ⥤ FintypeCat.Skeleton.{u}`. Neither the morphisms nor the objects are the same. -/ structure LightDiagram' : Type u where /-- The diagram takes values in a small category equivalent to `FintypeCat`. -/ diagram : ℕᵒᵖ ⥤ FintypeCat.Skeleton.{u} /-- A `LightDiagram'` yields a `Profinite`. -/ def LightDiagram'.toProfinite (S : LightDiagram') : Profinite := limit (S.diagram ⋙ FintypeCat.Skeleton.equivalence.functor ⋙ FintypeCat.toProfinite.{u}) instance : Category LightDiagram' := InducedCategory.category LightDiagram'.toProfinite /-- The functor part of the equivalence of categories `LightDiagram' ≌ LightDiagram`. -/ def LightDiagram'.toLightFunctor : LightDiagram'.{u} ⥤ LightDiagram.{u} where obj X := ⟨X.diagram ⋙ Skeleton.equivalence.functor, _, limit.isLimit _⟩ map f := f instance : LightDiagram'.toLightFunctor.{u}.Faithful := ⟨id⟩ instance : LightDiagram'.toLightFunctor.{u}.Full where map_surjective f := ⟨f, rfl⟩ instance : LightDiagram'.toLightFunctor.{u}.EssSurj where mem_essImage Y := ⟨⟨Y.diagram ⋙ Skeleton.equivalence.inverse⟩, ⟨lightDiagramToProfinite.preimageIso ( (Limits.lim.mapIso (Functor.isoWhiskerRight ((Functor.isoWhiskerLeft Y.diagram Skeleton.equivalence.counitIso)) toProfinite)) ≪≫ (limit.isLimit _).conePointUniqueUpToIso Y.isLimit)⟩⟩ instance : LightDiagram'.toLightFunctor.IsEquivalence where /-- The equivalence between `LightDiagram` and a small category. -/ def LightDiagram.equivSmall : LightDiagram.{u} ≌ LightDiagram'.{u} := LightDiagram'.toLightFunctor.asEquivalence.symm instance : EssentiallySmall.{u} LightDiagram.{u} where equiv_smallCategory := ⟨LightDiagram', inferInstance, ⟨LightDiagram.equivSmall⟩⟩ instance : EssentiallySmall.{u} LightProfinite.{u} where equiv_smallCategory := ⟨LightDiagram', inferInstance, ⟨LightProfinite.equivDiagram.trans LightDiagram.equivSmall⟩⟩ end EssentiallySmall
.lake/packages/mathlib/Mathlib/Topology/Category/LightProfinite/Limits.lean	import Mathlib.Topology.Category.CompHausLike.Limits import Mathlib.Topology.Category.LightProfinite.Basic /-! # Explicit limits and colimits This file applies the general API for explicit limits and colimits in `CompHausLike P` (see the file `Mathlib/Topology/Category/CompHausLike/Limits.lean`) to the special case of `LightProfinite`. -/ namespace LightProfinite universe u w open CategoryTheory Limits CompHausLike instance : HasExplicitPullbacks (fun Y ↦ TotallyDisconnectedSpace Y ∧ SecondCountableTopology Y) where hasProp _ _ := { hasProp := ⟨show TotallyDisconnectedSpace {_xy : _ \| _} from inferInstance, show SecondCountableTopology {_xy : _ \| _} from inferInstance⟩ } instance : HasExplicitFiniteCoproducts.{w, u} (fun Y ↦ TotallyDisconnectedSpace Y ∧ SecondCountableTopology Y) where hasProp _ := { hasProp := ⟨show TotallyDisconnectedSpace (Σ (_a : _), _) from inferInstance, show SecondCountableTopology (Σ (_a : _), _) from inferInstance⟩ } /-- A one-element space is terminal in `Profinite` -/ abbrev isTerminalPUnit : IsTerminal (LightProfinite.of PUnit.{u + 1}) := CompHausLike.isTerminalPUnit example : FinitaryExtensive LightProfinite.{u} := inferInstance noncomputable example : PreservesFiniteCoproducts lightProfiniteToCompHaus.{u} := inferInstance end LightProfinite
.lake/packages/mathlib/Mathlib/Topology/Category/LightProfinite/Extend.lean	import Mathlib.Topology.Category.LightProfinite.AsLimit import Mathlib.Topology.Category.Profinite.Extend /-! # Extending cones in `LightProfinite` Let `(Sₙ)_{n : ℕᵒᵖ}` be a sequential inverse system of finite sets and let `S` be its limit in `Profinite`. Let `G` be a functor from `LightProfinite` to a category `C` and suppose that `G` preserves the limit described above. Suppose further that the projection maps `S ⟶ Sₙ` are epimorphic for all `n`. Then `G.obj S` is isomorphic to a limit indexed by `StructuredArrow S toLightProfinite` (see `LightProfinite.Extend.isLimitCone`). We also provide the dual result for a functor of the form `G : LightProfiniteᵒᵖ ⥤ C`. We apply this to define `LightProfinite.diagram'`, `LightProfinite.asLimitCone'`, and `LightProfinite.asLimit'`, analogues to their unprimed versions in `Mathlib/Topology/Category/LightProfinite/AsLimit.lean`, in which the indexing category is `StructuredArrow S toLightProfinite` instead of `ℕᵒᵖ`. -/ universe u open CategoryTheory Limits FintypeCat Functor attribute [local instance] FintypeCat.discreteTopology namespace LightProfinite variable {F : ℕᵒᵖ ⥤ FintypeCat.{u}} (c : Cone <\| F ⋙ toLightProfinite) namespace Extend /-- Given a sequential cone in `LightProfinite` consisting of finite sets, we obtain a functor from the indexing category to `StructuredArrow c.pt toLightProfinite`. -/ @[simps] def functor : ℕᵒᵖ ⥤ StructuredArrow c.pt toLightProfinite where obj i := StructuredArrow.mk (c.π.app i) map f := StructuredArrow.homMk (F.map f) (c.w f) -- We check that the original diagram factors through `LightProfinite.Extend.functor`. example : functor c ⋙ StructuredArrow.proj c.pt toLightProfinite ≅ F := Iso.refl _ /-- Given a sequential cone in `LightProfinite` consisting of finite sets, we obtain a functor from the opposite of the indexing category to `CostructuredArrow toProfinite.op ⟨c.pt⟩`. -/ @[simps! obj map] def functorOp : ℕ ⥤ CostructuredArrow toLightProfinite.op ⟨c.pt⟩ := (functor c).rightOp ⋙ StructuredArrow.toCostructuredArrow _ _ -- We check that the opposite of the original diagram factors through `Profinite.Extend.functorOp`. example : functorOp c ⋙ CostructuredArrow.proj toLightProfinite.op ⟨c.pt⟩ ≅ F.rightOp := Iso.refl _ -- We check that `Profinite.Extend.functor` factors through `LightProfinite.Extend.functor`, -- via the equivalence `StructuredArrow.post _ _ lightToProfinite`. example : functor c ⋙ (StructuredArrow.post _ _ lightToProfinite) = Profinite.Extend.functor (lightToProfinite.mapCone c) := rfl /-- If the projection maps in the cone are epimorphic and the cone is limiting, then `LightProfinite.Extend.functor` is initial. -/ theorem functor_initial (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Initial (functor c) := by rw [initial_iff_comp_equivalence _ (StructuredArrow.post _ _ lightToProfinite)] have : ∀ i, Epi ((lightToProfinite.mapCone c).π.app i) := fun i ↦ inferInstanceAs (Epi (lightToProfinite.map (c.π.app i))) exact Profinite.Extend.functor_initial _ (isLimitOfPreserves lightToProfinite hc) /-- If the projection maps in the cone are epimorphic and the cone is limiting, then `LightProfinite.Extend.functorOp` is final. -/ theorem functorOp_final (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Final (functorOp c) := by have := functor_initial c hc have : ((StructuredArrow.toCostructuredArrow toLightProfinite c.pt)).IsEquivalence := (inferInstance : (structuredArrowOpEquivalence _ _).functor.IsEquivalence ) have : (functor c).rightOp.Final := inferInstanceAs ((opOpEquivalence ℕ).inverse ⋙ (functor c).op).Final exact Functor.final_comp (functor c).rightOp _ section Limit variable {C : Type} [Category C] (G : LightProfinite ⥤ C) /-- Given a functor `G` from `LightProfinite` and `S : LightProfinite`, we obtain a cone on `(StructuredArrow.proj S toLightProfinite ⋙ toLightProfinite ⋙ G)` with cone point `G.obj S`. Whiskering this cone with `LightProfinite.Extend.functor c` gives `G.mapCone c` as we check in the example below. -/ def cone (S : LightProfinite) : Cone (StructuredArrow.proj S toLightProfinite ⋙ toLightProfinite ⋙ G) where pt := G.obj S π := { app := fun i ↦ G.map i.hom naturality := fun _ _ f ↦ (by have := f.w simp only [const_obj_obj, StructuredArrow.left_eq_id, const_obj_map, Category.id_comp, StructuredArrow.w] at this simp only [const_obj_obj, comp_obj, StructuredArrow.proj_obj, const_obj_map, Category.id_comp, Functor.comp_map, StructuredArrow.proj_map, ← map_comp, StructuredArrow.w]) } example : G.mapCone c = (cone G c.pt).whisker (functor c) := rfl /-- If `c` and `G.mapCone c` are limit cones and the projection maps in `c` are epimorphic, then `cone G c.pt` is a limit cone. -/ noncomputable def isLimitCone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsLimit <\| G.mapCone c) : IsLimit (cone G c.pt) := (functor_initial c hc).isLimitWhiskerEquiv _ _ hc' end Limit section Colimit variable {C : Type} [Category C] (G : LightProfiniteᵒᵖ ⥤ C) /-- Given a functor `G` from `LightProfiniteᵒᵖ` and `S : LightProfinite`, we obtain a cocone on `(CostructuredArrow.proj toLightProfinite.op ⟨S⟩ ⋙ toLightProfinite.op ⋙ G)` with cocone point `G.obj ⟨S⟩`. Whiskering this cocone with `LightProfinite.Extend.functorOp c` gives `G.mapCocone c.op` as we check in the example below. -/ @[simps] def cocone (S : LightProfinite) : Cocone (CostructuredArrow.proj toLightProfinite.op ⟨S⟩ ⋙ toLightProfinite.op ⋙ G) where pt := G.obj ⟨S⟩ ι := { app := fun i ↦ G.map i.hom naturality := fun _ _ f ↦ (by have := f.w simp only [op_obj, const_obj_obj, op_map, CostructuredArrow.right_eq_id, const_obj_map, Category.comp_id] at this simp only [comp_obj, CostructuredArrow.proj_obj, op_obj, const_obj_obj, Functor.comp_map, CostructuredArrow.proj_map, op_map, ← map_comp, this, const_obj_map, Category.comp_id]) } example : G.mapCocone c.op = (cocone G c.pt).whisker ((opOpEquivalence ℕ).functor ⋙ functorOp c) := rfl /-- If `c` is a limit cone, `G.mapCocone c.op` is a colimit cone and the projection maps in `c` are epimorphic, then `cocone G c.pt` is a colimit cone. -/ noncomputable def isColimitCocone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsColimit <\| G.mapCocone c.op) : IsColimit (cocone G c.pt) := haveI := functorOp_final c hc (Functor.final_comp (opOpEquivalence ℕ).functor (functorOp c)).isColimitWhiskerEquiv _ _ hc' end Colimit end Extend open Extend section LightProfiniteAsLimit variable (S : LightProfinite.{u}) /-- A functor `StructuredArrow S toLightProfinite ⥤ FintypeCat` whose limit in `LightProfinite` is isomorphic to `S`. -/ abbrev fintypeDiagram' : StructuredArrow S toLightProfinite ⥤ FintypeCat := StructuredArrow.proj S toLightProfinite /-- An abbreviation for `S.fintypeDiagram' ⋙ toLightProfinite`. -/ abbrev diagram' : StructuredArrow S toLightProfinite ⥤ LightProfinite := S.fintypeDiagram' ⋙ toLightProfinite /-- A cone over `S.diagram'` whose cone point is `S`. -/ def asLimitCone' : Cone (S.diagram') := cone (𝟭 _) S instance (i : ℕᵒᵖ) : Epi (S.asLimitCone.π.app i) := (epi_iff_surjective _).mpr (S.proj_surjective _) /-- `S.asLimitCone'` is a limit cone. -/ noncomputable def asLimit' : IsLimit S.asLimitCone' := isLimitCone _ (𝟭 _) S.asLimit S.asLimit /-- A bundled version of `S.asLimitCone'` and `S.asLimit'`. -/ noncomputable def lim' : LimitCone S.diagram' := ⟨S.asLimitCone', S.asLimit'⟩ end LightProfiniteAsLimit end LightProfinite
.lake/packages/mathlib/Mathlib/Topology/Category/LightProfinite/AsLimit.lean	import Mathlib.Topology.Category.LightProfinite.Basic /-! # Light profinite sets as limits of finite sets. We show that any light profinite set is isomorphic to a sequential limit of finite sets. The limit cone for `S : LightProfinite` is `S.asLimitCone`, the fact that it's a limit is `S.asLimit`. We also prove that the projection and transition maps in this limit are surjective. -/ noncomputable section open CategoryTheory Limits CompHausLike namespace LightProfinite universe u variable (S : LightProfinite.{u}) /-- The functor `ℕᵒᵖ ⥤ FintypeCat` whose limit is isomorphic to `S`. -/ abbrev fintypeDiagram : ℕᵒᵖ ⥤ FintypeCat := S.toLightDiagram.diagram /-- An abbreviation for `S.fintypeDiagram ⋙ FintypeCat.toProfinite`. -/ abbrev diagram : ℕᵒᵖ ⥤ LightProfinite := S.fintypeDiagram ⋙ FintypeCat.toLightProfinite /-- A cone over `S.diagram` whose cone point is isomorphic to `S`. (Auxiliary definition, use `S.asLimitCone` instead.) -/ def asLimitConeAux : Cone S.diagram := let c : Cone (S.diagram ⋙ lightToProfinite) := S.toLightDiagram.cone let hc : IsLimit c := S.toLightDiagram.isLimit liftLimit hc /-- An auxiliary isomorphism of cones used to prove that `S.asLimitConeAux` is a limit cone. -/ def isoMapCone : lightToProfinite.mapCone S.asLimitConeAux ≅ S.toLightDiagram.cone := let c : Cone (S.diagram ⋙ lightToProfinite) := S.toLightDiagram.cone let hc : IsLimit c := S.toLightDiagram.isLimit liftedLimitMapsToOriginal hc /-- `S.asLimitConeAux` is indeed a limit cone. (Auxiliary definition, use `S.asLimit` instead.) -/ def asLimitAux : IsLimit S.asLimitConeAux := let hc : IsLimit (lightToProfinite.mapCone S.asLimitConeAux) := S.toLightDiagram.isLimit.ofIsoLimit S.isoMapCone.symm isLimitOfReflects lightToProfinite hc /-- A cone over `S.diagram` whose cone point is `S`. -/ def asLimitCone : Cone S.diagram where pt := S π := { app := fun n ↦ (lightToProfiniteFullyFaithful.preimageIso <\| (Cones.forget _).mapIso S.isoMapCone).inv ≫ S.asLimitConeAux.π.app n naturality := fun _ _ _ ↦ by simp only [Category.assoc, S.asLimitConeAux.w]; rfl } /-- `S.asLimitCone` is indeed a limit cone. -/ def asLimit : IsLimit S.asLimitCone := S.asLimitAux.ofIsoLimit <\| Cones.ext (lightToProfiniteFullyFaithful.preimageIso <\| (Cones.forget _).mapIso S.isoMapCone) (fun _ ↦ by rw [← @Iso.inv_comp_eq]; rfl) /-- A bundled version of `S.asLimitCone` and `S.asLimit`. -/ def lim : Limits.LimitCone S.diagram := ⟨S.asLimitCone, S.asLimit⟩ /-- The projection from `S` to the `n`th component of `S.diagram`. -/ abbrev proj (n : ℕ) : S ⟶ S.diagram.obj ⟨n⟩ := S.asLimitCone.π.app ⟨n⟩ lemma lightToProfinite_map_proj_eq (n : ℕ) : lightToProfinite.map (S.proj n) = (lightToProfinite.obj S).asLimitCone.π.app _ := by simp only [Functor.comp_obj, toCompHausLike_map] let c : Cone (S.diagram ⋙ lightToProfinite) := S.toLightDiagram.cone let hc : IsLimit c := S.toLightDiagram.isLimit exact liftedLimitMapsToOriginal_inv_map_π hc _ lemma proj_surjective (n : ℕ) : Function.Surjective (S.proj n) := by change Function.Surjective (lightToProfinite.map (S.proj n)) rw [lightToProfinite_map_proj_eq] exact DiscreteQuotient.proj_surjective _ /-- An abbreviation for the `n`th component of `S.diagram`. -/ abbrev component (n : ℕ) : LightProfinite := S.diagram.obj ⟨n⟩ /-- The transition map from `S_{n+1}` to `S_n` in `S.diagram`. -/ abbrev transitionMap (n : ℕ) : S.component (n + 1) ⟶ S.component n := S.diagram.map ⟨homOfLE (Nat.le_succ _)⟩ /-- The transition map from `S_m` to `S_n` in `S.diagram`, when `m ≤ n`. -/ abbrev transitionMapLE {n m : ℕ} (h : n ≤ m) : S.component m ⟶ S.component n := S.diagram.map ⟨homOfLE h⟩ lemma proj_comp_transitionMap (n : ℕ) : S.proj (n + 1) ≫ S.diagram.map ⟨homOfLE (Nat.le_succ _)⟩ = S.proj n := S.asLimitCone.w (homOfLE (Nat.le_succ n)).op lemma proj_comp_transitionMap' (n : ℕ) : S.transitionMap n ∘ S.proj (n + 1) = S.proj n := by rw [← S.proj_comp_transitionMap n] rfl lemma proj_comp_transitionMapLE {n m : ℕ} (h : n ≤ m) : S.proj m ≫ S.diagram.map ⟨homOfLE h⟩ = S.proj n := S.asLimitCone.w (homOfLE h).op lemma proj_comp_transitionMapLE' {n m : ℕ} (h : n ≤ m) : S.transitionMapLE h ∘ S.proj m = S.proj n := by rw [← S.proj_comp_transitionMapLE h] rfl lemma surjective_transitionMap (n : ℕ) : Function.Surjective (S.transitionMap n) := by apply Function.Surjective.of_comp (g := S.proj (n + 1)) simpa only [proj_comp_transitionMap'] using S.proj_surjective n lemma surjective_transitionMapLE {n m : ℕ} (h : n ≤ m) : Function.Surjective (S.transitionMapLE h) := by apply Function.Surjective.of_comp (g := S.proj m) simpa only [proj_comp_transitionMapLE'] using S.proj_surjective n end LightProfinite
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Yoneda.lean	import Mathlib.CategoryTheory.Limits.Preserves.Finite import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products import Mathlib.CategoryTheory.Limits.Types.Products import Mathlib.Topology.Category.TopCat.Limits.Products /-! # Yoneda presheaves on topologically concrete categories This file develops some API for "topologically concrete" categories, defining universe polymorphic "Yoneda presheaves" on such categories. -/ universe w w' v u open CategoryTheory Opposite Limits variable {C : Type u} [Category.{v} C] (F : C ⥤ TopCat.{w}) (Y : Type w') [TopologicalSpace Y] namespace ContinuousMap /-- A universe polymorphic "Yoneda presheaf" on `C` given by continuous maps into a topological space `Y`. -/ @[simps] def yonedaPresheaf : Cᵒᵖ ⥤ Type (max w w') where obj X := C(F.obj (unop X), Y) map f g := ContinuousMap.comp g (F.map f.unop).hom /-- A universe polymorphic Yoneda presheaf on `TopCat` given by continuous maps into a topological space `Y`. -/ @[simps] def yonedaPresheaf' : TopCat.{w}ᵒᵖ ⥤ Type (max w w') where obj X := C((unop X).1, Y) map f g := ContinuousMap.comp g f.unop.hom theorem comp_yonedaPresheaf' : yonedaPresheaf F Y = F.op ⋙ yonedaPresheaf' Y := rfl theorem piComparison_fac {α : Type} (X : α → TopCat) : piComparison (yonedaPresheaf'.{w, w'} Y) (fun x ↦ op (X x)) = (yonedaPresheaf' Y).map ((opCoproductIsoProduct X).inv ≫ (TopCat.sigmaIsoSigma X).inv.op) ≫ (equivEquivIso (sigmaEquiv Y (fun x ↦ (X x).1))).inv ≫ (Types.productIso _).inv := by rw [← Category.assoc, Iso.eq_comp_inv] ext simp only [yonedaPresheaf', unop_op, piComparison, types_comp_apply, Types.productIso_hom_comp_eval_apply, Types.pi_lift_π_apply, comp_apply, TopCat.coe_of, unop_comp, Quiver.Hom.unop_op, sigmaEquiv, equivEquivIso_hom, Equiv.toIso_inv, Equiv.coe_fn_symm_mk, comp_assoc, sigmaMk_apply, ← opCoproductIsoProduct_inv_comp_ι] rfl /-- The universe polymorphic Yoneda presheaf on `TopCat` preserves finite products. -/ noncomputable instance : PreservesFiniteProducts (yonedaPresheaf'.{w, w'} Y) where preserves _ := { preservesLimit := fun {K} => have : ∀ {α : Type} (X : α → TopCat), PreservesLimit (Discrete.functor (fun x ↦ op (X x))) (yonedaPresheaf'.{w, w'} Y) := fun X => @PreservesProduct.of_iso_comparison _ _ _ _ (yonedaPresheaf' Y) _ (fun x ↦ op (X x)) _ _ (by rw [piComparison_fac]; infer_instance) let i : K ≅ Discrete.functor (fun i ↦ op (unop (K.obj ⟨i⟩))) := Discrete.natIsoFunctor preservesLimit_of_iso_diagram _ i.symm } end ContinuousMap
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/EpiMono.lean	import Mathlib.Topology.Category.TopCat.Adjunctions import Mathlib.CategoryTheory.Functor.EpiMono /-! # Epi- and monomorphisms in `Top` This file shows that a continuous function is an epimorphism in the category of topological spaces if and only if it is surjective, and that a continuous function is a monomorphism in the category of topological spaces if and only if it is injective. -/ universe u open CategoryTheory open TopCat namespace TopCat theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Function.Surjective f := by suffices Epi f ↔ Epi ((forget TopCat).map f) by rw [this, CategoryTheory.epi_iff_surjective] rfl constructor · intro infer_instance · apply Functor.epi_of_epi_map theorem mono_iff_injective {X Y : TopCat.{u}} (f : X ⟶ Y) : Mono f ↔ Function.Injective f := by suffices Mono f ↔ Mono ((forget TopCat).map f) by rw [this, CategoryTheory.mono_iff_injective] rfl constructor · intro infer_instance · apply Functor.mono_of_mono_map end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Opens.lean	import Mathlib.CategoryTheory.Category.GaloisConnection import Mathlib.CategoryTheory.EqToHom import Mathlib.Topology.Category.TopCat.EpiMono import Mathlib.Topology.Sets.Opens /-! # The category of open sets in a topological space. We define `toTopCat : Opens X ⥤ TopCat` and `map (f : X ⟶ Y) : Opens Y ⥤ Opens X`, given by taking preimages of open sets. Unfortunately `Opens` isn't (usefully) a functor `TopCat ⥤ Cat`. (One can in fact define such a functor, but using it results in unresolvable `Eq.rec` terms in goals.) Really it's a 2-functor from (spaces, continuous functions, equalities) to (categories, functors, natural isomorphisms). We don't attempt to set up the full theory here, but do provide the natural isomorphisms `mapId : map (𝟙 X) ≅ 𝟭 (Opens X)` and `mapComp : map (f ≫ g) ≅ map g ⋙ map f`. Beyond that, there's a collection of simp lemmas for working with these constructions. -/ open CategoryTheory TopologicalSpace Opposite Topology universe u namespace TopologicalSpace.Opens variable {X Y Z : TopCat.{u}} {U V W : Opens X} /-! Since `Opens X` has a partial order, it automatically receives a `Category` instance. Unfortunately, because we do not allow morphisms in `Prop`, the morphisms `U ⟶ V` are not just proofs `U ≤ V`, but rather `ULift (PLift (U ≤ V))`. -/ instance opensHom.instFunLike : FunLike (U ⟶ V) U V where coe f := Set.inclusion f.le coe_injective' := by rintro ⟨⟨_⟩⟩ _ _; congr! lemma apply_def (f : U ⟶ V) (x : U) : f x = ⟨x, f.le x.2⟩ := rfl @[simp] lemma apply_mk (f : U ⟶ V) (x : X) (hx) : f ⟨x, hx⟩ = ⟨x, f.le hx⟩ := rfl @[simp] lemma val_apply (f : U ⟶ V) (x : U) : (f x : X) = x := rfl @[simp, norm_cast] lemma coe_id (f : U ⟶ U) : ⇑f = id := rfl lemma id_apply (f : U ⟶ U) (x : U) : f x = x := rfl @[simp] lemma comp_apply (f : U ⟶ V) (g : V ⟶ W) (x : U) : (f ≫ g) x = g (f x) := rfl /-! We now construct as morphisms various inclusions of open sets. -/ -- This is tedious, but necessary because we decided not to allow Prop as morphisms in a category... /-- The inclusion `U ⊓ V ⟶ U` as a morphism in the category of open sets. -/ noncomputable def infLELeft (U V : Opens X) : U ⊓ V ⟶ U := inf_le_left.hom /-- The inclusion `U ⊓ V ⟶ V` as a morphism in the category of open sets. -/ noncomputable def infLERight (U V : Opens X) : U ⊓ V ⟶ V := inf_le_right.hom /-- The inclusion `U i ⟶ iSup U` as a morphism in the category of open sets. -/ noncomputable def leSupr {ι : Type} (U : ι → Opens X) (i : ι) : U i ⟶ iSup U := (le_iSup U i).hom /-- The inclusion `⊥ ⟶ U` as a morphism in the category of open sets. -/ noncomputable def botLE (U : Opens X) : ⊥ ⟶ U := bot_le.hom /-- The inclusion `U ⟶ ⊤` as a morphism in the category of open sets. -/ noncomputable def leTop (U : Opens X) : U ⟶ ⊤ := le_top.hom -- We do not mark this as a simp lemma because it breaks open `x`. -- Nevertheless, it is useful in `SheafOfFunctions`. theorem infLELeft_apply (U V : Opens X) (x) : (infLELeft U V) x = ⟨x.1, (@inf_le_left _ _ U V : _ ≤ _) x.2⟩ := rfl @[simp] theorem infLELeft_apply_mk (U V : Opens X) (x) (m) : (infLELeft U V) ⟨x, m⟩ = ⟨x, (@inf_le_left _ _ U V : _ ≤ _) m⟩ := rfl @[simp] theorem leSupr_apply_mk {ι : Type} (U : ι → Opens X) (i : ι) (x) (m) : (leSupr U i) ⟨x, m⟩ = ⟨x, (le_iSup U i :) m⟩ := rfl /-- The functor from open sets in `X` to `TopCat`, realising each open set as a topological space itself. -/ def toTopCat (X : TopCat.{u}) : Opens X ⥤ TopCat where obj U := TopCat.of U map i := TopCat.ofHom ⟨fun x ↦ ⟨x.1, i.le x.2⟩, IsEmbedding.subtypeVal.continuous_iff.2 continuous_induced_dom⟩ @[simp] theorem toTopCat_map (X : TopCat.{u}) {U V : Opens X} {f : U ⟶ V} {x} {h} : ((toTopCat X).map f) ⟨x, h⟩ = ⟨x, f.le h⟩ := rfl /-- The inclusion map from an open subset to the whole space, as a morphism in `TopCat`. -/ @[simps! -fullyApplied] def inclusion' {X : TopCat.{u}} (U : Opens X) : (toTopCat X).obj U ⟶ X := TopCat.ofHom { toFun := _ continuous_toFun := continuous_subtype_val } @[simp] theorem coe_inclusion' {X : TopCat} {U : Opens X} : (inclusion' U : U → X) = Subtype.val := rfl theorem isOpenEmbedding {X : TopCat.{u}} (U : Opens X) : IsOpenEmbedding (inclusion' U) := U.2.isOpenEmbedding_subtypeVal /-- The inclusion of the top open subset (i.e. the whole space) is an isomorphism. -/ def inclusionTopIso (X : TopCat.{u}) : (toTopCat X).obj ⊤ ≅ X where hom := inclusion' ⊤ inv := TopCat.ofHom ⟨fun x => ⟨x, trivial⟩, continuous_def.2 fun _ ⟨_, hS, hSU⟩ => hSU ▸ hS⟩ /-- `Opens.map f` gives the functor from open sets in Y to open set in X, given by taking preimages under f. -/ def map (f : X ⟶ Y) : Opens Y ⥤ Opens X where obj U := ⟨f ⁻¹' (U : Set Y), U.isOpen.preimage f.hom.continuous⟩ map i := ⟨⟨fun _ h => i.le h⟩⟩ @[simp] theorem map_coe (f : X ⟶ Y) (U : Opens Y) : ((map f).obj U : Set X) = f ⁻¹' (U : Set Y) := rfl @[simp] theorem map_obj (f : X ⟶ Y) (U) (p) : (map f).obj ⟨U, p⟩ = ⟨f ⁻¹' U, p.preimage f.hom.continuous⟩ := rfl @[simp] lemma map_homOfLE (f : X ⟶ Y) {U V : Opens Y} (e : U ≤ V) : (TopologicalSpace.Opens.map f).map (homOfLE e) = homOfLE (show (Opens.map f).obj U ≤ (Opens.map f).obj V from fun _ hx ↦ e hx) := rfl @[simp] theorem map_id_obj (U : Opens X) : (map (𝟙 X)).obj U = U := let ⟨_, _⟩ := U rfl @[simp] theorem map_id_obj' (U) (p) : (map (𝟙 X)).obj ⟨U, p⟩ = ⟨U, p⟩ := rfl theorem map_id_obj_unop (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).obj (unop U) = unop U := by simp theorem op_map_id_obj (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).op.obj U = U := by simp @[simp] lemma map_top (f : X ⟶ Y) : (Opens.map f).obj ⊤ = ⊤ := rfl /-- The inclusion `U ⟶ (map f).obj ⊤` as a morphism in the category of open sets. -/ noncomputable def leMapTop (f : X ⟶ Y) (U : Opens X) : U ⟶ (map f).obj ⊤ := leTop U @[simp] theorem map_comp_obj (f : X ⟶ Y) (g : Y ⟶ Z) (U) : (map (f ≫ g)).obj U = (map f).obj ((map g).obj U) := rfl @[simp] theorem map_comp_obj' (f : X ⟶ Y) (g : Y ⟶ Z) (U) (p) : (map (f ≫ g)).obj ⟨U, p⟩ = (map f).obj ((map g).obj ⟨U, p⟩) := rfl @[simp] theorem map_comp_map (f : X ⟶ Y) (g : Y ⟶ Z) {U V} (i : U ⟶ V) : (map (f ≫ g)).map i = (map f).map ((map g).map i) := rfl @[simp] theorem map_comp_obj_unop (f : X ⟶ Y) (g : Y ⟶ Z) (U) : (map (f ≫ g)).obj (unop U) = (map f).obj ((map g).obj (unop U)) := rfl @[simp] theorem op_map_comp_obj (f : X ⟶ Y) (g : Y ⟶ Z) (U) : (map (f ≫ g)).op.obj U = (map f).op.obj ((map g).op.obj U) := rfl theorem map_iSup (f : X ⟶ Y) {ι : Type*} (U : ι → Opens Y) : (map f).obj (iSup U) = iSup ((map f).obj ∘ U) := by ext simp section variable (X) /-- The functor `Opens X ⥤ Opens X` given by taking preimages under the identity function is naturally isomorphic to the identity functor. -/ @[simps] def mapId : map (𝟙 X) ≅ 𝟭 (Opens X) where hom := { app := fun U => eqToHom (map_id_obj U) } inv := { app := fun U => eqToHom (map_id_obj U).symm } theorem map_id_eq : map (𝟙 X) = 𝟭 (Opens X) := by rfl end /-- The natural isomorphism between taking preimages under `f ≫ g`, and the composite of taking preimages under `g`, then preimages under `f`. -/ @[simps] def mapComp (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) ≅ map g ⋙ map f where hom := { app := fun U => eqToHom (map_comp_obj f g U) } inv := { app := fun U => eqToHom (map_comp_obj f g U).symm } theorem map_comp_eq (f : X ⟶ Y) (g : Y ⟶ Z) : map (f ≫ g) = map g ⋙ map f := rfl -- We could make `f g` implicit here, but it's nice to be able to see when -- they are the identity (often!) /-- If two continuous maps `f g : X ⟶ Y` are equal, then the functors `Opens Y ⥤ Opens X` they induce are isomorphic. -/ def mapIso (f g : X ⟶ Y) (h : f = g) : map f ≅ map g := NatIso.ofComponents fun U => eqToIso (by rw [congr_arg map h]) theorem map_eq (f g : X ⟶ Y) (h : f = g) : map f = map g := by subst h rfl @[simp] theorem mapIso_refl (f : X ⟶ Y) (h) : mapIso f f h = Iso.refl (map _) := rfl @[simp] theorem mapIso_hom_app (f g : X ⟶ Y) (h : f = g) (U : Opens Y) : (mapIso f g h).hom.app U = eqToHom (by rw [h]) := rfl @[simp] theorem mapIso_inv_app (f g : X ⟶ Y) (h : f = g) (U : Opens Y) : (mapIso f g h).inv.app U = eqToHom (by rw [h]) := rfl /-- A homeomorphism of spaces gives an equivalence of categories of open sets. TODO: define `OrderIso.equivalence`, use it. -/ @[simps] def mapMapIso {X Y : TopCat.{u}} (H : X ≅ Y) : Opens Y ≌ Opens X where functor := map H.hom inverse := map H.inv unitIso := NatIso.ofComponents fun U => eqToIso (by simp [map, Set.preimage_preimage]) counitIso := NatIso.ofComponents fun U => eqToIso (by simp [map, Set.preimage_preimage]) end TopologicalSpace.Opens /-- An open map `f : X ⟶ Y` induces a functor `Opens X ⥤ Opens Y`. -/ @[simps obj_coe] def IsOpenMap.functor {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) : Opens X ⥤ Opens Y where obj U := ⟨f '' (U : Set X), hf (U : Set X) U.2⟩ map h := ⟨⟨Set.image_mono h.down.down⟩⟩ /-- An open map `f : X ⟶ Y` induces an adjunction between `Opens X` and `Opens Y`. -/ def IsOpenMap.adjunction {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) : hf.functor ⊣ Opens.map f where unit := { app := fun _ => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ } counit := { app := fun _ => homOfLE fun _ ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV } instance IsOpenMap.functorFullOfMono {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) [H : Mono f] : hf.functor.Full where map_surjective i := ⟨homOfLE fun x hx => by obtain ⟨y, hy, eq⟩ := i.le ⟨x, hx, rfl⟩ exact (TopCat.mono_iff_injective f).mp H eq ▸ hy, rfl⟩ instance IsOpenMap.functor_faithful {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) : hf.functor.Faithful where lemma Topology.IsOpenEmbedding.functor_obj_injective {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenEmbedding f) : Function.Injective hf.isOpenMap.functor.obj := fun _ _ e ↦ Opens.ext (Set.image_injective.mpr hf.injective (congr_arg (↑· : Opens Y → Set Y) e)) namespace Topology.IsInducing /-- Given an inducing map `X ⟶ Y` and some `U : Opens X`, this is the union of all open sets whose preimage is `U`. This is right adjoint to `Opens.map`. -/ @[nolint unusedArguments] def functorObj {X Y : TopCat} {f : X ⟶ Y} (_ : IsInducing f) (U : Opens X) : Opens Y := sSup { s : Opens Y \| (Opens.map f).obj s = U } lemma map_functorObj {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) (U : Opens X) : (Opens.map f).obj (hf.functorObj U) = U := by apply le_antisymm · rintro x ⟨_, ⟨s, rfl⟩, _, ⟨rfl : _ = U, rfl⟩, hx : f x ∈ s⟩; exact hx · intro x hx obtain ⟨U, hU⟩ := U obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU exact Opens.mem_sSup.mpr ⟨⟨_, ht⟩, rfl, hx⟩ lemma mem_functorObj_iff {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) (U : Opens X) {x : X} : f x ∈ hf.functorObj U ↔ x ∈ U := by conv_rhs => rw [← hf.map_functorObj U] rfl lemma le_functorObj_iff {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) {U : Opens X} {V : Opens Y} : V ≤ hf.functorObj U ↔ (Opens.map f).obj V ≤ U := by obtain ⟨U, hU⟩ := U obtain ⟨t, ht, rfl⟩ := hf.isOpen_iff.mp hU constructor · exact fun i x hx ↦ (hf.mem_functorObj_iff ((Opens.map f).obj ⟨t, ht⟩)).mp (i hx) · intro h x hx refine Opens.mem_sSup.mpr ⟨⟨_, V.2.union ht⟩, Opens.ext ?_, Set.mem_union_left t hx⟩ dsimp rwa [Set.union_eq_right] /-- An inducing map `f : X ⟶ Y` induces a Galois insertion between `Opens Y` and `Opens X`. -/ def opensGI {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) : GaloisInsertion (Opens.map f).obj hf.functorObj := ⟨_, fun _ _ ↦ hf.le_functorObj_iff.symm, fun U ↦ (hf.map_functorObj U).ge, fun _ _ ↦ rfl⟩ /-- An inducing map `f : X ⟶ Y` induces a functor `Opens X ⥤ Opens Y`. -/ @[simps] def functor {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) : Opens X ⥤ Opens Y where obj := hf.functorObj map {U V} h := homOfLE (hf.le_functorObj_iff.mpr ((hf.map_functorObj U).trans_le h.le)) /-- An inducing map `f : X ⟶ Y` induces an adjunction between `Opens Y` and `Opens X`. -/ def adjunction {X Y : TopCat} {f : X ⟶ Y} (hf : IsInducing f) : Opens.map f ⊣ hf.functor := hf.opensGI.gc.adjunction end Topology.IsInducing namespace TopologicalSpace.Opens open TopologicalSpace @[simp] theorem isOpenEmbedding_obj_top {X : TopCat} (U : Opens X) : U.isOpenEmbedding.isOpenMap.functor.obj ⊤ = U := by ext1 exact Set.image_univ.trans Subtype.range_coe @[simp] theorem inclusion'_map_eq_top {X : TopCat} (U : Opens X) : (Opens.map U.inclusion').obj U = ⊤ := by ext1 exact Subtype.coe_preimage_self _ @[simp] theorem adjunction_counit_app_self {X : TopCat} (U : Opens X) : U.isOpenEmbedding.isOpenMap.adjunction.counit.app U = eqToHom (by simp) := Subsingleton.elim _ _ theorem inclusion'_top_functor (X : TopCat) : (@Opens.isOpenEmbedding X ⊤).isOpenMap.functor = map (inclusionTopIso X).inv := by refine CategoryTheory.Functor.ext ?_ ?_ · intro U ext x exact ⟨fun ⟨⟨_, _⟩, h, rfl⟩ => h, fun h => ⟨⟨x, trivial⟩, h, rfl⟩⟩ · subsingleton theorem functor_obj_map_obj {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) (U : Opens Y) : hf.functor.obj ((Opens.map f).obj U) = hf.functor.obj ⊤ ⊓ U := by ext constructor · rintro ⟨x, hx, rfl⟩ exact ⟨⟨x, trivial, rfl⟩, hx⟩ · rintro ⟨⟨x, -, rfl⟩, hx⟩ exact ⟨x, hx, rfl⟩ lemma set_range_inclusion' {X : TopCat} (U : Opens X) : Set.range (inclusion' U) = (U : Set X) := by ext x constructor · rintro ⟨x, rfl⟩ exact x.2 · intro h exact ⟨⟨x, h⟩, rfl⟩ @[simp] theorem functor_map_eq_inf {X : TopCat} (U V : Opens X) : U.isOpenEmbedding.isOpenMap.functor.obj ((Opens.map U.inclusion').obj V) = V ⊓ U := by ext1 simp only [IsOpenMap.coe_functor_obj, map_coe, coe_inf, Set.image_preimage_eq_inter_range, set_range_inclusion' U] theorem map_functor_eq' {X U : TopCat} (f : U ⟶ X) (hf : IsOpenEmbedding f) (V) : ((Opens.map f).obj <\| hf.isOpenMap.functor.obj V) = V := Opens.ext <\| Set.preimage_image_eq _ hf.injective @[simp] theorem map_functor_eq {X : TopCat} {U : Opens X} (V : Opens U) : ((Opens.map U.inclusion').obj <\| U.isOpenEmbedding.isOpenMap.functor.obj V) = V := TopologicalSpace.Opens.map_functor_eq' _ U.isOpenEmbedding V @[simp] theorem adjunction_counit_map_functor {X : TopCat} {U : Opens X} (V : Opens U) : U.isOpenEmbedding.isOpenMap.adjunction.counit.app (U.isOpenEmbedding.isOpenMap.functor.obj V) = eqToHom (by dsimp; rw [map_functor_eq V]) := by subsingleton end TopologicalSpace.Opens
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean	import Mathlib.CategoryTheory.EffectiveEpi.RegularEpi import Mathlib.Topology.Category.TopCat.Limits.Pullbacks /-! # Effective epimorphisms in `TopCat` This file proves the result `TopCat.effectiveEpi_iff_isQuotientMap`: The effective epimorphisms in `TopCat` are precisely the quotient maps. -/ universe u open CategoryTheory Limits Topology namespace TopCat /-- Implementation: If `π` is a morphism in `TopCat` which is a quotient map, then it is an effective epimorphism. The theorem `TopCat.effectiveEpi_iff_isQuotientMap` should be used instead of this definition. -/ noncomputable def effectiveEpiStructOfQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) (hπ : IsQuotientMap π) : EffectiveEpiStruct π where /- `IsQuotientMap.lift` gives the required morphism -/ desc e h := ofHom <\| hπ.lift e.hom fun a b hab ↦ CategoryTheory.congr_fun (h (ofHom ⟨fun _ ↦ a, continuous_const⟩) (ofHom ⟨fun _ ↦ b, continuous_const⟩) (by ext; exact hab)) a /- `IsQuotientMap.lift_comp` gives the factorisation -/ fac e h := hom_ext (hπ.lift_comp e.hom fun a b hab ↦ CategoryTheory.congr_fun (h (ofHom ⟨fun _ ↦ a, continuous_const⟩) (ofHom ⟨fun _ ↦ b, continuous_const⟩) (by ext; exact hab)) a) /- Uniqueness follows from the fact that `IsQuotientMap.lift` is an equivalence (given by `IsQuotientMap.liftEquiv`). -/ uniq e h g hm := by suffices g = ofHom (hπ.liftEquiv ⟨e.hom, fun a b hab ↦ CategoryTheory.congr_fun (h (ofHom ⟨fun _ ↦ a, continuous_const⟩) (ofHom ⟨fun _ ↦ b, continuous_const⟩) (by ext; exact hab)) a⟩) by assumption apply hom_ext rw [hom_ofHom, ← Equiv.symm_apply_eq hπ.liftEquiv] ext simp only [IsQuotientMap.liftEquiv_symm_apply_coe, ContinuousMap.comp_apply, ← hm] rfl /-- The effective epimorphisms in `TopCat` are precisely the quotient maps. -/ theorem effectiveEpi_iff_isQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) : EffectiveEpi π ↔ IsQuotientMap π := by /- The backward direction is given by `effectiveEpiStructOfQuotientMap` above. -/ refine ⟨fun _ ↦ ?_, fun hπ ↦ ⟨⟨effectiveEpiStructOfQuotientMap π hπ⟩⟩⟩ /- Since `TopCat` has pullbacks, `π` is in fact a `RegularEpi`. This means that it exhibits `B` as a coequalizer of two maps into `X`. It suffices to prove that `π` followed by the isomorphism to an arbitrary coequalizer is a quotient map. -/ have hπ : RegularEpi π := inferInstance exact isQuotientMap_of_isColimit_cofork _ hπ.isColimit end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Basic.lean	import Mathlib.CategoryTheory.Elementwise import Mathlib.Topology.ContinuousMap.Basic /-! # Category instance for topological spaces We introduce the bundled category `TopCat` of topological spaces together with the functors `TopCat.discrete` and `TopCat.trivial` from the category of types to `TopCat` which equip a type with the corresponding discrete, resp. trivial, topology. For a proof that these functors are left, resp. right adjoint to the forgetful functor, see `Mathlib/Topology/Category/TopCat/Adjunctions.lean`. -/ assert_not_exists Module open CategoryTheory TopologicalSpace Topology universe u /-- The category of topological spaces. -/ structure TopCat where /-- The object in `TopCat` associated to a type equipped with the appropriate typeclasses. -/ of :: /-- The underlying type. -/ carrier : Type u [str : TopologicalSpace carrier] attribute [instance] TopCat.str initialize_simps_projections TopCat (-str) namespace TopCat instance : CoeSort (TopCat) (Type u) := ⟨TopCat.carrier⟩ attribute [coe] TopCat.carrier lemma coe_of (X : Type u) [TopologicalSpace X] : (of X : Type u) = X := rfl lemma of_carrier (X : TopCat.{u}) : of X = X := rfl variable {X} in /-- The type of morphisms in `TopCat`. -/ @[ext] structure Hom (X Y : TopCat.{u}) where private mk :: /-- The underlying `ContinuousMap`. -/ hom' : C(X, Y) instance : Category TopCat where Hom X Y := Hom X Y id X := ⟨ContinuousMap.id X⟩ comp f g := ⟨g.hom'.comp f.hom'⟩ instance : ConcreteCategory.{u} TopCat (fun X Y => C(X, Y)) where hom := Hom.hom' ofHom f := ⟨f⟩ /-- Turn a morphism in `TopCat` back into a `ContinuousMap`. -/ abbrev Hom.hom {X Y : TopCat.{u}} (f : Hom X Y) := ConcreteCategory.hom (C := TopCat) f /-- Typecheck a `ContinuousMap` as a morphism in `TopCat`. -/ abbrev ofHom {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) : of X ⟶ of Y := ConcreteCategory.ofHom (C := TopCat) f /-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/ def Hom.Simps.hom (X Y : TopCat) (f : Hom X Y) := f.hom initialize_simps_projections Hom (hom' → hom) /-! The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`. -/ @[simp] lemma hom_id {X : TopCat.{u}} : (𝟙 X : X ⟶ X).hom = ContinuousMap.id X := rfl @[simp] theorem id_app (X : TopCat.{u}) (x : ↑X) : (𝟙 X : X ⟶ X) x = x := rfl @[simp] theorem coe_id (X : TopCat.{u}) : (𝟙 X : X → X) = id := rfl @[simp] lemma hom_comp {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g).hom = g.hom.comp f.hom := rfl @[simp] theorem comp_app {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) : (f ≫ g : X → Z) x = g (f x) := rfl @[simp] theorem coe_comp {X Y Z : TopCat.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f := rfl @[ext] lemma hom_ext {X Y : TopCat} {f g : X ⟶ Y} (hf : f.hom = g.hom) : f = g := Hom.ext hf @[ext] lemma ext {X Y : TopCat} {f g : X ⟶ Y} (w : ∀ x : X, f x = g x) : f = g := ConcreteCategory.hom_ext _ _ w @[simp] lemma hom_ofHom {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) : (ofHom f).hom = f := rfl @[simp] lemma ofHom_hom {X Y : TopCat} (f : X ⟶ Y) : ofHom (Hom.hom f) = f := rfl @[simp] lemma ofHom_id {X : Type u} [TopologicalSpace X] : ofHom (ContinuousMap.id X) = 𝟙 (of X) := rfl @[simp] lemma ofHom_comp {X Y Z : Type u} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : C(X, Y)) (g : C(Y, Z)) : ofHom (g.comp f) = ofHom f ≫ ofHom g := rfl lemma ofHom_apply {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] (f : C(X, Y)) (x : X) : (ofHom f) x = f x := rfl lemma hom_inv_id_apply {X Y : TopCat} (f : X ≅ Y) (x : X) : f.inv (f.hom x) = x := by simp lemma inv_hom_id_apply {X Y : TopCat} (f : X ≅ Y) (y : Y) : f.hom (f.inv y) = y := by simp /-- Replace a function coercion for a morphism `TopCat.of X ⟶ TopCat.of Y` with the definitionally equal function coercion for a continuous map `C(X, Y)`. -/ @[simp] theorem coe_of_of {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {x} : @DFunLike.coe (TopCat.of X ⟶ TopCat.of Y) ((CategoryTheory.forget TopCat).obj (TopCat.of X)) (fun _ ↦ (CategoryTheory.forget TopCat).obj (TopCat.of Y)) HasForget.instFunLike (ofHom f) x = @DFunLike.coe C(X, Y) X (fun _ ↦ Y) _ f x := rfl instance inhabited : Inhabited TopCat := ⟨TopCat.of Empty⟩ /-- The discrete topology on any type. -/ def discrete : Type u ⥤ TopCat.{u} where obj X := @of X ⊥ map f := @ofHom _ _ ⊥ ⊥ <\| @ContinuousMap.mk _ _ ⊥ ⊥ f continuous_bot instance {X : Type u} : DiscreteTopology (discrete.obj X) := ⟨rfl⟩ /-- The trivial topology on any type. -/ def trivial : Type u ⥤ TopCat.{u} where obj X := @of X ⊤ map f := @ofHom _ _ ⊤ ⊤ <\| @ContinuousMap.mk _ _ ⊤ ⊤ f continuous_top /-- Any homeomorphisms induces an isomorphism in `Top`. -/ @[simps] def isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : X ≅ Y where hom := ofHom f inv := ofHom f.symm /-- Any isomorphism in `Top` induces a homeomorphism. -/ @[simps] def homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : X ≃ₜ Y where toFun := f.hom invFun := f.inv left_inv x := by simp right_inv x := by simp continuous_toFun := f.hom.hom.continuous continuous_invFun := f.inv.hom.continuous @[simp] theorem of_isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : homeoOfIso (isoOfHomeo f) = f := by ext rfl @[simp] theorem of_homeoOfIso {X Y : TopCat.{u}} (f : X ≅ Y) : isoOfHomeo (homeoOfIso f) = f := by ext rfl lemma isIso_of_bijective_of_isOpenMap {X Y : TopCat.{u}} (f : X ⟶ Y) (hfbij : Function.Bijective f) (hfcl : IsOpenMap f) : IsIso f := let e : X ≃ₜ Y := (Equiv.ofBijective f hfbij).toHomeomorphOfContinuousOpen f.hom.continuous hfcl inferInstanceAs <\| IsIso (TopCat.isoOfHomeo e).hom lemma isIso_of_bijective_of_isClosedMap {X Y : TopCat.{u}} (f : X ⟶ Y) (hfbij : Function.Bijective f) (hfcl : IsClosedMap f) : IsIso f := let e : X ≃ₜ Y := (Equiv.ofBijective f hfbij).toHomeomorphOfContinuousClosed f.hom.continuous hfcl inferInstanceAs <\| IsIso (TopCat.isoOfHomeo e).hom lemma isIso_iff_isHomeomorph {X Y : TopCat.{u}} (f : X ⟶ Y) : IsIso f ↔ IsHomeomorph f := ⟨fun _ ↦ (homeoOfIso (asIso f)).isHomeomorph, fun H ↦ isIso_of_bijective_of_isOpenMap _ H.bijective H.isOpenMap⟩ theorem isOpenEmbedding_iff_comp_isIso {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] : IsOpenEmbedding (f ≫ g) ↔ IsOpenEmbedding f := (TopCat.homeoOfIso (asIso g)).isOpenEmbedding.of_comp_iff f @[simp] theorem isOpenEmbedding_iff_comp_isIso' {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] : IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding f := by simp only exact isOpenEmbedding_iff_comp_isIso f g theorem isOpenEmbedding_iff_isIso_comp {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] : IsOpenEmbedding (f ≫ g) ↔ IsOpenEmbedding g := by constructor · intro h convert h.comp (TopCat.homeoOfIso (asIso f).symm).isOpenEmbedding exact congr_arg (DFunLike.coe ∘ ConcreteCategory.hom) (IsIso.inv_hom_id_assoc f g).symm · exact fun h => h.comp (TopCat.homeoOfIso (asIso f)).isOpenEmbedding @[simp] theorem isOpenEmbedding_iff_isIso_comp' {X Y Z : TopCat} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] : IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding g := by simp only exact isOpenEmbedding_iff_isIso_comp f g end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/ULift.lean	import Mathlib.Topology.Category.TopCat.Basic import Mathlib.Topology.Homeomorph.Lemmas /-! # Lifting topological spaces to a higher universe In this file, we construct the functor `uliftFunctor.{v, u} : TopCat.{u} ⥤ TopCat.{max u v}` which sends a topological space `X : Type u` to a homeomorphic space in `Type (max u v)`. -/ universe v u open CategoryTheory namespace TopCat -- Note: no `@[simps!]` attribute here in order to get good simplification lemmas -- like `uliftFunctorObjHomeo_naturality_apply` below. We should access -- `uliftFunctor.obj X` via the homeomorphism `X.uliftFunctorObjHomeo`. /-- The functor which sends a topological space in `Type u` to a homeomorphic space in `Type (max u v)`. -/ def uliftFunctor : TopCat.{u} ⥤ TopCat.{max u v} where obj X := TopCat.of (ULift.{v} X) map {X Y} f := ofHom ⟨ULift.map f, by continuity⟩ /-- Given `X : TopCat.{u}`, this is the homeomorphism `X ≃ₜ uliftFunctor.{v}.obj X`. -/ def uliftFunctorObjHomeo (X : TopCat.{u}) : X ≃ₜ uliftFunctor.{v}.obj X := Homeomorph.ulift.symm @[simp] lemma uliftFunctorObjHomeo_naturality_apply {X Y : TopCat.{u}} (f : X ⟶ Y) (x : X) : uliftFunctor.{v}.map f (X.uliftFunctorObjHomeo x) = Y.uliftFunctorObjHomeo (f x) := rfl @[simp] lemma uliftFunctorObjHomeo_symm_naturality_apply {X Y : TopCat.{u}} (f : X ⟶ Y) (x : uliftFunctor.{v}.obj X) : Y.uliftFunctorObjHomeo.symm (uliftFunctor.{v}.map f x) = f (X.uliftFunctorObjHomeo.symm x) := rfl /-- The `ULift` functor on categories of topological spaces is compatible with the one defined on categories of types. -/ @[simps!] def uliftFunctorCompForgetIso : uliftFunctor.{v, u} ⋙ forget TopCat.{max u v} ≅ forget TopCat.{u} ⋙ CategoryTheory.uliftFunctor.{v, u} := Iso.refl _ /-- The `ULift` functor on categories of topological spaces is fully faithful. -/ def uliftFunctorFullyFaithful : uliftFunctor.{v, u}.FullyFaithful where preimage f := ofHom ⟨ULift.down ∘ f ∘ ULift.up, by continuity⟩ instance : uliftFunctor.{v, u}.Full := uliftFunctorFullyFaithful.full instance : uliftFunctor.{v, u}.Faithful := uliftFunctorFullyFaithful.faithful end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/OpenNhds.lean	import Mathlib.Topology.Category.TopCat.Opens import Mathlib.Data.Set.Subsingleton /-! # The category of open neighborhoods of a point Given an object `X` of the category `TopCat` of topological spaces and a point `x : X`, this file builds the type `OpenNhds x` of open neighborhoods of `x` in `X` and endows it with the partial order given by inclusion and the corresponding category structure (as a full subcategory of the poset category `Set X`). This is used in `Topology.Sheaves.Stalks` to build the stalk of a sheaf at `x` as a limit over `OpenNhds x`. ## Main declarations Besides `OpenNhds`, the main constructions here are: * `inclusion (x : X)`: the obvious functor `OpenNhds x ⥤ Opens X` * `functorNhds`: An open map `f : X ⟶ Y` induces a functor `OpenNhds x ⥤ OpenNhds (f x)` * `adjunctionNhds`: An open map `f : X ⟶ Y` induces an adjunction between `OpenNhds x` and `OpenNhds (f x)`. -/ open CategoryTheory TopologicalSpace Opposite Topology universe u variable {X Y : TopCat.{u}} (f : X ⟶ Y) namespace TopologicalSpace /-- The type of open neighbourhoods of a point `x` in a (bundled) topological space. -/ def OpenNhds (x : X) := ObjectProperty.FullSubcategory fun U : Opens X => x ∈ U namespace OpenNhds variable {x : X} {U V W : OpenNhds x} instance partialOrder (x : X) : PartialOrder (OpenNhds x) where le U V := U.1 ≤ V.1 le_refl _ := le_rfl le_trans _ _ _ := le_trans le_antisymm _ _ i j := ObjectProperty.FullSubcategory.ext <\| le_antisymm i j instance (x : X) : Lattice (OpenNhds x) := { OpenNhds.partialOrder x with inf := fun U V => ⟨U.1 ⊓ V.1, ⟨U.2, V.2⟩⟩ le_inf := fun U V W => @le_inf _ _ U.1.1 V.1.1 W.1.1 inf_le_left := fun U V => @inf_le_left _ _ U.1.1 V.1.1 inf_le_right := fun U V => @inf_le_right _ _ U.1.1 V.1.1 sup := fun U V => ⟨U.1 ⊔ V.1, Set.mem_union_left V.1.1 U.2⟩ sup_le := fun U V W => @sup_le _ _ U.1.1 V.1.1 W.1.1 le_sup_left := fun U V => @le_sup_left _ _ U.1.1 V.1.1 le_sup_right := fun U V => @le_sup_right _ _ U.1.1 V.1.1 } instance (x : X) : OrderTop (OpenNhds x) where top := ⟨⊤, trivial⟩ le_top _ := by dsimp [LE.le]; exact le_top instance (x : X) : Inhabited (OpenNhds x) := ⟨⊤⟩ instance openNhdsCategory (x : X) : Category.{u} (OpenNhds x) := inferInstance instance opensNhds.instFunLike : FunLike (U ⟶ V) U.1 V.1 where coe f := Set.inclusion f.le coe_injective' := by rintro ⟨⟨_⟩⟩ _ _; congr! @[simp] lemma apply_mk (f : U ⟶ V) (y : X) (hy) : f ⟨y, hy⟩ = ⟨y, f.le hy⟩ := rfl @[simp] lemma val_apply (f : U ⟶ V) (y : U.1) : (f y : X) = y := rfl @[simp, norm_cast] lemma coe_id (f : U ⟶ U) : ⇑f = id := rfl lemma id_apply (f : U ⟶ U) (y : U.1) : f y = y := rfl @[simp] lemma comp_apply (f : U ⟶ V) (g : V ⟶ W) (x : U.1) : (f ≫ g) x = g (f x) := rfl /-- The inclusion `U ⊓ V ⟶ U` as a morphism in the category of open sets. -/ def infLELeft {x : X} (U V : OpenNhds x) : U ⊓ V ⟶ U := homOfLE inf_le_left /-- The inclusion `U ⊓ V ⟶ V` as a morphism in the category of open sets. -/ def infLERight {x : X} (U V : OpenNhds x) : U ⊓ V ⟶ V := homOfLE inf_le_right /-- The inclusion functor from open neighbourhoods of `x` to open sets in the ambient topological space. -/ def inclusion (x : X) : OpenNhds x ⥤ Opens X := ObjectProperty.ι _ @[simp] theorem inclusion_obj (x : X) (U) (p) : (inclusion x).obj ⟨U, p⟩ = U := rfl theorem isOpenEmbedding {x : X} (U : OpenNhds x) : IsOpenEmbedding U.1.inclusion' := U.1.isOpenEmbedding /-- The preimage functor from neighborhoods of `f x` to neighborhoods of `x`. -/ def map (x : X) : OpenNhds (f x) ⥤ OpenNhds x where obj U := ⟨(Opens.map f).obj U.1, U.2⟩ map i := (Opens.map f).map i @[simp] theorem map_obj (x : X) (U) (q) : (map f x).obj ⟨U, q⟩ = ⟨(Opens.map f).obj U, q⟩ := rfl @[simp] theorem map_id_obj (x : X) (U) : (map (𝟙 X) x).obj U = U := rfl @[simp] theorem map_id_obj' (x : X) (U) (p) (q) : (map (𝟙 X) x).obj ⟨⟨U, p⟩, q⟩ = ⟨⟨U, p⟩, q⟩ := rfl @[simp] theorem map_id_obj_unop (x : X) (U : (OpenNhds x)ᵒᵖ) : (map (𝟙 X) x).obj (unop U) = unop U := by simp @[simp] theorem op_map_id_obj (x : X) (U : (OpenNhds x)ᵒᵖ) : (map (𝟙 X) x).op.obj U = U := by simp /-- `Opens.map f` and `OpenNhds.map f` form a commuting square (up to natural isomorphism) with the inclusion functors into `Opens X`. -/ @[simps! hom_app inv_app] def inclusionMapIso (x : X) : inclusion (f x) ⋙ Opens.map f ≅ map f x ⋙ inclusion x := NatIso.ofComponents fun U => { hom := 𝟙 _, inv := 𝟙 _ } @[simp] theorem inclusionMapIso_hom (x : X) : (inclusionMapIso f x).hom = 𝟙 _ := rfl @[simp] theorem inclusionMapIso_inv (x : X) : (inclusionMapIso f x).inv = 𝟙 _ := rfl end OpenNhds end TopologicalSpace namespace IsOpenMap open TopologicalSpace variable {f} /-- An open map `f : X ⟶ Y` induces a functor `OpenNhds x ⥤ OpenNhds (f x)`. -/ @[simps] def functorNhds (h : IsOpenMap f) (x : X) : OpenNhds x ⥤ OpenNhds (f x) where obj U := ⟨h.functor.obj U.1, ⟨x, U.2, rfl⟩⟩ map i := h.functor.map i /-- An open map `f : X ⟶ Y` induces an adjunction between `OpenNhds x` and `OpenNhds (f x)`. -/ def adjunctionNhds (h : IsOpenMap f) (x : X) : IsOpenMap.functorNhds h x ⊣ OpenNhds.map f x where unit := { app := fun _ => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ } counit := { app := fun _ => homOfLE fun _ ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV } end IsOpenMap namespace Topology.IsInducing open TopologicalSpace variable {f} /-- An inducing map `f : X ⟶ Y` induces a functor `open_nhds x ⥤ open_nhds (f x)`. -/ @[simps] def functorNhds (h : IsInducing f) (x : X) : OpenNhds x ⥤ OpenNhds (f x) where obj U := ⟨h.functor.obj U.1, (h.mem_functorObj_iff U.1).mpr U.2⟩ map := h.functor.map /-- An inducing map `f : X ⟶ Y` induces an adjunction between `open_nhds x` and `open_nhds (f x)`. -/ def adjunctionNhds (h : IsInducing f) (x : X) : OpenNhds.map f x ⊣ h.functorNhds x where unit := { app := fun U => homOfLE (h.adjunction.unit.app U.1).le } counit := { app := fun U => homOfLE (h.adjunction.counit.app U.1).le } end Topology.IsInducing
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Sphere.lean	import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.Topology.Category.TopCat.EpiMono /-! # Euclidean spheres This file defines the `n`-sphere `𝕊 n`, the `n`-disk `𝔻 n`, its boundary `∂𝔻 n` and its interior `𝔹 n` as objects in `TopCat`. -/ universe u namespace TopCat open CategoryTheory /-- The `n`-disk is the set of points in ℝⁿ whose norm is at most `1`, endowed with the subspace topology. -/ noncomputable def disk (n : ℕ) : TopCat.{u} := TopCat.of <\| ULift <\| Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) 1 /-- The boundary of the `n`-disk. -/ noncomputable def diskBoundary (n : ℕ) : TopCat.{u} := TopCat.of <\| ULift <\| Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1 /-- The `n`-sphere is the set of points in ℝⁿ⁺¹ whose norm equals `1`, endowed with the subspace topology. -/ noncomputable def sphere (n : ℕ) : TopCat.{u} := diskBoundary (n + 1) /-- The `n`-ball is the set of points in ℝⁿ whose norm is strictly less than `1`, endowed with the subspace topology. -/ noncomputable def ball (n : ℕ) : TopCat.{u} := TopCat.of <\| ULift <\| Metric.ball (0 : EuclideanSpace ℝ (Fin n)) 1 /-- `𝔻 n` denotes the `n`-disk. -/ scoped prefix:arg "𝔻 " => disk /-- `∂𝔻 n` denotes the boundary of the `n`-disk. -/ scoped prefix:arg "∂𝔻 " => diskBoundary /-- `𝕊 n` denotes the `n`-sphere. -/ scoped prefix:arg "𝕊 " => sphere /-- `𝔹 n` denotes the `n`-ball, the interior of the `n`-disk. -/ scoped prefix:arg "𝔹 " => ball /-- The inclusion `∂𝔻 n ⟶ 𝔻 n` of the boundary of the `n`-disk. -/ def diskBoundaryInclusion (n : ℕ) : ∂𝔻 n ⟶ 𝔻 n := ofHom { toFun := fun ⟨p, hp⟩ ↦ ⟨p, le_of_eq hp⟩ continuous_toFun := ⟨fun t ⟨s, ⟨r, hro, hrs⟩, hst⟩ ↦ by rw [isOpen_induced_iff, ← hst, ← hrs] tauto⟩ } /-- The inclusion `𝔹 n ⟶ 𝔻 n` of the interior of the `n`-disk. -/ def ballInclusion (n : ℕ) : 𝔹 n ⟶ 𝔻 n := ofHom { toFun := fun ⟨p, hp⟩ ↦ ⟨p, Metric.ball_subset_closedBall hp⟩ continuous_toFun := ⟨fun t ⟨s, ⟨r, hro, hrs⟩, hst⟩ ↦ by rw [isOpen_induced_iff, ← hst, ← hrs] tauto⟩ } instance {n : ℕ} : Mono (diskBoundaryInclusion n) := mono_iff_injective _ \|>.mpr <\| by intro ⟨x, hx⟩ ⟨y, hy⟩ h obtain rfl : x = y := by simpa [diskBoundaryInclusion, disk] using h congr instance {n : ℕ} : Mono (ballInclusion n) := TopCat.mono_iff_injective _ \|>.mpr <\| by intro ⟨x, hx⟩ ⟨y, hy⟩ h obtain rfl : x = y := by simpa [ballInclusion, disk] using h congr instance (n : ℕ) : CompactSpace (𝔻 n) := by convert Homeomorph.compactSpace Homeomorph.ulift.symm infer_instance instance (n : ℕ) : CompactSpace (∂𝔻 n) := by convert Homeomorph.compactSpace Homeomorph.ulift.symm infer_instance end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Adjunctions.lean	import Mathlib.Topology.Category.TopCat.Basic import Mathlib.CategoryTheory.Adjunction.Basic /-! # Adjunctions regarding the category of topological spaces This file shows that the forgetful functor from topological spaces to types has a left and right adjoint, given by `TopCat.discrete`, resp. `TopCat.trivial`, the functors which equip a type with the discrete, resp. trivial, topology. -/ universe u open CategoryTheory open TopCat namespace TopCat /-- Equipping a type with the discrete topology is left adjoint to the forgetful functor `Top ⥤ Type`. -/ @[simps! unit counit] def adj₁ : discrete ⊣ forget TopCat.{u} where unit := { app := fun _ => id } counit := { app := fun X => TopCat.ofHom (X := discrete.obj X) ⟨id, continuous_bot⟩ } /-- Equipping a type with the trivial topology is right adjoint to the forgetful functor `Top ⥤ Type`. -/ @[simps! unit counit] def adj₂ : forget TopCat.{u} ⊣ trivial where unit := { app := fun X => TopCat.ofHom (Y := trivial.obj X) ⟨id, continuous_top⟩ } counit := { app := fun _ => id } instance : (forget TopCat.{u}).IsRightAdjoint := ⟨_, ⟨adj₁⟩⟩ instance : (forget TopCat.{u}).IsLeftAdjoint := ⟨_, ⟨adj₂⟩⟩ end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Limits/Products.lean	import Mathlib.Topology.Category.TopCat.EpiMono import Mathlib.Topology.Category.TopCat.Limits.Basic import Mathlib.CategoryTheory.Limits.Shapes.Products import Mathlib.CategoryTheory.Limits.ConcreteCategory.Basic import Mathlib.Data.Set.Subsingleton import Mathlib.Tactic.CategoryTheory.Elementwise import Mathlib.Topology.Homeomorph.Lemmas /-! # Products and coproducts in the category of topological spaces -/ open CategoryTheory Limits Set TopologicalSpace Topology universe v u w noncomputable section namespace TopCat variable {J : Type v} [Category.{w} J] /-- The projection from the product as a bundled continuous map. -/ abbrev piπ {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : TopCat.of (∀ i, α i) ⟶ α i := ofHom ⟨fun f => f i, continuous_apply i⟩ /-- The explicit fan of a family of topological spaces given by the pi type. -/ @[simps! pt π_app] def piFan {ι : Type v} (α : ι → TopCat.{max v u}) : Fan α := Fan.mk (TopCat.of (∀ i, α i)) (piπ.{v,u} α) /-- The constructed fan is indeed a limit -/ def piFanIsLimit {ι : Type v} (α : ι → TopCat.{max v u}) : IsLimit (piFan α) where lift S := ofHom { toFun := fun s i => S.π.app ⟨i⟩ s continuous_toFun := continuous_pi (fun i => (S.π.app ⟨i⟩).hom.2) } uniq := by intro S m h ext x funext i simp [ContinuousMap.coe_mk, ← h ⟨i⟩] fac _ _ := rfl /-- The product is homeomorphic to the product of the underlying spaces, equipped with the product topology. -/ def piIsoPi {ι : Type v} (α : ι → TopCat.{max v u}) : ∏ᶜ α ≅ TopCat.of (∀ i, α i) := (limit.isLimit _).conePointUniqueUpToIso (piFanIsLimit.{v, u} α) @[reassoc (attr := simp)] theorem piIsoPi_inv_π {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : (piIsoPi α).inv ≫ Pi.π α i = piπ α i := by simp [piIsoPi] theorem piIsoPi_inv_π_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : ∀ i, α i) : (Pi.π α i :) ((piIsoPi α).inv x) = x i := ConcreteCategory.congr_hom (piIsoPi_inv_π α i) x theorem piIsoPi_hom_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : (∏ᶜ α : TopCat.{max v u})) : (piIsoPi α).hom x i = (Pi.π α i :) x := rfl /-- The inclusion to the coproduct as a bundled continuous map. -/ abbrev sigmaι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : α i ⟶ TopCat.of (Σ i, α i) := by refine ofHom (ContinuousMap.mk ?_ ?_) · dsimp apply Sigma.mk i · dsimp; continuity /-- The explicit cofan of a family of topological spaces given by the sigma type. -/ @[simps! pt ι_app] def sigmaCofan {ι : Type v} (α : ι → TopCat.{max v u}) : Cofan α := Cofan.mk (TopCat.of (Σ i, α i)) (sigmaι α) /-- The constructed cofan is indeed a colimit -/ def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCat.{max v u}) : IsColimit (sigmaCofan β) where desc S := ofHom { toFun := fun (s : of (Σ i, β i)) => S.ι.app ⟨s.1⟩ s.2 continuous_toFun := by continuity } uniq := by intro S m h ext ⟨i, x⟩ simp only [← h] congr fac s j := by cases j cat_disch /-- The coproduct is homeomorphic to the disjoint union of the topological spaces. -/ def sigmaIsoSigma {ι : Type v} (α : ι → TopCat.{max v u}) : ∐ α ≅ TopCat.of (Σ i, α i) := (colimit.isColimit _).coconePointUniqueUpToIso (sigmaCofanIsColimit.{v, u} α) @[reassoc (attr := simp)] theorem sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) : Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i := by simp [sigmaIsoSigma] theorem sigmaIsoSigma_hom_ι_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : α i) : (sigmaIsoSigma α).hom ((Sigma.ι α i :) x) = Sigma.mk i x := ConcreteCategory.congr_hom (sigmaIsoSigma_hom_ι α i) x theorem sigmaIsoSigma_inv_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) (x : α i) : (sigmaIsoSigma α).inv ⟨i, x⟩ = (Sigma.ι α i :) x := by rw [← sigmaIsoSigma_hom_ι_apply, ← comp_app, ← comp_app, Iso.hom_inv_id, Category.comp_id] section Prod /-- The first projection from the product. -/ abbrev prodFst {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ X := ofHom { toFun := Prod.fst } /-- The second projection from the product. -/ abbrev prodSnd {X Y : TopCat.{u}} : TopCat.of (X × Y) ⟶ Y := ofHom { toFun := Prod.snd } /-- The explicit binary cofan of `X, Y` given by `X × Y`. -/ def prodBinaryFan (X Y : TopCat.{u}) : BinaryFan X Y := BinaryFan.mk prodFst prodSnd /-- The constructed binary fan is indeed a limit -/ def prodBinaryFanIsLimit (X Y : TopCat.{u}) : IsLimit (prodBinaryFan X Y) where lift := fun S : BinaryFan X Y => ofHom { toFun := fun s => (S.fst s, S.snd s) continuous_toFun := by continuity } fac := by rintro S (_ \| _) <;> {dsimp; ext; rfl} uniq := by intro S m h ext x -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): used to be part of `ext x` refine Prod.ext ?_ ?_ · specialize h ⟨WalkingPair.left⟩ apply_fun fun e => e x at h exact h · specialize h ⟨WalkingPair.right⟩ apply_fun fun e => e x at h exact h /-- The homeomorphism between `X ⨯ Y` and the set-theoretic product of `X` and `Y`, equipped with the product topology. -/ def prodIsoProd (X Y : TopCat.{u}) : X ⨯ Y ≅ TopCat.of (X × Y) := (limit.isLimit _).conePointUniqueUpToIso (prodBinaryFanIsLimit X Y) @[reassoc (attr := simp)] theorem prodIsoProd_hom_fst (X Y : TopCat.{u}) : (prodIsoProd X Y).hom ≫ prodFst = Limits.prod.fst := by simp [← Iso.eq_inv_comp, prodIsoProd] rfl @[reassoc (attr := simp)] theorem prodIsoProd_hom_snd (X Y : TopCat.{u}) : (prodIsoProd X Y).hom ≫ prodSnd = Limits.prod.snd := by simp [← Iso.eq_inv_comp, prodIsoProd] rfl -- Note that `(x : X ⨯ Y)` would mean `(x : ↑X × ↑Y)` below: theorem prodIsoProd_hom_apply {X Y : TopCat.{u}} (x : ↑(X ⨯ Y)) : (prodIsoProd X Y).hom x = ((Limits.prod.fst : X ⨯ Y ⟶ _) x, (Limits.prod.snd : X ⨯ Y ⟶ _) x) := rfl @[reassoc (attr := simp), elementwise] theorem prodIsoProd_inv_fst (X Y : TopCat.{u}) : (prodIsoProd X Y).inv ≫ Limits.prod.fst = prodFst := by simp [Iso.inv_comp_eq] @[reassoc (attr := simp), elementwise] theorem prodIsoProd_inv_snd (X Y : TopCat.{u}) : (prodIsoProd X Y).inv ≫ Limits.prod.snd = prodSnd := by simp [Iso.inv_comp_eq] theorem prod_topology {X Y : TopCat.{u}} : (X ⨯ Y).str = induced (Limits.prod.fst : X ⨯ Y ⟶ _) X.str ⊓ induced (Limits.prod.snd : X ⨯ Y ⟶ _) Y.str := by let homeo := homeoOfIso (prodIsoProd X Y) refine homeo.isInducing.eq_induced.trans ?_ change induced homeo (_ ⊓ _) = _ simp [induced_compose] rfl theorem range_prod_map {W X Y Z : TopCat.{u}} (f : W ⟶ Y) (g : X ⟶ Z) : Set.range (Limits.prod.map f g) = (Limits.prod.fst : Y ⨯ Z ⟶ _) ⁻¹' Set.range f ∩ (Limits.prod.snd : Y ⨯ Z ⟶ _) ⁻¹' Set.range g := by ext x constructor · rintro ⟨y, rfl⟩ simp_rw [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range, ← ConcreteCategory.comp_apply, Limits.prod.map_fst, Limits.prod.map_snd, ConcreteCategory.comp_apply, exists_apply_eq_apply, and_self_iff] · rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩ use (prodIsoProd W X).inv (x₁, x₂) apply Concrete.limit_ext rintro ⟨⟨⟩⟩ · rw [← ConcreteCategory.comp_apply] erw [Limits.prod.map_fst] rw [ConcreteCategory.comp_apply, TopCat.prodIsoProd_inv_fst_apply] exact hx₁ · rw [← ConcreteCategory.comp_apply] erw [Limits.prod.map_snd] rw [ConcreteCategory.comp_apply, TopCat.prodIsoProd_inv_snd_apply] exact hx₂ theorem isInducing_prodMap {W X Y Z : TopCat.{u}} {f : W ⟶ X} {g : Y ⟶ Z} (hf : IsInducing f) (hg : IsInducing g) : IsInducing (Limits.prod.map f g) := by constructor simp_rw [prod_topology, induced_inf, induced_compose, ← coe_comp, prod.map_fst, prod.map_snd, coe_comp, ← induced_compose (g := f), ← induced_compose (g := g)] rw [← hf.eq_induced, ← hg.eq_induced] theorem isEmbedding_prodMap {W X Y Z : TopCat.{u}} {f : W ⟶ X} {g : Y ⟶ Z} (hf : IsEmbedding f) (hg : IsEmbedding g) : IsEmbedding (Limits.prod.map f g) := ⟨isInducing_prodMap hf.isInducing hg.isInducing, by haveI := (TopCat.mono_iff_injective _).mpr hf.injective haveI := (TopCat.mono_iff_injective _).mpr hg.injective exact (TopCat.mono_iff_injective _).mp inferInstance⟩ end Prod /-- The binary coproduct cofan in `TopCat`. -/ protected def binaryCofan (X Y : TopCat.{u}) : BinaryCofan X Y := BinaryCofan.mk (ofHom ⟨Sum.inl, by continuity⟩) (ofHom ⟨Sum.inr, by continuity⟩) /-- The constructed binary coproduct cofan in `TopCat` is the coproduct. -/ def binaryCofanIsColimit (X Y : TopCat.{u}) : IsColimit (TopCat.binaryCofan X Y) := by refine Limits.BinaryCofan.isColimitMk (fun s => ofHom { toFun := Sum.elim s.inl s.inr, continuous_toFun := ?_ }) ?_ ?_ ?_ · continuity · intro s ext rfl · intro s ext rfl · intro s m h₁ h₂ ext (x \| x) exacts [ConcreteCategory.congr_hom h₁ x, ConcreteCategory.congr_hom h₂ x] theorem binaryCofan_isColimit_iff {X Y : TopCat} (c : BinaryCofan X Y) : Nonempty (IsColimit c) ↔ IsOpenEmbedding c.inl ∧ IsOpenEmbedding c.inr ∧ IsCompl (range c.inl) (range c.inr) := by classical constructor · rintro ⟨h⟩ rw [← show _ = c.inl from h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.left⟩, ← show _ = c.inr from h.comp_coconePointUniqueUpToIso_inv (binaryCofanIsColimit X Y) ⟨WalkingPair.right⟩] dsimp refine ⟨(homeoOfIso <\| h.coconePointUniqueUpToIso (binaryCofanIsColimit X Y)).symm.isOpenEmbedding.comp .inl, (homeoOfIso <\| h.coconePointUniqueUpToIso (binaryCofanIsColimit X Y)).symm.isOpenEmbedding.comp .inr, ?_⟩ rw [Set.range_comp, ← eq_compl_iff_isCompl] conv_rhs => rw [Set.range_comp] erw [← Set.image_compl_eq (homeoOfIso <\| h.coconePointUniqueUpToIso (binaryCofanIsColimit X Y)).symm.bijective, Set.compl_range_inr, Set.image_comp] · rintro ⟨h₁, h₂, h₃⟩ have : ∀ x, x ∈ Set.range c.inl ∨ x ∈ Set.range c.inr := by rw [eq_compl_iff_isCompl.mpr h₃.symm] exact fun _ => or_not refine ⟨BinaryCofan.IsColimit.mk _ ?_ ?_ ?_ ?_⟩ · intro T f g refine ofHom (ContinuousMap.mk ?_ ?_) · exact fun x => if h : x ∈ Set.range c.inl then f ((Equiv.ofInjective _ h₁.injective).symm ⟨x, h⟩) else g ((Equiv.ofInjective _ h₂.injective).symm ⟨x, (this x).resolve_left h⟩) rw [continuous_iff_continuousAt] intro x by_cases h : x ∈ Set.range c.inl · revert h x apply (IsOpen.continuousOn_iff _).mp · rw [continuousOn_iff_continuous_restrict] convert_to Continuous (f ∘ h₁.isEmbedding.toHomeomorph.symm) · ext ⟨x, hx⟩ exact dif_pos hx continuity · exact h₁.isOpen_range · revert h x apply (IsOpen.continuousOn_iff _).mp · rw [continuousOn_iff_continuous_restrict] have : ∀ a, a ∉ Set.range c.inl → a ∈ Set.range c.inr := by rintro a (h : a ∈ (Set.range c.inl)ᶜ) rwa [eq_compl_iff_isCompl.mpr h₃.symm] convert_to Continuous (g ∘ h₂.isEmbedding.toHomeomorph.symm ∘ Subtype.map _ this) · ext ⟨x, hx⟩ exact dif_neg hx apply Continuous.comp · exact g.hom.continuous_toFun · apply Continuous.comp · continuity · rw [IsEmbedding.subtypeVal.isInducing.continuous_iff] exact continuous_subtype_val · change IsOpen (Set.range c.inl)ᶜ rw [← eq_compl_iff_isCompl.mpr h₃.symm] exact h₂.isOpen_range · intro T f g ext x dsimp rw [dif_pos ⟨x, rfl⟩] conv_lhs => rw [Equiv.ofInjective_symm_apply] · intro T f g ext x dsimp rw [dif_neg] · exact congr_arg g (Equiv.ofInjective_symm_apply _ _) · rintro ⟨y, e⟩ have : c.inr x ∈ Set.range c.inl ⊓ Set.range c.inr := ⟨⟨_, e⟩, ⟨_, rfl⟩⟩ rwa [disjoint_iff.mp h₃.1] at this · rintro T _ _ m rfl rfl ext x change m x = dite _ _ _ split_ifs <;> exact congr_arg _ (Equiv.apply_ofInjective_symm _ ⟨_, _⟩).symm end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Limits/Konig.lean	import Mathlib.CategoryTheory.Filtered.Basic import Mathlib.Topology.Category.TopCat.Limits.Basic /-! # Topological Kőnig's lemma A topological version of Kőnig's lemma is that the inverse limit of nonempty compact Hausdorff spaces is nonempty. (Note: this can be generalized further to inverse limits of nonempty compact T0 spaces, where all the maps are closed maps; see [Stone1979] --- however there is an erratum for Theorem 4 that the element in the inverse limit can have cofinally many components that are not closed points.) We give this in a more general form, which is that cofiltered limits of nonempty compact Hausdorff spaces are nonempty (`nonempty_limitCone_of_compact_t2_cofiltered_system`). This also applies to inverse limits, where `{J : Type u} [Preorder J] [IsDirected J (≤)]` and `F : Jᵒᵖ ⥤ TopCat`. The theorem is specialized to nonempty finite types (which are compact Hausdorff with the discrete topology) in lemmas `nonempty_sections_of_finite_cofiltered_system` and `nonempty_sections_of_finite_inverse_system` in `Mathlib/CategoryTheory/CofilteredSystem.lean`. (See <https://stacks.math.columbia.edu/tag/086J> for the Set version.) -/ open CategoryTheory open CategoryTheory.Limits universe v u w noncomputable section namespace TopCat section TopologicalKonig variable {J : Type u} [SmallCategory J] variable (F : J ⥤ TopCat.{v}) private abbrev FiniteDiagramArrow {J : Type u} [SmallCategory J] (G : Finset J) := Σ' (X Y : J) (_ : X ∈ G) (_ : Y ∈ G), X ⟶ Y private abbrev FiniteDiagram (J : Type u) [SmallCategory J] := Σ G : Finset J, Finset (FiniteDiagramArrow G) /-- Partial sections of a cofiltered limit are sections when restricted to a finite subset of objects and morphisms of `J`. -/ def partialSections {J : Type u} [SmallCategory J] (F : J ⥤ TopCat.{v}) {G : Finset J} (H : Finset (FiniteDiagramArrow G)) : Set (∀ j, F.obj j) := {u \| ∀ {f : FiniteDiagramArrow G} (_ : f ∈ H), F.map f.2.2.2.2 (u f.1) = u f.2.1} theorem partialSections.nonempty [IsCofilteredOrEmpty J] [h : ∀ j : J, Nonempty (F.obj j)] {G : Finset J} (H : Finset (FiniteDiagramArrow G)) : (partialSections F H).Nonempty := by classical cases isEmpty_or_nonempty J · exact ⟨isEmptyElim, fun {j} => IsEmpty.elim' inferInstance j.1⟩ haveI : IsCofiltered J := ⟨⟩ use fun j : J => if hj : j ∈ G then F.map (IsCofiltered.infTo G H hj) (h (IsCofiltered.inf G H)).some else (h _).some rintro ⟨X, Y, hX, hY, f⟩ hf dsimp only rwa [dif_pos hX, dif_pos hY, ← comp_app, ← F.map_comp, @IsCofiltered.infTo_commutes _ _ _ G H] theorem partialSections.directed : Directed Superset fun G : FiniteDiagram J => partialSections F G.2 := by classical intro A B let ιA : FiniteDiagramArrow A.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f => ⟨f.1, f.2.1, Finset.mem_union_left _ f.2.2.1, Finset.mem_union_left _ f.2.2.2.1, f.2.2.2.2⟩ let ιB : FiniteDiagramArrow B.1 → FiniteDiagramArrow (A.1 ⊔ B.1) := fun f => ⟨f.1, f.2.1, Finset.mem_union_right _ f.2.2.1, Finset.mem_union_right _ f.2.2.2.1, f.2.2.2.2⟩ refine ⟨⟨A.1 ⊔ B.1, A.2.image ιA ⊔ B.2.image ιB⟩, ?_, ?_⟩ · rintro u hu f hf have : ιA f ∈ A.2.image ιA ⊔ B.2.image ιB := by apply Finset.mem_union_left rw [Finset.mem_image] exact ⟨f, hf, rfl⟩ exact hu this · rintro u hu f hf have : ιB f ∈ A.2.image ιA ⊔ B.2.image ιB := by apply Finset.mem_union_right rw [Finset.mem_image] exact ⟨f, hf, rfl⟩ exact hu this theorem partialSections.closed [∀ j : J, T2Space (F.obj j)] {G : Finset J} (H : Finset (FiniteDiagramArrow G)) : IsClosed (partialSections F H) := by have : partialSections F H = ⋂ (f : FiniteDiagramArrow G) (_ : f ∈ H), {u \| F.map f.2.2.2.2 (u f.1) = u f.2.1} := by ext1 simp only [Set.mem_iInter, Set.mem_setOf_eq] rfl rw [this] apply isClosed_biInter intro f _ apply isClosed_eq <;> fun_prop /-- Cofiltered limits of nonempty compact Hausdorff spaces are nonempty topological spaces. -/ theorem nonempty_limitCone_of_compact_t2_cofiltered_system (F : J ⥤ TopCat.{max v u}) [IsCofilteredOrEmpty J] [∀ j : J, Nonempty (F.obj j)] [∀ j : J, CompactSpace (F.obj j)] [∀ j : J, T2Space (F.obj j)] : Nonempty (TopCat.limitCone F).pt := by classical obtain ⟨u, hu⟩ := IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed (fun G => partialSections F _) (partialSections.directed F) (fun G => partialSections.nonempty F _) (fun G => IsClosed.isCompact (partialSections.closed F _)) fun G => partialSections.closed F _ use u intro X Y f let G : FiniteDiagram J := ⟨{X, Y}, {⟨X, Y, by grind, by grind, f⟩}⟩ exact hu _ ⟨G, rfl⟩ (Finset.mem_singleton_self _) end TopologicalKonig end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Limits/Basic.lean	import Mathlib.Topology.Category.TopCat.Adjunctions import Mathlib.CategoryTheory.Limits.Types.Limits import Mathlib.CategoryTheory.Limits.Types.Colimits import Mathlib.CategoryTheory.Limits.Shapes.Terminal import Mathlib.CategoryTheory.Adjunction.Limits /-! # The category of topological spaces has all limits and colimits Further, these limits and colimits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types. -/ open TopologicalSpace CategoryTheory CategoryTheory.Limits Opposite universe v u u' w noncomputable section local notation "forget" => forget TopCat namespace TopCat section Limits variable {J : Type v} [Category.{w} J] attribute [local fun_prop] continuous_subtype_val /-- A choice of limit cone for a functor `F : J ⥤ TopCat`. Generally you should just use `limit.cone F`, unless you need the actual definition (which is in terms of `Types.limitCone`). -/ def limitCone (F : J ⥤ TopCat.{max v u}) : Cone F where pt := TopCat.of { u : ∀ j : J, F.obj j \| ∀ {i j : J} (f : i ⟶ j), F.map f (u i) = u j } π := { app := fun j => ofHom { toFun := fun u => u.val j -- Porting note: `continuity` from the original mathlib3 proof failed here. continuous_toFun := Continuous.comp (continuous_apply _) (continuous_subtype_val) } naturality := fun X Y f => by ext a exact (a.2 f).symm } /-- The chosen cone `TopCat.limitCone F` for a functor `F : J ⥤ TopCat` is a limit cone. Generally you should just use `limit.isLimit F`, unless you need the actual definition (which is in terms of `Types.limitConeIsLimit`). -/ def limitConeIsLimit (F : J ⥤ TopCat.{max v u}) : IsLimit (limitCone.{v,u} F) where lift S := ofHom { toFun := fun x => ⟨fun _ => S.π.app _ x, fun f => by dsimp rw [← S.w f] rfl⟩ continuous_toFun := Continuous.subtype_mk (continuous_pi fun j => (S.π.app j).hom.2) fun x i j f => by dsimp rw [← S.w f] rfl } uniq S m h := by ext a simp [← h] rfl section variable {F : J ⥤ TopCat.{u}} (c : Cone (F ⋙ forget)) /-- Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)` of the underlying functor to types, this is the type `c.pt` with the infimum of the induced topologies by the maps `c.π.app j`. -/ def conePtOfConeForget : Type _ := c.pt instance topologicalSpaceConePtOfConeForget : TopologicalSpace (conePtOfConeForget c) := (⨅ j, (F.obj j).str.induced (c.π.app j)) /-- Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)` of the underlying functor to types, this is a cone for `F` whose point is `c.pt` with the infimum of the induced topologies by the maps `c.π.app j`. -/ @[simps pt π_app] def coneOfConeForget : Cone F where pt := of (conePtOfConeForget c) π := { app j := ofHom (ContinuousMap.mk (c.π.app j) (by rw [continuous_iff_le_induced] exact iInf_le (fun j ↦ (F.obj j).str.induced (c.π.app j)) j)) naturality j j' φ := by ext apply congr_fun (c.π.naturality φ) } /-- Given a functor `F : J ⥤ TopCat` and a cone `c : Cone (F ⋙ forget)` of the underlying functor to types, the limit of `F` is `c.pt` equipped with the infimum of the induced topologies by the maps `c.π.app j`. -/ def isLimitConeOfForget (c : Cone (F ⋙ forget)) (hc : IsLimit c) : IsLimit (coneOfConeForget c) := by refine IsLimit.ofFaithful forget (ht := hc) (fun s ↦ ofHom (ContinuousMap.mk (hc.lift ((forget).mapCone s)) ?_)) (fun _ ↦ rfl) rw [continuous_iff_coinduced_le] dsimp [topologicalSpaceConePtOfConeForget] rw [le_iInf_iff] intro j rw [coinduced_le_iff_le_induced, induced_compose] convert continuous_iff_le_induced.1 (s.π.app j).hom.continuous exact hc.fac ((forget).mapCone s) j end section IsLimit variable {F : J ⥤ TopCat.{u}} (c : Cone F) (hc : IsLimit c) include hc theorem induced_of_isLimit : c.pt.str = ⨅ j, (F.obj j).str.induced (c.π.app j) := by let c' := coneOfConeForget ((forget).mapCone c) let hc' : IsLimit c' := isLimitConeOfForget _ (isLimitOfPreserves forget hc) let e := IsLimit.conePointUniqueUpToIso hc' hc have he (j : J) : e.inv ≫ c'.π.app j = c.π.app j := IsLimit.conePointUniqueUpToIso_inv_comp hc' hc j apply (homeoOfIso e.symm).induced_eq.symm.trans dsimp [coneOfConeForget_pt, c', topologicalSpaceConePtOfConeForget] conv_rhs => simp only [← he] simp [← induced_compose, homeoOfIso, c'] end IsLimit variable (F : J ⥤ TopCat.{u}) theorem limit_topology [HasLimit F] : (limit F).str = ⨅ j, (F.obj j).str.induced (limit.π F j) := induced_of_isLimit _ (limit.isLimit _) lemma hasLimit_iff_small_sections : HasLimit F ↔ Small.{u} ((F ⋙ forget).sections) := by rw [← Types.hasLimit_iff_small_sections] constructor <;> intro · infer_instance · exact ⟨⟨_, isLimitConeOfForget _ (limit.isLimit _)⟩⟩ instance topCat_hasLimitsOfShape (J : Type v) [Category J] [Small.{u} J] : HasLimitsOfShape J TopCat.{u} where has_limit := fun F => by rw [hasLimit_iff_small_sections] infer_instance instance topCat_hasLimitsOfSize [UnivLE.{v, u}] : HasLimitsOfSize.{w, v} TopCat.{u} where instance topCat_hasLimits : HasLimits TopCat.{u} := TopCat.topCat_hasLimitsOfSize.{u, u} instance forget_preservesLimitsOfSize : PreservesLimitsOfSize.{w, v} (forget : TopCat.{u} ⥤ _) where instance forget_preservesLimits : PreservesLimits (forget : TopCat.{u} ⥤ _) where end Limits section Colimits variable {J : Type v} [Category.{w} J] {F : J ⥤ TopCat.{u}} section variable (c : Cocone (F ⋙ forget)) /-- Given a functor `F : J ⥤ TopCat` and a cocone `c : Cocone (F ⋙ forget)` of the underlying cocone of types, this is the type `c.pt` with the supremum of the topologies that are coinduced by the maps `c.ι.app j`. -/ def coconePtOfCoconeForget : Type _ := c.pt instance topologicalSpaceCoconePtOfCoconeForget : TopologicalSpace (coconePtOfCoconeForget c) := (⨆ j, (F.obj j).str.coinduced (c.ι.app j)) /-- Given a functor `F : J ⥤ TopCat` and a cocone `c : Cocone (F ⋙ forget)` of the underlying cocone of types, this is a cocone for `F` whose point is `c.pt` with the supremum of the coinduced topologies by the maps `c.ι.app j`. -/ @[simps pt ι_app] def coconeOfCoconeForget : Cocone F where pt := of (coconePtOfCoconeForget c) ι := { app j := ofHom (ContinuousMap.mk (c.ι.app j) (by rw [continuous_iff_coinduced_le] exact le_iSup (fun j ↦ (F.obj j).str.coinduced (c.ι.app j)) j)) naturality j j' φ := by ext apply congr_fun (c.ι.naturality φ) } /-- Given a functor `F : J ⥤ TopCat` and a cocone `c : Cocone (F ⋙ forget)` of the underlying cocone of types, the colimit of `F` is `c.pt` equipped with the supremum of the coinduced topologies by the maps `c.ι.app j`. -/ def isColimitCoconeOfForget (c : Cocone (F ⋙ forget)) (hc : IsColimit c) : IsColimit (coconeOfCoconeForget c) := by refine IsColimit.ofFaithful forget (ht := hc) (fun s ↦ ofHom (ContinuousMap.mk (hc.desc ((forget).mapCocone s)) ?_)) (fun _ ↦ rfl) rw [continuous_iff_le_induced] dsimp [topologicalSpaceCoconePtOfCoconeForget] rw [iSup_le_iff] intro j rw [coinduced_le_iff_le_induced, induced_compose] convert continuous_iff_le_induced.1 (s.ι.app j).hom.continuous exact hc.fac ((forget).mapCocone s) j end section IsColimit variable (c : Cocone F) (hc : IsColimit c) include hc theorem coinduced_of_isColimit : c.pt.str = ⨆ j, (F.obj j).str.coinduced (c.ι.app j) := by let c' := coconeOfCoconeForget ((forget).mapCocone c) let hc' : IsColimit c' := isColimitCoconeOfForget _ (isColimitOfPreserves forget hc) let e := IsColimit.coconePointUniqueUpToIso hc' hc have he (j : J) : c'.ι.app j ≫ e.hom = c.ι.app j := IsColimit.comp_coconePointUniqueUpToIso_hom hc' hc j apply (homeoOfIso e).coinduced_eq.symm.trans dsimp [coconeOfCoconeForget_pt, c', topologicalSpaceCoconePtOfCoconeForget] simp only [coinduced_iSup] conv_rhs => simp only [← he] rfl lemma isOpen_iff_of_isColimit (X : Set c.pt) : IsOpen X ↔ ∀ (j : J), IsOpen (c.ι.app j ⁻¹' X) := by trans (⨆ (j : J), (F.obj j).str.coinduced (c.ι.app j)).IsOpen X · rw [← coinduced_of_isColimit c hc, isOpen_fold] · simp only [← isOpen_coinduced] apply isOpen_iSup_iff lemma isClosed_iff_of_isColimit (X : Set c.pt) : IsClosed X ↔ ∀ (j : J), IsClosed (c.ι.app j ⁻¹' X) := by simp only [← isOpen_compl_iff, isOpen_iff_of_isColimit _ hc, Functor.const_obj_obj, Set.preimage_compl] lemma continuous_iff_of_isColimit {X : Type u'} [TopologicalSpace X] (f : c.pt → X) : Continuous f ↔ ∀ (j : J), Continuous (f ∘ c.ι.app j) := by simp only [continuous_def, isOpen_iff_of_isColimit _ hc] tauto end IsColimit variable (F) theorem colimit_topology (F : J ⥤ TopCat.{u}) [HasColimit F] : (colimit F).str = ⨆ j, (F.obj j).str.coinduced (colimit.ι F j) := coinduced_of_isColimit _ (colimit.isColimit _) theorem colimit_isOpen_iff (F : J ⥤ TopCat.{u}) [HasColimit F] (U : Set ((colimit F : _) : Type u)) : IsOpen U ↔ ∀ j, IsOpen (colimit.ι F j ⁻¹' U) := by apply isOpen_iff_of_isColimit _ (colimit.isColimit _) lemma hasColimit_iff_small_colimitType : HasColimit F ↔ Small.{u} (F ⋙ forget).ColimitType := by rw [← Types.hasColimit_iff_small_colimitType] constructor <;> intro · infer_instance · exact ⟨⟨_, isColimitCoconeOfForget _ (colimit.isColimit _)⟩⟩ instance topCat_hasColimitsOfShape (J : Type v) [Category J] [Small.{u} J] : HasColimitsOfShape J TopCat.{u} where has_colimit := fun F => by rw [hasColimit_iff_small_colimitType] infer_instance instance topCat_hasColimitsOfSize [UnivLE.{v, u}] : HasColimitsOfSize.{w, v} TopCat.{u} where instance topCat_hasColimits : HasColimits TopCat.{u} := TopCat.topCat_hasColimitsOfSize.{u, u} instance forget_preservesColimitsOfSize : PreservesColimitsOfSize.{w, v} (forget : TopCat.{u} ⥤ _) where instance forget_preservesColimits : PreservesColimits (forget : TopCat.{u} ⥤ Type u) where end Colimits /-- The terminal object of `Top` is `PUnit`. -/ def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) := haveI : ∀ X, Unique (X ⟶ TopCat.of PUnit.{u + 1}) := fun X => ⟨⟨ofHom ⟨fun _ => PUnit.unit, continuous_const⟩⟩, fun f => by ext⟩ Limits.IsTerminal.ofUnique _ /-- The terminal object of `Top` is `PUnit`. -/ def terminalIsoPUnit : ⊤_ TopCat.{u} ≅ TopCat.of PUnit := terminalIsTerminal.uniqueUpToIso isTerminalPUnit /-- The initial object of `Top` is `PEmpty`. -/ def isInitialPEmpty : IsInitial (TopCat.of PEmpty.{u + 1}) := haveI : ∀ X, Unique (TopCat.of PEmpty.{u + 1} ⟶ X) := fun X => ⟨⟨ofHom ⟨fun x => x.elim, by continuity⟩⟩, fun f => by ext ⟨⟩⟩ Limits.IsInitial.ofUnique _ /-- The initial object of `Top` is `PEmpty`. -/ def initialIsoPEmpty : ⊥_ TopCat.{u} ≅ TopCat.of PEmpty := initialIsInitial.uniqueUpToIso isInitialPEmpty end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean	import Mathlib.Topology.Category.TopCat.Limits.Basic import Mathlib.CategoryTheory.Filtered.Basic /-! # Cofiltered limits in the category of topological spaces Given a compatible collection of topological bases for the factors in a cofiltered limit which contain `Set.univ` and are closed under intersections, the induced naive collection of sets in the limit is, in fact, a topological basis. -/ open TopologicalSpace Topology open CategoryTheory open CategoryTheory.Limits universe u v w noncomputable section namespace TopCat section CofilteredLimit variable {J : Type v} [Category.{w} J] [IsCofiltered J] (F : J ⥤ TopCat.{max v u}) (C : Cone F) /-- Given a compatible collection of topological bases for the factors in a cofiltered limit which contain `Set.univ` and are closed under intersections, the induced naive collection of sets in the limit is, in fact, a topological basis. -/ theorem isTopologicalBasis_cofiltered_limit (hC : IsLimit C) (T : ∀ j, Set (Set (F.obj j))) (hT : ∀ j, IsTopologicalBasis (T j)) (univ : ∀ i : J, Set.univ ∈ T i) (inter : ∀ (i) (U1 U2 : Set (F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i) (compat : ∀ (i j : J) (f : i ⟶ j) (V : Set (F.obj j)) (_hV : V ∈ T j), F.map f ⁻¹' V ∈ T i) : IsTopologicalBasis {U : Set C.pt \| ∃ (j : _) (V : Set (F.obj j)), V ∈ T j ∧ U = C.π.app j ⁻¹' V} := by classical convert IsTopologicalBasis.iInf_induced hT fun j (x : C.pt) => C.π.app j x using 1 · exact induced_of_isLimit C hC ext U0 constructor · rintro ⟨j, V, hV, rfl⟩ let U : ∀ i, Set (F.obj i) := fun i => if h : i = j then by rw [h]; exact V else Set.univ refine ⟨U, {j}, ?_, ?_⟩ · simp only [Finset.mem_singleton] rintro i rfl simpa [U] · simp [U] · rintro ⟨U, G, h1, h2⟩ obtain ⟨j, hj⟩ := IsCofiltered.inf_objs_exists G let g : ∀ e ∈ G, j ⟶ e := fun _ he => (hj he).some let Vs : J → Set (F.obj j) := fun e => if h : e ∈ G then F.map (g e h) ⁻¹' U e else Set.univ let V : Set (F.obj j) := ⋂ (e : J) (_he : e ∈ G), Vs e refine ⟨j, V, ?_, ?_⟩ · -- An intermediate claim used to apply induction along `G : Finset J` later on. have : ∀ (S : Set (Set (F.obj j))) (E : Finset J) (P : J → Set (F.obj j)) (_univ : Set.univ ∈ S) (_inter : ∀ A B : Set (F.obj j), A ∈ S → B ∈ S → A ∩ B ∈ S) (_cond : ∀ (e : J) (_he : e ∈ E), P e ∈ S), (⋂ (e) (_he : e ∈ E), P e) ∈ S := by intro S E induction E using Finset.induction_on with \| empty => intro P he _hh simpa \| insert a E _ha hh1 => intro hh2 hh3 hh4 hh5 rw [Finset.set_biInter_insert] refine hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 ?_) intro e he exact hh5 e (Finset.mem_insert_of_mem he) -- use the intermediate claim to finish off the goal using `univ` and `inter`. refine this _ _ _ (univ _) (inter _) ?_ intro e he dsimp [Vs] rw [dif_pos he] exact compat j e (g e he) (U e) (h1 e he) · -- conclude... rw [h2] change _ = (C.π.app j)⁻¹' ⋂ (e : J) (_ : e ∈ G), Vs e rw [Set.preimage_iInter] apply congrArg ext1 e rw [Set.preimage_iInter] apply congrArg ext1 he simp [Vs, dif_pos he, ← Set.preimage_comp, ← coe_comp] end CofilteredLimit end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean	import Mathlib.Topology.Category.TopCat.Limits.Products /-! # Pullbacks and pushouts in the category of topological spaces -/ open TopologicalSpace Topology open CategoryTheory open CategoryTheory.Limits universe v u w noncomputable section namespace TopCat variable {J : Type v} [Category.{w} J] section Pullback variable {X Y Z : TopCat.{u}} /-- The first projection from the pullback. -/ abbrev pullbackFst (f : X ⟶ Z) (g : Y ⟶ Z) : TopCat.of { p : X × Y // f p.1 = g p.2 } ⟶ X := ofHom ⟨Prod.fst ∘ Subtype.val, by fun_prop⟩ lemma pullbackFst_apply (f : X ⟶ Z) (g : Y ⟶ Z) (x) : pullbackFst f g x = x.1.1 := rfl /-- The second projection from the pullback. -/ abbrev pullbackSnd (f : X ⟶ Z) (g : Y ⟶ Z) : TopCat.of { p : X × Y // f p.1 = g p.2 } ⟶ Y := ofHom ⟨Prod.snd ∘ Subtype.val, by fun_prop⟩ lemma pullbackSnd_apply (f : X ⟶ Z) (g : Y ⟶ Z) (x) : pullbackSnd f g x = x.1.2 := rfl /-- The explicit pullback cone of `X, Y` given by `{ p : X × Y // f p.1 = g p.2 }`. -/ def pullbackCone (f : X ⟶ Z) (g : Y ⟶ Z) : PullbackCone f g := PullbackCone.mk (pullbackFst f g) (pullbackSnd f g) (by dsimp [pullbackFst, pullbackSnd, Function.comp_def] ext ⟨x, h⟩ simpa) /-- The constructed cone is a limit. -/ def pullbackConeIsLimit (f : X ⟶ Z) (g : Y ⟶ Z) : IsLimit (pullbackCone f g) := PullbackCone.isLimitAux' _ (by intro S constructor; swap · exact ofHom { toFun := fun x => ⟨⟨S.fst x, S.snd x⟩, by simpa using ConcreteCategory.congr_hom S.condition x⟩ continuous_toFun := by fun_prop } refine ⟨?_, ?_, ?_⟩ · delta pullbackCone ext a dsimp · delta pullbackCone ext a dsimp · intro m h₁ h₂ ext x -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): used to be `ext x`. apply Subtype.ext apply Prod.ext · simpa using ConcreteCategory.congr_hom h₁ x · simpa using ConcreteCategory.congr_hom h₂ x) /-- The pullback of two maps can be identified as a subspace of `X × Y`. -/ def pullbackIsoProdSubtype (f : X ⟶ Z) (g : Y ⟶ Z) : pullback f g ≅ TopCat.of { p : X × Y // f p.1 = g p.2 } := (limit.isLimit _).conePointUniqueUpToIso (pullbackConeIsLimit f g) @[reassoc (attr := simp)] theorem pullbackIsoProdSubtype_inv_fst (f : X ⟶ Z) (g : Y ⟶ Z) : (pullbackIsoProdSubtype f g).inv ≫ pullback.fst _ _ = pullbackFst f g := by simp [pullbackCone, pullbackIsoProdSubtype] theorem pullbackIsoProdSubtype_inv_fst_apply (f : X ⟶ Z) (g : Y ⟶ Z) (x : { p : X × Y // f p.1 = g p.2 }) : pullback.fst f g ((pullbackIsoProdSubtype f g).inv x) = (x : X × Y).fst := ConcreteCategory.congr_hom (pullbackIsoProdSubtype_inv_fst f g) x @[reassoc (attr := simp)] theorem pullbackIsoProdSubtype_inv_snd (f : X ⟶ Z) (g : Y ⟶ Z) : (pullbackIsoProdSubtype f g).inv ≫ pullback.snd _ _ = pullbackSnd f g := by simp [pullbackCone, pullbackIsoProdSubtype] theorem pullbackIsoProdSubtype_inv_snd_apply (f : X ⟶ Z) (g : Y ⟶ Z) (x : { p : X × Y // f p.1 = g p.2 }) : pullback.snd f g ((pullbackIsoProdSubtype f g).inv x) = (x : X × Y).snd := ConcreteCategory.congr_hom (pullbackIsoProdSubtype_inv_snd f g) x theorem pullbackIsoProdSubtype_hom_fst (f : X ⟶ Z) (g : Y ⟶ Z) : (pullbackIsoProdSubtype f g).hom ≫ pullbackFst f g = pullback.fst _ _ := by rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_fst] theorem pullbackIsoProdSubtype_hom_snd (f : X ⟶ Z) (g : Y ⟶ Z) : (pullbackIsoProdSubtype f g).hom ≫ pullbackSnd f g = pullback.snd _ _ := by rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_snd] theorem pullbackIsoProdSubtype_hom_apply {f : X ⟶ Z} {g : Y ⟶ Z} (x : ↑(pullback f g)) : (pullbackIsoProdSubtype f g).hom x = ⟨⟨pullback.fst f g x, pullback.snd f g x⟩, by simpa using CategoryTheory.congr_fun pullback.condition x⟩ := rfl theorem pullback_topology {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) : (pullback f g).str = induced (pullback.fst f g) X.str ⊓ induced (pullback.snd f g) Y.str := by let homeo := homeoOfIso (pullbackIsoProdSubtype f g) refine homeo.isInducing.eq_induced.trans ?_ change induced homeo (induced _ ( (induced Prod.fst X.str) ⊓ (induced Prod.snd Y.str))) = _ simp only [induced_compose, induced_inf] rfl theorem range_pullback_to_prod {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) : Set.range (prod.lift (pullback.fst f g) (pullback.snd f g)) = { x \| (Limits.prod.fst ≫ f) x = (Limits.prod.snd ≫ g) x } := by ext x constructor · rintro ⟨y, rfl⟩ simp only [← ConcreteCategory.comp_apply, Set.mem_setOf_eq] simp [pullback.condition] · rintro (h : f (_, _).1 = g (_, _).2) use (pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, h⟩ apply Concrete.limit_ext rintro ⟨⟨⟩⟩ <;> rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply, limit.lift_π] <;> -- This used to be `simp` before https://github.com/leanprover/lean4/pull/2644 cat_disch /-- The pullback along an embedding is (isomorphic to) the preimage. -/ noncomputable def pullbackHomeoPreimage {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → Z) (hf : Continuous f) (g : Y → Z) (hg : IsEmbedding g) : { p : X × Y // f p.1 = g p.2 } ≃ₜ f ⁻¹' Set.range g where toFun := fun x ↦ ⟨x.1.1, _, x.2.symm⟩ invFun := fun x ↦ ⟨⟨x.1, Exists.choose x.2⟩, (Exists.choose_spec x.2).symm⟩ left_inv := by intro x ext <;> dsimp apply hg.injective convert x.prop exact Exists.choose_spec (p := fun y ↦ g y = f (↑x : X × Y).1) _ continuous_toFun := by fun_prop continuous_invFun := by apply Continuous.subtype_mk refine continuous_subtype_val.prodMk <\| hg.isInducing.continuous_iff.mpr ?_ convert hf.comp continuous_subtype_val ext x exact Exists.choose_spec x.2 theorem isInducing_pullback_to_prod {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) : IsInducing <\| ⇑(prod.lift (pullback.fst f g) (pullback.snd f g)) := ⟨by simp [prod_topology, pullback_topology, induced_compose, ← coe_comp]⟩ theorem isEmbedding_pullback_to_prod {X Y Z : TopCat.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) : IsEmbedding <\| ⇑(prod.lift (pullback.fst f g) (pullback.snd f g)) := ⟨isInducing_pullback_to_prod f g, (TopCat.mono_iff_injective _).mp inferInstance⟩ /-- If the map `S ⟶ T` is mono, then there is a description of the image of `W ×ₛ X ⟶ Y ×ₜ Z`. -/ theorem range_pullback_map {W X Y Z S T : TopCat} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : Set.range (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) = (pullback.fst g₁ g₂) ⁻¹' Set.range i₁ ∩ (pullback.snd g₁ g₂) ⁻¹' Set.range i₂ := by ext constructor · rintro ⟨y, rfl⟩ simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range] rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply] simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app] exact ⟨exists_apply_eq_apply _ _, exists_apply_eq_apply _ _⟩ rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩ have : f₁ x₁ = f₂ x₂ := by apply (TopCat.mono_iff_injective _).mp H₃ rw [← ConcreteCategory.comp_apply, eq₁, ← ConcreteCategory.comp_apply, eq₂, ConcreteCategory.comp_apply, ConcreteCategory.comp_apply, hx₁, hx₂, ← ConcreteCategory.comp_apply, pullback.condition, ConcreteCategory.comp_apply] use (pullbackIsoProdSubtype f₁ f₂).inv ⟨⟨x₁, x₂⟩, this⟩ apply Concrete.limit_ext rintro (_ \| _ \| _) <;> rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply] · simp [hx₁, ← limit.w _ WalkingCospan.Hom.inl] · simp [hx₁] · simp [hx₂] theorem pullback_fst_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) : Set.range (pullback.fst f g) = { x : X \| ∃ y : Y, f x = g y } := by ext x constructor · rintro ⟨y, rfl⟩ use pullback.snd f g y exact CategoryTheory.congr_fun pullback.condition y · rintro ⟨y, eq⟩ use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩ rw [pullbackIsoProdSubtype_inv_fst_apply] theorem pullback_snd_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) : Set.range (pullback.snd f g) = { y : Y \| ∃ x : X, f x = g y } := by ext y constructor · rintro ⟨x, rfl⟩ use pullback.fst f g x exact CategoryTheory.congr_fun pullback.condition x · rintro ⟨x, eq⟩ use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩ rw [pullbackIsoProdSubtype_inv_snd_apply] /-- If there is a diagram where the morphisms `W ⟶ Y` and `X ⟶ Z` are embeddings, then the induced morphism `W ×ₛ X ⟶ Y ×ₜ Z` is also an embedding. ``` W ⟶ Y ↘ ↘ S ⟶ T ↗ ↗ X ⟶ Z ``` -/ theorem pullback_map_isEmbedding {W X Y Z S T : TopCat.{u}} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : IsEmbedding i₁) (H₂ : IsEmbedding i₂) (i₃ : S ⟶ T) (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : IsEmbedding (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) := by refine .of_comp (ContinuousMap.continuous_toFun _) (show Continuous (prod.lift (pullback.fst g₁ g₂) (pullback.snd g₁ g₂)) from ContinuousMap.continuous_toFun _) ?_ suffices IsEmbedding (prod.lift (pullback.fst f₁ f₂) (pullback.snd f₁ f₂) ≫ Limits.prod.map i₁ i₂) by simpa [← coe_comp] using this rw [coe_comp] exact (isEmbedding_prodMap H₁ H₂).comp (isEmbedding_pullback_to_prod _ _) /-- If there is a diagram where the morphisms `W ⟶ Y` and `X ⟶ Z` are open embeddings, and `S ⟶ T` is mono, then the induced morphism `W ×ₛ X ⟶ Y ×ₜ Z` is also an open embedding. ``` W ⟶ Y ↘ ↘ S ⟶ T ↗ ↗ X ⟶ Z ``` -/ theorem pullback_map_isOpenEmbedding {W X Y Z S T : TopCat.{u}} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : IsOpenEmbedding i₁) (H₂ : IsOpenEmbedding i₂) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : IsOpenEmbedding (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) := by constructor · apply pullback_map_isEmbedding f₁ f₂ g₁ g₂ H₁.isEmbedding H₂.isEmbedding i₃ eq₁ eq₂ · rw [range_pullback_map] apply IsOpen.inter <;> apply Continuous.isOpen_preimage · apply ContinuousMap.continuous_toFun · exact H₁.isOpen_range · apply ContinuousMap.continuous_toFun · exact H₂.isOpen_range lemma snd_isEmbedding_of_left {X Y S : TopCat} {f : X ⟶ S} (H : IsEmbedding f) (g : Y ⟶ S) : IsEmbedding <\| ⇑(pullback.snd f g) := by convert (homeoOfIso (asIso (pullback.snd (𝟙 S) g))).isEmbedding.comp (pullback_map_isEmbedding (i₂ := 𝟙 Y) f g (𝟙 S) g H (homeoOfIso (Iso.refl _)).isEmbedding (𝟙 _) rfl (by simp)) simp [homeoOfIso, ← coe_comp] theorem fst_isEmbedding_of_right {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S} (H : IsEmbedding g) : IsEmbedding <\| ⇑(pullback.fst f g) := by convert (homeoOfIso (asIso (pullback.fst f (𝟙 S)))).isEmbedding.comp (pullback_map_isEmbedding (i₁ := 𝟙 X) f g f (𝟙 _) (homeoOfIso (Iso.refl _)).isEmbedding H (𝟙 _) rfl (by simp)) simp [homeoOfIso, ← coe_comp] theorem isEmbedding_of_pullback {X Y S : TopCat} {f : X ⟶ S} {g : Y ⟶ S} (H₁ : IsEmbedding f) (H₂ : IsEmbedding g) : IsEmbedding (limit.π (cospan f g) WalkingCospan.one) := by convert H₂.comp (snd_isEmbedding_of_left H₁ g) rw [← coe_comp, ← limit.w _ WalkingCospan.Hom.inr] rfl theorem snd_isOpenEmbedding_of_left {X Y S : TopCat} {f : X ⟶ S} (H : IsOpenEmbedding f) (g : Y ⟶ S) : IsOpenEmbedding <\| ⇑(pullback.snd f g) := by convert (homeoOfIso (asIso (pullback.snd (𝟙 S) g))).isOpenEmbedding.comp (pullback_map_isOpenEmbedding (i₂ := 𝟙 Y) f g (𝟙 _) g H (homeoOfIso (Iso.refl _)).isOpenEmbedding (𝟙 _) rfl (by simp)) simp [homeoOfIso, ← coe_comp] theorem fst_isOpenEmbedding_of_right {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S} (H : IsOpenEmbedding g) : IsOpenEmbedding <\| ⇑(pullback.fst f g) := by convert (homeoOfIso (asIso (pullback.fst f (𝟙 S)))).isOpenEmbedding.comp (pullback_map_isOpenEmbedding (i₁ := 𝟙 X) f g f (𝟙 _) (homeoOfIso (Iso.refl _)).isOpenEmbedding H (𝟙 _) rfl (by simp)) simp [homeoOfIso, ← coe_comp] /-- If `X ⟶ S`, `Y ⟶ S` are open embeddings, then so is `X ×ₛ Y ⟶ S`. -/ theorem isOpenEmbedding_of_pullback {X Y S : TopCat} {f : X ⟶ S} {g : Y ⟶ S} (H₁ : IsOpenEmbedding f) (H₂ : IsOpenEmbedding g) : IsOpenEmbedding (limit.π (cospan f g) WalkingCospan.one) := by convert H₂.comp (snd_isOpenEmbedding_of_left H₁ g) rw [← coe_comp, ← limit.w _ WalkingCospan.Hom.inr] rfl theorem fst_iso_of_right_embedding_range_subset {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S} (hg : IsEmbedding g) (H : Set.range f ⊆ Set.range g) : IsIso (pullback.fst f g) := by let esto : (pullback f g : TopCat) ≃ₜ X := (fst_isEmbedding_of_right f hg).toHomeomorph.trans { toFun := Subtype.val invFun := fun x => ⟨x, by rw [pullback_fst_range] exact ⟨_, (H (Set.mem_range_self x)).choose_spec.symm⟩⟩ } convert (isoOfHomeo esto).isIso_hom theorem snd_iso_of_left_embedding_range_subset {X Y S : TopCat} {f : X ⟶ S} (hf : IsEmbedding f) (g : Y ⟶ S) (H : Set.range g ⊆ Set.range f) : IsIso (pullback.snd f g) := by let esto : (pullback f g : TopCat) ≃ₜ Y := (snd_isEmbedding_of_left hf g).toHomeomorph.trans { toFun := Subtype.val invFun := fun x => ⟨x, by rw [pullback_snd_range] exact ⟨_, (H (Set.mem_range_self x)).choose_spec⟩⟩ } convert (isoOfHomeo esto).isIso_hom theorem pullback_snd_image_fst_preimage (f : X ⟶ Z) (g : Y ⟶ Z) (U : Set X) : (pullback.snd f g) '' ((pullback.fst f g) ⁻¹' U) = g ⁻¹' (f '' U) := by ext x constructor · rintro ⟨y, hy, rfl⟩ exact ⟨(pullback.fst f g) y, hy, CategoryTheory.congr_fun pullback.condition y⟩ · rintro ⟨y, hy, eq⟩ -- next 5 lines were -- `exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq⟩, by simpa, by simp⟩` before https://github.com/leanprover-community/mathlib4/pull/13170 refine ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq⟩, ?_, ?_⟩ · simp only [coe_of, Set.mem_preimage] convert hy rw [pullbackIsoProdSubtype_inv_fst_apply] · rw [pullbackIsoProdSubtype_inv_snd_apply] theorem pullback_fst_image_snd_preimage (f : X ⟶ Z) (g : Y ⟶ Z) (U : Set Y) : (pullback.fst f g) '' ((pullback.snd f g) ⁻¹' U) = f ⁻¹' (g '' U) := by ext x constructor · rintro ⟨y, hy, rfl⟩ exact ⟨(pullback.snd f g) y, hy, (CategoryTheory.congr_fun pullback.condition y).symm⟩ · rintro ⟨y, hy, eq⟩ -- next 5 lines were -- `exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq.symm⟩, by simpa, by simp⟩` -- before https://github.com/leanprover-community/mathlib4/pull/13170 refine ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq.symm⟩, ?_, ?_⟩ · simp only [coe_of, Set.mem_preimage] convert hy rw [pullbackIsoProdSubtype_inv_snd_apply] · rw [pullbackIsoProdSubtype_inv_fst_apply] end Pullback section variable {X Y : TopCat.{u}} {f g : X ⟶ Y} lemma isOpen_iff_of_isColimit_cofork (c : Cofork f g) (hc : IsColimit c) (U : Set c.pt) : IsOpen U ↔ IsOpen (c.π ⁻¹' U) := by rw [isOpen_iff_of_isColimit _ hc] constructor · intro h exact h .one · rintro h (_ \| _) · rw [← c.w .left] exact Continuous.isOpen_preimage f.hom.continuous (c.π ⁻¹' U) h · exact h lemma isQuotientMap_of_isColimit_cofork (c : Cofork f g) (hc : IsColimit c) : IsQuotientMap c.π := by rw [isQuotientMap_iff] constructor · simpa only [← epi_iff_surjective] using epi_of_isColimit_cofork hc · exact isOpen_iff_of_isColimit_cofork c hc theorem coequalizer_isOpen_iff (U : Set ((coequalizer f g :) : Type u)) : IsOpen U ↔ IsOpen (coequalizer.π f g ⁻¹' U) := isOpen_iff_of_isColimit_cofork _ (coequalizerIsCoequalizer f g) _ end end TopCat
.lake/packages/mathlib/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean	import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products import Mathlib.Topology.Category.CompHausLike.Limits /-! # The sigma-comparison map This file defines the map `CompHausLike.sigmaComparison` associated to a presheaf `X` on `CompHausLike P`, and a finite family `S₁,...,Sₙ` of spaces in `CompHausLike P`, where `P` is stable under taking finite disjoint unions. The map `sigmaComparison` is the canonical map `X(S₁ ⊔ ... ⊔ Sₙ) ⟶ X(S₁) × ... × X(Sₙ)` induced by the inclusion maps `Sᵢ ⟶ S₁ ⊔ ... ⊔ Sₙ`, and it is an isomorphism when `X` preserves finite products. -/ universe u w open CategoryTheory Limits namespace CompHausLike variable {P : TopCat.{u} → Prop} [HasExplicitFiniteCoproducts.{u} P] (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w) [PreservesFiniteProducts X] {α : Type u} [Finite α] (σ : α → Type u) [∀ a, TopologicalSpace (σ a)] [∀ a, CompactSpace (σ a)] [∀ a, T2Space (σ a)] [∀ a, HasProp P (σ a)] instance : HasProp P (Σ (a : α), (σ a)) := HasExplicitFiniteCoproducts.hasProp (fun a ↦ of P (σ a)) /-- The comparison map from the value of a condensed set on a finite coproduct to the product of the values on the components. -/ def sigmaComparison : X.obj ⟨(of P ((a : α) × σ a))⟩ ⟶ ((a : α) → X.obj ⟨of P (σ a)⟩) := fun x a ↦ X.map (ofHom _ ⟨Sigma.mk a, continuous_sigmaMk⟩).op x theorem sigmaComparison_eq_comp_isos : sigmaComparison X σ = (X.mapIso (opCoproductIsoProduct' (finiteCoproduct.isColimit.{u, u} (fun a ↦ of P (σ a))) (productIsProduct fun x ↦ Opposite.op (of P (σ x))))).hom ≫ (PreservesProduct.iso X fun a ↦ ⟨of P (σ a)⟩).hom ≫ (Types.productIso.{u, max u w} fun a ↦ X.obj ⟨of P (σ a)⟩).hom := by ext x a simp only [Cofan.mk_pt, Fan.mk_pt, Functor.mapIso_hom, PreservesProduct.iso_hom, types_comp_apply, Types.productIso_hom_comp_eval_apply] have := congrFun (piComparison_comp_π X (fun a ↦ ⟨of P (σ a)⟩) a) simp only [types_comp_apply] at this rw [this, ← FunctorToTypes.map_comp_apply] simp only [sigmaComparison] apply congrFun congr 2 rw [← opCoproductIsoProduct_inv_comp_ι] simp only [Opposite.unop_op, unop_comp, Quiver.Hom.unop_op, Category.assoc] simp only [opCoproductIsoProduct, ← unop_comp, opCoproductIsoProduct'_comp_self] erw [IsColimit.fac] rfl instance isIsoSigmaComparison : IsIso <\| sigmaComparison X σ := by rw [sigmaComparison_eq_comp_isos] infer_instance end CompHausLike
.lake/packages/mathlib/Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean	import Mathlib.CategoryTheory.Sites.Coherent.Comparison import Mathlib.Topology.Category.CompHausLike.Limits /-! # Effective epimorphisms in `CompHausLike` In any category of compact Hausdorff spaces, continuous surjections are effective epimorphisms. We deduce that if the converse holds and explicit pullbacks exist, then `CompHausLike P` is preregular. If furthermore explicit finite coproducts exist, then `CompHausLike P` is precoherent. -/ universe u open CategoryTheory Limits Topology namespace CompHausLike variable {P : TopCat.{u} → Prop} /-- If `π` is a surjective morphism in `CompHausLike P`, then it is an effective epi. -/ noncomputable def effectiveEpiStruct {B X : CompHausLike P} (π : X ⟶ B) (hπ : Function.Surjective π) : EffectiveEpiStruct π where desc e h := ofHom _ ((IsQuotientMap.of_surjective_continuous hπ π.hom.continuous).lift e.hom fun a b hab ↦ CategoryTheory.congr_fun (h (ofHom _ ⟨fun _ ↦ a, continuous_const⟩) (ofHom _ ⟨fun _ ↦ b, continuous_const⟩) (by ext; exact hab)) a) fac e h := TopCat.hom_ext ((IsQuotientMap.of_surjective_continuous hπ π.hom.continuous).lift_comp e.hom fun a b hab ↦ CategoryTheory.congr_fun (h (ofHom _ ⟨fun _ ↦ a, continuous_const⟩) (ofHom _ ⟨fun _ ↦ b, continuous_const⟩) (by ext; exact hab)) a) uniq e h g hm := by suffices g = ofHom _ ((IsQuotientMap.of_surjective_continuous hπ π.hom.continuous).liftEquiv ⟨e.hom, fun a b hab ↦ CategoryTheory.congr_fun (h (ofHom _ ⟨fun _ ↦ a, continuous_const⟩) (ofHom _ ⟨fun _ ↦ b, continuous_const⟩) (by ext; exact hab)) a⟩) by assumption apply ConcreteCategory.ext rw [hom_ofHom, ← Equiv.symm_apply_eq (IsQuotientMap.of_surjective_continuous hπ π.hom.continuous).liftEquiv] ext simp only [IsQuotientMap.liftEquiv_symm_apply_coe, ContinuousMap.comp_apply, ← hm] rfl theorem preregular [HasExplicitPullbacks P] (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective f) : Preregular (CompHausLike P) where exists_fac := by intro X Y Z f π hπ refine ⟨pullback f π, pullback.fst f π, ⟨⟨effectiveEpiStruct _ ?_⟩⟩, pullback.snd f π, (pullback.condition _ _).symm⟩ intro y obtain ⟨z, hz⟩ := hs π hπ (f y) exact ⟨⟨(y, z), hz.symm⟩, rfl⟩ theorem precoherent [HasExplicitPullbacks P] [HasExplicitFiniteCoproducts.{0} P] (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective f) : Precoherent (CompHausLike P) := by have : Preregular (CompHausLike P) := preregular hs infer_instance end CompHausLike
.lake/packages/mathlib/Mathlib/Topology/Category/CompHausLike/Basic.lean	import Mathlib.Topology.Category.TopCat.Basic import Mathlib.CategoryTheory.Functor.EpiMono import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic /-! # Categories of Compact Hausdorff Spaces We construct the category of compact Hausdorff spaces satisfying an additional property `P`. ## Implementation We define a structure `CompHausLike` which takes as an argument a predicate `P` on topological spaces. It consists of the data of a topological space, satisfying the additional properties of being compact and Hausdorff, and satisfying `P`. We give a category structure to `CompHausLike P` induced by the forgetful functor to topological spaces. It used to be the case (before https://github.com/leanprover-community/mathlib4/pull/12930 was merged) that several different categories of compact Hausdorff spaces, possibly satisfying some extra property, were defined from scratch in this way. For example, one would define a structure `CompHaus` as follows: ```lean structure CompHaus where toTop : TopCat [is_compact : CompactSpace toTop] [is_hausdorff : T2Space toTop] ``` and give it the category structure induced from topological spaces. Then the category of profinite spaces was defined as follows: ```lean structure Profinite where toCompHaus : CompHaus [isTotallyDisconnected : TotallyDisconnectedSpace toCompHaus] ``` The categories `Stonean` consisting of extremally disconnected compact Hausdorff spaces and `LightProfinite` consisting of totally disconnected, second countable compact Hausdorff spaces were defined in a similar way. This resulted in code duplication, and reducing this duplication was part of the motivation for introducing `CompHausLike`. Using `CompHausLike`, we can now define `CompHaus := CompHausLike (fun _ ↦ True)` `Profinite := CompHausLike (fun X ↦ TotallyDisconnectedSpace X)`. `Stonean := CompHausLike (fun X ↦ ExtremallyDisconnected X)`. `LightProfinite := CompHausLike (fun X ↦ TotallyDisconnectedSpace X ∧ SecondCountableTopology X)`. These four categories are important building blocks of condensed objects (see the files `Condensed.Basic` and `Condensed.Light.Basic`). These categories share many properties and often, one wants to argue about several of them simultaneously. This is the other part of the motivation for introducing `CompHausLike`. On paper, one would say "let `C` be on of the categories `CompHaus` or `Profinite`, then the following holds: ...". This was not possible in Lean using the old definitions. Using the new definitions, this becomes a matter of identifying what common property of `CompHaus` and `Profinite` is used in the proof in question, and then proving the theorem for `CompHausLike P` satisfying that property, and it will automatically apply to both `CompHaus` and `Profinite`. -/ universe u open CategoryTheory variable (P : TopCat.{u} → Prop) /-- The type of Compact Hausdorff topological spaces satisfying an additional property `P`. -/ structure CompHausLike where /-- The underlying topological space of an object of `CompHausLike P`. -/ toTop : TopCat /-- The underlying topological space is compact. -/ [is_compact : CompactSpace toTop] /-- The underlying topological space is T2. -/ [is_hausdorff : T2Space toTop] /-- The underlying topological space satisfies P. -/ prop : P toTop namespace CompHausLike attribute [instance] is_compact is_hausdorff instance : CoeSort (CompHausLike P) (Type u) := ⟨fun X => X.toTop⟩ instance category : Category (CompHausLike P) := InducedCategory.category toTop instance concreteCategory : ConcreteCategory (CompHausLike P) (C(·, ·)) := InducedCategory.concreteCategory toTop instance hasForget₂ : HasForget₂ (CompHausLike P) TopCat := InducedCategory.hasForget₂ _ variable (X : Type u) [TopologicalSpace X] [CompactSpace X] [T2Space X] /-- This wraps the predicate `P : TopCat → Prop` in a typeclass. -/ class HasProp : Prop where hasProp : P (TopCat.of X) instance (X : CompHausLike P) : HasProp P X := ⟨X.4⟩ variable [HasProp P X] /-- A constructor for objects of the category `CompHausLike P`, taking a type, and bundling the compact Hausdorff topology found by typeclass inference. -/ abbrev of : CompHausLike P where toTop := TopCat.of X is_compact := ‹_› is_hausdorff := ‹_› prop := HasProp.hasProp theorem coe_of : (CompHausLike.of P X : Type _) = X := rfl @[simp] theorem coe_id (X : CompHausLike P) : (𝟙 X : X → X) = id := rfl @[simp] theorem coe_comp {X Y Z : CompHausLike P} (f : X ⟶ Y) (g : Y ⟶ Z) : (f ≫ g : X → Z) = g ∘ f := rfl section variable {X} {Y : Type u} [TopologicalSpace Y] [CompactSpace Y] [T2Space Y] [HasProp P Y] variable {Z : Type u} [TopologicalSpace Z] [CompactSpace Z] [T2Space Z] [HasProp P Z] /-- Typecheck a continuous map as a morphism in the category `CompHausLike P`. -/ abbrev ofHom (f : C(X, Y)) : of P X ⟶ of P Y := ConcreteCategory.ofHom f @[simp] lemma hom_ofHom (f : C(X, Y)) : ConcreteCategory.hom (ofHom P f) = f := rfl @[simp] lemma ofHom_id : ofHom P (ContinuousMap.id X) = 𝟙 (of _ X) := rfl @[simp] lemma ofHom_comp (f : C(X, Y)) (g : C(Y, Z)) : ofHom P (g.comp f) = ofHom _ f ≫ ofHom _ g := rfl end variable {P} /-- If `P` implies `P'`, then there is a functor from `CompHausLike P` to `CompHausLike P'`. -/ @[simps map] def toCompHausLike {P P' : TopCat → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) : CompHausLike P ⥤ CompHausLike P' where obj X := have : HasProp P' X := ⟨(h _ X.prop)⟩ CompHausLike.of _ X map f := f section variable {P P' : TopCat → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) /-- If `P` implies `P'`, then the functor from `CompHausLike P` to `CompHausLike P'` is fully faithful. -/ def fullyFaithfulToCompHausLike : (toCompHausLike h).FullyFaithful := fullyFaithfulInducedFunctor _ instance : (toCompHausLike h).Full := (fullyFaithfulToCompHausLike h).full instance : (toCompHausLike h).Faithful := (fullyFaithfulToCompHausLike h).faithful end variable (P) /-- The fully faithful embedding of `CompHausLike P` in `TopCat`. -/ @[simps! map] def compHausLikeToTop : CompHausLike.{u} P ⥤ TopCat.{u} := inducedFunctor _ -- The `Full, Faithful` instances should be constructed by a deriving handler. -- https://github.com/leanprover-community/mathlib4/issues/380 example {P P' : TopCat → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) : toCompHausLike h ⋙ compHausLikeToTop P' = compHausLikeToTop P := rfl /-- The functor from `CompHausLike P` to `TopCat` is fully faithful. -/ def fullyFaithfulCompHausLikeToTop : (compHausLikeToTop P).FullyFaithful := fullyFaithfulInducedFunctor _ instance : (compHausLikeToTop P).Full := inferInstanceAs (inducedFunctor _).Full instance : (compHausLikeToTop P).Faithful := inferInstanceAs (inducedFunctor _).Faithful instance (X : CompHausLike P) : CompactSpace ((compHausLikeToTop P).obj X) := inferInstanceAs (CompactSpace X.toTop) instance (X : CompHausLike P) : T2Space ((compHausLikeToTop P).obj X) := inferInstanceAs (T2Space X.toTop) variable {P} theorem epi_of_surjective {X Y : CompHausLike.{u} P} (f : X ⟶ Y) (hf : Function.Surjective f) : Epi f := by rw [← CategoryTheory.epi_iff_surjective] at hf exact (forget (CompHausLike P)).epi_of_epi_map hf theorem mono_iff_injective {X Y : CompHausLike.{u} P} (f : X ⟶ Y) : Mono f ↔ Function.Injective f := by constructor · intro hf x₁ x₂ h let g₁ : X ⟶ X := ofHom _ ⟨fun _ => x₁, continuous_const⟩ let g₂ : X ⟶ X := ofHom _ ⟨fun _ => x₂, continuous_const⟩ have : g₁ ≫ f = g₂ ≫ f := by ext; exact h exact CategoryTheory.congr_fun ((cancel_mono _).mp this) x₁ · rw [← CategoryTheory.mono_iff_injective] apply (forget (CompHausLike P)).mono_of_mono_map /-- Any continuous function on compact Hausdorff spaces is a closed map. -/ theorem isClosedMap {X Y : CompHausLike.{u} P} (f : X ⟶ Y) : IsClosedMap f := fun _ hC => (hC.isCompact.image f.hom.continuous).isClosed /-- Any continuous bijection of compact Hausdorff spaces is an isomorphism. -/ theorem isIso_of_bijective {X Y : CompHausLike.{u} P} (f : X ⟶ Y) (bij : Function.Bijective f) : IsIso f := by let E := Equiv.ofBijective _ bij have hE : Continuous E.symm := by rw [continuous_iff_isClosed] intro S hS rw [← E.image_eq_preimage_symm] exact isClosedMap f S hS refine ⟨⟨ofHom _ ⟨E.symm, hE⟩, ?_, ?_⟩⟩ · ext x apply E.symm_apply_apply · ext x apply E.apply_symm_apply instance forget_reflectsIsomorphisms : (forget (CompHausLike.{u} P)).ReflectsIsomorphisms := ⟨by intro A B f hf; rw [isIso_iff_bijective] at hf; exact isIso_of_bijective _ hf⟩ /-- Any continuous bijection of compact Hausdorff spaces induces an isomorphism. -/ noncomputable def isoOfBijective {X Y : CompHausLike.{u} P} (f : X ⟶ Y) (bij : Function.Bijective f) : X ≅ Y := letI := isIso_of_bijective _ bij asIso f /-- Construct an isomorphism from a homeomorphism. -/ @[simps!] def isoOfHomeo {X Y : CompHausLike.{u} P} (f : X ≃ₜ Y) : X ≅ Y := (fullyFaithfulCompHausLikeToTop P).preimageIso (TopCat.isoOfHomeo f) /-- Construct a homeomorphism from an isomorphism. -/ @[simps!] def homeoOfIso {X Y : CompHausLike.{u} P} (f : X ≅ Y) : X ≃ₜ Y := TopCat.homeoOfIso <\| (compHausLikeToTop P).mapIso f /-- The equivalence between isomorphisms in `CompHaus` and homeomorphisms of topological spaces. -/ @[simps] def isoEquivHomeo {X Y : CompHausLike.{u} P} : (X ≅ Y) ≃ (X ≃ₜ Y) where toFun := homeoOfIso invFun := isoOfHomeo /-- A constant map as a morphism in `CompHausLike` -/ def const {P : TopCat.{u} → Prop} (T : CompHausLike.{u} P) {S : CompHausLike.{u} P} (s : S) : T ⟶ S := ofHom _ (ContinuousMap.const _ s) lemma const_comp {P : TopCat.{u} → Prop} {S T U : CompHausLike.{u} P} (s : S) (g : S ⟶ U) : T.const s ≫ g = T.const (g s) := rfl end CompHausLike
.lake/packages/mathlib/Mathlib/Topology/Category/CompHausLike/Limits.lean	import Mathlib.CategoryTheory.Extensive import Mathlib.CategoryTheory.Limits.Preserves.Finite import Mathlib.Topology.Category.CompHausLike.Basic /-! # Explicit limits and colimits This file collects some constructions of explicit limits and colimits in `CompHausLike P`, which may be useful due to their definitional properties. ## Main definitions * `HasExplicitFiniteCoproducts`: A typeclass describing the property that forming all finite disjoint unions is stable under the property `P`. - Given this property, we deduce that `CompHausLike P` has finite coproducts and the inclusion functors to other `CompHausLike P'` and to `TopCat` preserve them. * `HasExplicitPullbacks`: A typeclass describing the property that forming all "explicit pullbacks" is stable under the property `P`. Here, explicit pullbacks are defined as a subset of the product. - Given this property, we deduce that `CompHausLike P` has pullbacks and the inclusion functors to other `CompHausLike P'` and to `TopCat` preserve them. - We also define a variant `HasExplicitPullbacksOfInclusions` which is says that explicit pullbacks along inclusion maps into finite disjoint unions exist. `Stonean` has this property but not the stronger one. ## Main results * Given `[HasExplicitPullbacksOfInclusions P]` (which is implied by `[HasExplicitPullbacks P]`), we provide an instance `FinitaryExtensive (CompHausLike P)`. -/ open CategoryTheory Limits Topology namespace CompHausLike universe w u section FiniteCoproducts variable {P : TopCat.{max u w} → Prop} {α : Type w} [Finite α] (X : α → CompHausLike P) /-- A typeclass describing the property that forming the disjoint union is stable under the property `P`. -/ abbrev HasExplicitFiniteCoproduct := HasProp P (Σ (a : α), X a) variable [HasExplicitFiniteCoproduct X] /-- The coproduct of a finite family of objects in `CompHaus`, constructed as the disjoint union with its usual topology. -/ def finiteCoproduct : CompHausLike P := CompHausLike.of P (Σ (a : α), X a) /-- The inclusion of one of the factors into the explicit finite coproduct. -/ def finiteCoproduct.ι (a : α) : X a ⟶ finiteCoproduct X := ofHom _ { toFun := fun x ↦ ⟨a, x⟩ continuous_toFun := continuous_sigmaMk (σ := fun a ↦ X a) } /-- To construct a morphism from the explicit finite coproduct, it suffices to specify a morphism from each of its factors. This is essentially the universal property of the coproduct. -/ def finiteCoproduct.desc {B : CompHausLike P} (e : (a : α) → (X a ⟶ B)) : finiteCoproduct X ⟶ B := ofHom _ { toFun := fun ⟨a, x⟩ ↦ e a x continuous_toFun := by apply continuous_sigma intro a; exact (e a).hom.continuous } @[reassoc (attr := simp)] lemma finiteCoproduct.ι_desc {B : CompHausLike P} (e : (a : α) → (X a ⟶ B)) (a : α) : finiteCoproduct.ι X a ≫ finiteCoproduct.desc X e = e a := rfl lemma finiteCoproduct.hom_ext {B : CompHausLike P} (f g : finiteCoproduct X ⟶ B) (h : ∀ a : α, finiteCoproduct.ι X a ≫ f = finiteCoproduct.ι X a ≫ g) : f = g := by ext ⟨a, x⟩ specialize h a apply_fun (fun q ↦ q x) at h exact h /-- The coproduct cocone associated to the explicit finite coproduct. -/ abbrev finiteCoproduct.cofan : Limits.Cofan X := Cofan.mk (finiteCoproduct X) (finiteCoproduct.ι X) /-- The explicit finite coproduct cocone is a colimit cocone. -/ def finiteCoproduct.isColimit : Limits.IsColimit (finiteCoproduct.cofan X) := mkCofanColimit _ (fun s ↦ desc _ fun a ↦ s.inj a) (fun _ _ ↦ ι_desc _ _ _) fun _ _ hm ↦ finiteCoproduct.hom_ext _ _ _ fun a ↦ (ConcreteCategory.hom_ext _ _ fun t ↦ congrFun (congrArg _ (hm a)) t) lemma finiteCoproduct.ι_injective (a : α) : Function.Injective (finiteCoproduct.ι X a) := by intro x y hxy exact eq_of_heq (Sigma.ext_iff.mp hxy).2 lemma finiteCoproduct.ι_jointly_surjective (R : finiteCoproduct X) : ∃ (a : α) (r : X a), R = finiteCoproduct.ι X a r := ⟨R.fst, R.snd, rfl⟩ lemma finiteCoproduct.ι_desc_apply {B : CompHausLike P} {π : (a : α) → X a ⟶ B} (a : α) : ∀ x, finiteCoproduct.desc X π (finiteCoproduct.ι X a x) = π a x := by tauto instance : HasCoproduct X where exists_colimit := ⟨finiteCoproduct.cofan X, finiteCoproduct.isColimit X⟩ variable (P) in /-- A typeclass describing the property that forming all finite disjoint unions is stable under the property `P`. -/ class HasExplicitFiniteCoproducts : Prop where hasProp {α : Type w} [Finite α] (X : α → CompHausLike.{max u w} P) : HasExplicitFiniteCoproduct X /- This linter complains that the universes `u` and `w` only occur together, but `w` appears by itself in the indexing type of the coproduct. In almost all cases, `w` will be either `0` or `u`, but we want to allow both possibilities. -/ attribute [nolint checkUnivs] HasExplicitFiniteCoproducts attribute [instance] HasExplicitFiniteCoproducts.hasProp instance [HasExplicitFiniteCoproducts.{w} P] (α : Type w) [Finite α] : HasColimitsOfShape (Discrete α) (CompHausLike P) where has_colimit _ := hasColimit_of_iso Discrete.natIsoFunctor instance [HasExplicitFiniteCoproducts.{w} P] : HasFiniteCoproducts (CompHausLike.{max u w} P) where out n := by let α := ULift.{w} (Fin n) let e : Discrete α ≌ Discrete (Fin n) := Discrete.equivalence Equiv.ulift exact hasColimitsOfShape_of_equivalence e variable {P : TopCat.{u} → Prop} [HasExplicitFiniteCoproducts.{0} P] example : HasFiniteCoproducts (CompHausLike.{u} P) := inferInstance /-- The inclusion maps into the explicit finite coproduct are open embeddings. -/ lemma finiteCoproduct.isOpenEmbedding_ι (a : α) : IsOpenEmbedding (finiteCoproduct.ι X a) := .sigmaMk (σ := fun a ↦ X a) /-- The inclusion maps into the abstract finite coproduct are open embeddings. -/ lemma Sigma.isOpenEmbedding_ι (a : α) : IsOpenEmbedding (Sigma.ι X a) := by refine IsOpenEmbedding.of_comp _ (homeoOfIso ((colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit X))).isOpenEmbedding ?_ convert finiteCoproduct.isOpenEmbedding_ι X a ext x change (Sigma.ι X a ≫ _) x = _ simp /-- The functor to `TopCat` preserves finite coproducts if they exist. -/ instance (P) [HasExplicitFiniteCoproducts.{0} P] : PreservesFiniteCoproducts (compHausLikeToTop P) := by refine ⟨fun _ ↦ ⟨fun {F} ↦ ?_⟩⟩ suffices PreservesColimit (Discrete.functor (F.obj ∘ Discrete.mk)) (compHausLikeToTop P) from preservesColimit_of_iso_diagram _ Discrete.natIsoFunctor.symm apply preservesColimit_of_preserves_colimit_cocone (CompHausLike.finiteCoproduct.isColimit _) exact TopCat.sigmaCofanIsColimit _ /-- The functor to another `CompHausLike` preserves finite coproducts if they exist. -/ noncomputable instance {P' : TopCat.{u} → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) : PreservesFiniteCoproducts (toCompHausLike h) := by have : PreservesFiniteCoproducts (toCompHausLike h ⋙ compHausLikeToTop P') := inferInstanceAs (PreservesFiniteCoproducts (compHausLikeToTop _)) exact preservesFiniteCoproducts_of_reflects_of_preserves (toCompHausLike h) (compHausLikeToTop P') end FiniteCoproducts section Pullbacks variable {P : TopCat.{u} → Prop} {X Y B : CompHausLike P} (f : X ⟶ B) (g : Y ⟶ B) /-- A typeclass describing the property that an explicit pullback is stable under the property `P`. -/ abbrev HasExplicitPullback := HasProp P { xy : X × Y \| f xy.fst = g xy.snd } variable [HasExplicitPullback f g] -- (hP : P (TopCat.of { xy : X × Y \| f xy.fst = g xy.snd })) /-- The pullback of two morphisms `f,g` in `CompHaus`, constructed explicitly as the set of pairs `(x,y)` such that `f x = g y`, with the topology induced by the product. -/ def pullback : CompHausLike P := letI set := { xy : X × Y \| f xy.fst = g xy.snd } haveI : CompactSpace set := isCompact_iff_compactSpace.mp (isClosed_eq (by fun_prop) (by fun_prop)).isCompact CompHausLike.of P set /-- The projection from the pullback to the first component. -/ def pullback.fst : pullback f g ⟶ X := TopCat.ofHom { toFun := fun ⟨⟨x, _⟩, _⟩ ↦ x continuous_toFun := Continuous.comp continuous_fst continuous_subtype_val } /-- The projection from the pullback to the second component. -/ def pullback.snd : pullback f g ⟶ Y := TopCat.ofHom { toFun := fun ⟨⟨_,y⟩,_⟩ ↦ y continuous_toFun := Continuous.comp continuous_snd continuous_subtype_val } @[reassoc] lemma pullback.condition : pullback.fst f g ≫ f = pullback.snd f g ≫ g := by ext ⟨_,h⟩; exact h /-- Construct a morphism to the explicit pullback given morphisms to the factors which are compatible with the maps to the base. This is essentially the universal property of the pullback. -/ def pullback.lift {Z : CompHausLike P} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) : Z ⟶ pullback f g := TopCat.ofHom { toFun := fun z ↦ ⟨⟨a z, b z⟩, by apply_fun (fun q ↦ q z) at w; exact w⟩ continuous_toFun := by fun_prop } @[reassoc (attr := simp)] lemma pullback.lift_fst {Z : CompHausLike P} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) : pullback.lift f g a b w ≫ pullback.fst f g = a := rfl @[reassoc (attr := simp)] lemma pullback.lift_snd {Z : CompHausLike P} (a : Z ⟶ X) (b : Z ⟶ Y) (w : a ≫ f = b ≫ g) : pullback.lift f g a b w ≫ pullback.snd f g = b := rfl lemma pullback.hom_ext {Z : CompHausLike P} (a b : Z ⟶ pullback f g) (hfst : a ≫ pullback.fst f g = b ≫ pullback.fst f g) (hsnd : a ≫ pullback.snd f g = b ≫ pullback.snd f g) : a = b := by ext z apply_fun (fun q ↦ q z) at hfst hsnd apply Subtype.ext apply Prod.ext · exact hfst · exact hsnd /-- The pullback cone whose cone point is the explicit pullback. -/ @[simps! pt π] def pullback.cone : Limits.PullbackCone f g := Limits.PullbackCone.mk (pullback.fst f g) (pullback.snd f g) (pullback.condition f g) /-- The explicit pullback cone is a limit cone. -/ @[simps! lift] def pullback.isLimit : Limits.IsLimit (pullback.cone f g) := Limits.PullbackCone.isLimitAux _ (fun s ↦ pullback.lift f g s.fst s.snd s.condition) (fun _ ↦ pullback.lift_fst _ _ _ _ _) (fun _ ↦ pullback.lift_snd _ _ _ _ _) (fun _ _ hm ↦ pullback.hom_ext _ _ _ _ (hm .left) (hm .right)) instance : HasLimit (cospan f g) where exists_limit := ⟨⟨pullback.cone f g, pullback.isLimit f g⟩⟩ /-- The functor to `TopCat` creates pullbacks if they exist. -/ noncomputable instance : CreatesLimit (cospan f g) (compHausLikeToTop P) := by refine createsLimitOfFullyFaithfulOfIso (pullback f g) (((TopCat.pullbackConeIsLimit f g).conePointUniqueUpToIso (limit.isLimit _)) ≪≫ Limits.lim.mapIso (?_ ≪≫ (diagramIsoCospan _).symm)) exact Iso.refl _ /-- The functor to `TopCat` preserves pullbacks. -/ noncomputable instance : PreservesLimit (cospan f g) (compHausLikeToTop P) := preservesLimit_of_createsLimit_and_hasLimit _ _ /-- The functor to another `CompHausLike` preserves pullbacks. -/ noncomputable instance {P' : TopCat → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) : PreservesLimit (cospan f g) (toCompHausLike h) := by have : PreservesLimit (cospan f g) (toCompHausLike h ⋙ compHausLikeToTop P') := inferInstanceAs (PreservesLimit _ (compHausLikeToTop _)) exact preservesLimit_of_reflects_of_preserves (toCompHausLike h) (compHausLikeToTop P') variable (P) in /-- A typeclass describing the property that forming all explicit pullbacks is stable under the property `P`. -/ class HasExplicitPullbacks : Prop where hasProp {X Y B : CompHausLike P} (f : X ⟶ B) (g : Y ⟶ B) : HasExplicitPullback f g attribute [instance] HasExplicitPullbacks.hasProp instance [HasExplicitPullbacks P] : HasPullbacks (CompHausLike P) where has_limit F := hasLimit_of_iso (diagramIsoCospan F).symm variable (P) in /-- A typeclass describing the property that explicit pullbacks along inclusion maps into disjoint unions is stable under the property `P`. -/ class HasExplicitPullbacksOfInclusions [HasExplicitFiniteCoproducts.{0} P] : Prop where hasProp : ∀ {X Y Z : CompHausLike P} (f : Z ⟶ X ⨿ Y), HasExplicitPullback coprod.inl f attribute [instance] HasExplicitPullbacksOfInclusions.hasProp instance [HasExplicitPullbacks P] [HasExplicitFiniteCoproducts.{0} P] : HasExplicitPullbacksOfInclusions P where hasProp _ := inferInstance end Pullbacks section FiniteCoproducts variable {P : TopCat.{u} → Prop} [HasExplicitFiniteCoproducts.{0} P] instance [HasExplicitPullbacksOfInclusions P] : HasPullbacksOfInclusions (CompHausLike P) where hasPullbackInl _ := inferInstance theorem hasPullbacksOfInclusions (hP' : ∀ ⦃X Y B : CompHausLike.{u} P⦄ (f : X ⟶ B) (g : Y ⟶ B) (_ : IsOpenEmbedding f), HasExplicitPullback f g) : HasExplicitPullbacksOfInclusions P := { hasProp := by intro _ _ _ f apply hP' exact Sigma.isOpenEmbedding_ι _ _ } /-- The functor to `TopCat` preserves pullbacks of inclusions if they exist. -/ noncomputable instance [HasExplicitPullbacksOfInclusions P] : PreservesPullbacksOfInclusions (compHausLikeToTop P) := { preservesPullbackInl := by intro X Y Z f infer_instance } instance [HasExplicitPullbacksOfInclusions P] : FinitaryExtensive (CompHausLike P) := finitaryExtensive_of_preserves_and_reflects (compHausLikeToTop P) theorem finitaryExtensive (hP' : ∀ ⦃X Y B : CompHausLike.{u} P⦄ (f : X ⟶ B) (g : Y ⟶ B) (_ : IsOpenEmbedding f), HasExplicitPullback f g) : FinitaryExtensive (CompHausLike P) := have := hasPullbacksOfInclusions hP' finitaryExtensive_of_preserves_and_reflects (compHausLikeToTop P) end FiniteCoproducts section Terminal variable {P : TopCat.{u} → Prop} /-- A one-element space is terminal in `CompHaus` -/ def isTerminalPUnit [HasProp P PUnit.{u + 1}] : IsTerminal (CompHausLike.of P PUnit.{u + 1}) := haveI : ∀ X, Unique (X ⟶ CompHausLike.of P PUnit.{u + 1}) := fun _ ↦ ⟨⟨ofHom _ ⟨fun _ ↦ PUnit.unit, continuous_const⟩⟩, fun _ ↦ rfl⟩ Limits.IsTerminal.ofUnique _ end Terminal end CompHausLike

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