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nhds_infty_eq : 𝓝 (∞ : OnePoint X) = map (↑) (coclosedCompact X) ⊔ pure ∞ := by
rw [← nhdsNE_infty_eq, nhdsNE_sup_pure]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
nhds_infty_eq
| null |
tendsto_coe_infty : Tendsto (↑) (coclosedCompact X) (𝓝 (∞ : OnePoint X)) := by
rw [nhds_infty_eq]
exact Filter.Tendsto.mono_right tendsto_map le_sup_left
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
tendsto_coe_infty
| null |
hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} := by
rw [nhds_infty_eq]
exact (hasBasis_coclosedCompact.map _).sup_pure _
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
hasBasis_nhds_infty
| null |
comap_coe_nhds_infty : comap ((↑) : X → OnePoint X) (𝓝 ∞) = coclosedCompact X := by
simp [nhds_infty_eq, comap_sup, comap_map coe_injective]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
comap_coe_nhds_infty
| null |
le_nhds_infty {f : Filter (OnePoint X)} :
f ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' sᶜ ∪ {∞} ∈ f := by
simp only [hasBasis_nhds_infty.ge_iff, and_imp]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
le_nhds_infty
| null |
ultrafilter_le_nhds_infty {f : Ultrafilter (OnePoint X)} :
(f : Filter (OnePoint X)) ≤ 𝓝 ∞ ↔ ∀ s : Set X, IsClosed s → IsCompact s → (↑) '' s ∉ f := by
simp only [le_nhds_infty, ← compl_image_coe, Ultrafilter.mem_coe,
Ultrafilter.compl_mem_iff_notMem]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
ultrafilter_le_nhds_infty
| null |
tendsto_nhds_infty' {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ (↑)) (coclosedCompact X) l := by
simp [nhds_infty_eq, and_comm]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
tendsto_nhds_infty'
| null |
tendsto_nhds_infty {α : Type*} {f : OnePoint X → α} {l : Filter α} :
Tendsto f (𝓝 ∞) l ↔
∀ s ∈ l, f ∞ ∈ s ∧ ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
tendsto_nhds_infty'.trans <| by
simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and,
and_assoc]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
tendsto_nhds_infty
| null |
continuousAt_infty' {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
ContinuousAt f ∞ ↔ Tendsto (f ∘ (↑)) (coclosedCompact X) (𝓝 (f ∞)) :=
tendsto_nhds_infty'.trans <| and_iff_right (tendsto_pure_nhds _ _)
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousAt_infty'
| null |
continuousAt_infty {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} :
ContinuousAt f ∞ ↔
∀ s ∈ 𝓝 (f ∞), ∃ t : Set X, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ (↑)) tᶜ s :=
continuousAt_infty'.trans <| by simp only [hasBasis_coclosedCompact.tendsto_left_iff, and_assoc]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousAt_infty
| null |
continuousAt_coe {Y : Type*} [TopologicalSpace Y] {f : OnePoint X → Y} {x : X} :
ContinuousAt f x ↔ ContinuousAt (f ∘ (↑)) x := by
rw [ContinuousAt, nhds_coe_eq, tendsto_map'_iff, ContinuousAt]; rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousAt_coe
| null |
continuous_iff {Y : Type*} [TopologicalSpace Y] (f : OnePoint X → Y) : Continuous f ↔
Tendsto (fun x : X ↦ f x) (coclosedCompact X) (𝓝 (f ∞)) ∧ Continuous (fun x : X ↦ f x) := by
simp only [continuous_iff_continuousAt, OnePoint.forall, continuousAt_coe, continuousAt_infty',
Function.comp_def]
|
lemma
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuous_iff
| null |
continuousMapMk {Y : Type*} [TopologicalSpace Y] (f : C(X, Y)) (y : Y)
(h : Tendsto f (coclosedCompact X) (𝓝 y)) : C(OnePoint X, Y) where
toFun x := x.elim y f
continuous_toFun := by
rw [continuous_iff]
refine ⟨h, f.continuous⟩
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousMapMk
|
A constructor for continuous maps out of a one point compactification, given a continuous map from
the underlying space and a limit value at infinity.
|
continuous_iff_from_discrete {Y : Type*} [TopologicalSpace Y]
[DiscreteTopology X] (f : OnePoint X → Y) :
Continuous f ↔ Tendsto (fun x : X ↦ f x) cofinite (𝓝 (f ∞)) := by
simp [continuous_iff, cocompact_eq_cofinite, continuous_of_discreteTopology]
|
lemma
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuous_iff_from_discrete
| null |
continuousMapMkDiscrete {Y : Type*} [TopologicalSpace Y]
[DiscreteTopology X] (f : X → Y) (y : Y) (h : Tendsto f cofinite (𝓝 y)) :
C(OnePoint X, Y) :=
continuousMapMk ⟨f, continuous_of_discreteTopology⟩ y (by simpa [cocompact_eq_cofinite])
variable (X) in
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousMapMkDiscrete
|
A constructor for continuous maps out of a one point compactification of a discrete space, given a
map from the underlying space and a limit value at infinity.
|
noncomputable continuousMapDiscreteEquiv (Y : Type*) [DiscreteTopology X] [TopologicalSpace Y]
[T2Space Y] [Infinite X] :
C(OnePoint X, Y) ≃ { f : X → Y // ∃ L, Tendsto (fun x : X ↦ f x) cofinite (𝓝 L) } where
toFun f := ⟨(f ·), ⟨f ∞, continuous_iff_from_discrete _ |>.mp (map_continuous f)⟩⟩
invFun f :=
{ toFun := fun x => match x with
| ∞ => Classical.choose f.2
| some x => f.1 x
continuous_toFun := continuous_iff_from_discrete _ |>.mpr <| Classical.choose_spec f.2 }
left_inv f := by
ext x
refine OnePoint.rec ?_ ?_ x
· refine tendsto_nhds_unique ?_ (continuous_iff_from_discrete _ |>.mp <| map_continuous f)
let f' : { f : X → Y // ∃ L, Tendsto (fun x : X ↦ f x) cofinite (𝓝 L) } :=
⟨fun x ↦ f x, ⟨f ∞, continuous_iff_from_discrete f |>.mp <| map_continuous f⟩⟩
exact Classical.choose_spec f'.property
· simp
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousMapDiscreteEquiv
|
Continuous maps out of the one point compactification of an infinite discrete space to a Hausdorff
space correspond bijectively to "convergent" maps out of the discrete space.
|
continuous_iff_from_nat {Y : Type*} [TopologicalSpace Y] (f : OnePoint ℕ → Y) :
Continuous f ↔ Tendsto (fun x : ℕ ↦ f x) atTop (𝓝 (f ∞)) := by
rw [continuous_iff_from_discrete, Nat.cofinite_eq_atTop]
|
lemma
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuous_iff_from_nat
| null |
continuousMapMkNat {Y : Type*} [TopologicalSpace Y]
(f : ℕ → Y) (y : Y) (h : Tendsto f atTop (𝓝 y)) :
C(OnePoint ℕ, Y) :=
continuousMapMkDiscrete f y (by rwa [Nat.cofinite_eq_atTop])
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousMapMkNat
|
A constructor for continuous maps out of the one point compactification of `ℕ`, given a
sequence and a limit value at infinity.
|
noncomputable continuousMapNatEquiv (Y : Type*) [TopologicalSpace Y] [T2Space Y] :
C(OnePoint ℕ, Y) ≃ { f : ℕ → Y // ∃ L, Tendsto (f ·) atTop (𝓝 L) } := by
refine (continuousMapDiscreteEquiv ℕ Y).trans {
toFun := fun ⟨f, hf⟩ ↦ ⟨f, by rwa [← Nat.cofinite_eq_atTop]⟩
invFun := fun ⟨f, hf⟩ ↦ ⟨f, by rwa [Nat.cofinite_eq_atTop]⟩ }
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuousMapNatEquiv
|
Continuous maps out of the one point compactification of `ℕ` to a Hausdorff space `Y` correspond
bijectively to convergent sequences in `Y`.
|
denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint X) := by
rw [DenseRange, ← compl_infty]
exact dense_compl_singleton _
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
denseRange_coe
|
If `X` is not a compact space, then the natural embedding `X → OnePoint X` has dense range.
|
isDenseEmbedding_coe [NoncompactSpace X] : IsDenseEmbedding ((↑) : X → OnePoint X) :=
{ isOpenEmbedding_coe with dense := denseRange_coe }
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
isDenseEmbedding_coe
| null |
specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y :=
isOpenEmbedding_coe.isInducing.specializes_iff
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
specializes_coe
| null |
inseparable_coe {x y : X} : Inseparable (x : OnePoint X) y ↔ Inseparable x y :=
isOpenEmbedding_coe.isInducing.inseparable_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
inseparable_coe
| null |
not_specializes_infty_coe {x : X} : ¬Specializes ∞ (x : OnePoint X) :=
isClosed_infty.not_specializes rfl (coe_ne_infty x)
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
not_specializes_infty_coe
| null |
not_inseparable_infty_coe {x : X} : ¬Inseparable ∞ (x : OnePoint X) := fun h =>
not_specializes_infty_coe h.specializes
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
not_inseparable_infty_coe
| null |
not_inseparable_coe_infty {x : X} : ¬Inseparable (x : OnePoint X) ∞ := fun h =>
not_specializes_infty_coe h.specializes'
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
not_inseparable_coe_infty
| null |
inseparable_iff {x y : OnePoint X} :
Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x' : X, x = x' ∧ ∃ y' : X, y = y' ∧ Inseparable x' y' := by
induction x using OnePoint.rec <;> induction y using OnePoint.rec <;>
simp [not_inseparable_infty_coe, not_inseparable_coe_infty, coe_eq_coe, Inseparable.refl]
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
inseparable_iff
| null |
continuous_map_iff [TopologicalSpace Y] {f : X → Y} :
Continuous (OnePoint.map f) ↔
Continuous f ∧ Tendsto f (coclosedCompact X) (coclosedCompact Y) := by
simp_rw [continuous_iff, map_some, ← comap_coe_nhds_infty, tendsto_comap_iff, map_infty,
isOpenEmbedding_coe.isInducing.continuous_iff (Y := Y)]
exact and_comm
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuous_map_iff
| null |
continuous_map [TopologicalSpace Y] {f : X → Y} (hc : Continuous f)
(h : Tendsto f (coclosedCompact X) (coclosedCompact Y)) :
Continuous (OnePoint.map f) :=
continuous_map_iff.mpr ⟨hc, h⟩
/-!
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
continuous_map
| null |
not_continuous_cofiniteTopology_of_symm [Infinite X] [DiscreteTopology X] :
¬Continuous (@CofiniteTopology.of (OnePoint X)).symm := by
inhabit X
simp only [continuous_iff_continuousAt, ContinuousAt, not_forall]
use CofiniteTopology.of ↑(default : X)
simpa [nhds_coe_eq, nhds_discrete, CofiniteTopology.nhds_eq] using
(finite_singleton ((default : X) : OnePoint X)).infinite_compl
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
not_continuous_cofiniteTopology_of_symm
|
For any topological space `X`, its one point compactification is a compact space. -/
instance : CompactSpace (OnePoint X) where
isCompact_univ := by
have : Tendsto ((↑) : X → OnePoint X) (cocompact X) (𝓝 ∞) := by
rw [nhds_infty_eq]
exact (tendsto_map.mono_left cocompact_le_coclosedCompact).mono_right le_sup_left
rw [← insert_none_range_some X]
exact this.isCompact_insert_range_of_cocompact continuous_coe
/-- The one point compactification of a `T0Space` space is a `T0Space`. -/
instance [T0Space X] : T0Space (OnePoint X) := by
refine ⟨fun x y hxy => ?_⟩
rcases inseparable_iff.1 hxy with (⟨rfl, rfl⟩ | ⟨x, rfl, y, rfl, h⟩)
exacts [rfl, congr_arg some h.eq]
/-- The one point compactification of a `T1Space` space is a `T1Space`. -/
instance [T1Space X] : T1Space (OnePoint X) where
t1 z := by
induction z using OnePoint.rec
· exact isClosed_infty
· rw [← image_singleton, isClosed_image_coe]
exact ⟨isClosed_singleton, isCompact_singleton⟩
/-- The one point compactification of a weakly locally compact R₁ space
is a normal topological space. -/
instance [WeaklyLocallyCompactSpace X] [R1Space X] : NormalSpace (OnePoint X) := by
suffices R1Space (OnePoint X) by infer_instance
have key : ∀ z : X, Disjoint (𝓝 (some z)) (𝓝 ∞) := fun z ↦ by
rw [nhds_infty_eq, disjoint_sup_right, nhds_coe_eq, coclosedCompact_eq_cocompact,
disjoint_map coe_injective, ← principal_singleton, disjoint_principal_right, compl_infty]
exact ⟨disjoint_nhds_cocompact z, range_mem_map⟩
refine ⟨fun x y ↦ ?_⟩
induction x using OnePoint.rec <;> induction y using OnePoint.rec
· exact .inl le_rfl
· exact .inr (key _).symm
· exact .inr (key _)
· rw [nhds_coe_eq, nhds_coe_eq, disjoint_map coe_injective, specializes_coe]
apply specializes_or_disjoint_nhds
/-- The one point compactification of a weakly locally compact Hausdorff space is a T₄
(hence, Hausdorff and regular) topological space. -/
example [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := inferInstance
/-- If `X` is not a compact space, then `OnePoint X` is a connected space. -/
instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (OnePoint X) where
toPreconnectedSpace := isDenseEmbedding_coe.isDenseInducing.preconnectedSpace
toNonempty := inferInstance
/-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
`CofiniteTopology (OnePoint X)` to `OnePoint X` is not continuous.
|
noncomputable equivOfIsEmbeddingOfRangeEq :
OnePoint X ≃ₜ Y :=
have _i := hf.t2Space
have : Tendsto f (coclosedCompact X) (𝓝 y) := by
rw [coclosedCompact_eq_cocompact, hasBasis_cocompact.tendsto_left_iff]
intro N hN
obtain ⟨U, hU₁, hU₂, hU₃⟩ := mem_nhds_iff.mp hN
refine ⟨f⁻¹' Uᶜ, ?_, by simpa using (mapsTo_preimage f U).mono_right hU₁⟩
rw [hf.isCompact_iff, image_preimage_eq_iff.mpr (by simpa [hy])]
exact (isClosed_compl_iff.mpr hU₂).isCompact
let e : OnePoint X ≃ Y :=
{ toFun := fun p ↦ p.elim y f
invFun := fun q ↦ if hq : q = y then ∞ else ↑(show q ∈ range f from by simpa [hy]).choose
left_inv := fun p ↦ by
induction p using OnePoint.rec with
| infty => simp
| coe p =>
have hp : f p ≠ y := by simpa [hy] using mem_range_self (f := f) p
simpa [hp] using hf.injective (mem_range_self p).choose_spec
right_inv := fun q ↦ by
rcases eq_or_ne q y with rfl | hq
· simp
· have hq' : q ∈ range f := by simpa [hy]
simpa [hq] using hq'.choose_spec }
Continuous.homeoOfEquivCompactToT2 <| (continuous_iff e).mpr ⟨this, hf.continuous⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
equivOfIsEmbeddingOfRangeEq
|
If `f` embeds `X` into a compact Hausdorff space `Y`, and has exactly one point outside its
range, then `(Y, f)` is the one-point compactification of `X`.
|
equivOfIsEmbeddingOfRangeEq_apply_coe (x : X) :
equivOfIsEmbeddingOfRangeEq y f hf hy x = f x :=
rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
equivOfIsEmbeddingOfRangeEq_apply_coe
| null |
equivOfIsEmbeddingOfRangeEq_apply_infty :
equivOfIsEmbeddingOfRangeEq y f hf hy ∞ = y :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
equivOfIsEmbeddingOfRangeEq_apply_infty
| null |
@[simps]
onePointCongr (h : X ≃ₜ Y) : OnePoint X ≃ₜ OnePoint Y where
__ := h.toEquiv.withTopCongr
toFun := OnePoint.map h
invFun := OnePoint.map h.symm
continuous_toFun := continuous_map (map_continuous h) h.map_coclosedCompact.le
continuous_invFun := continuous_map (map_continuous h.symm) h.symm.map_coclosedCompact.le
|
def
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
onePointCongr
|
Extend a homeomorphism of topological spaces
to the homeomorphism of their one point compactifications.
|
Continuous.homeoOfEquivCompactToT2.t1_counterexample :
∃ (α β : Type) (_ : TopologicalSpace α) (_ : TopologicalSpace β),
CompactSpace α ∧ T1Space β ∧ ∃ f : α ≃ β, Continuous f ∧ ¬Continuous f.symm :=
⟨OnePoint ℕ, CofiniteTopology (OnePoint ℕ), inferInstance, inferInstance, inferInstance,
inferInstance, CofiniteTopology.of, CofiniteTopology.continuous_of,
OnePoint.not_continuous_cofiniteTopology_of_symm⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.Fintype.Option",
"Mathlib.Topology.Homeomorph.Lemmas",
"Mathlib.Topology.Sets.Opens"
] |
Mathlib/Topology/Compactification/OnePoint/Basic.lean
|
Continuous.homeoOfEquivCompactToT2.t1_counterexample
|
A concrete counterexample shows that `Continuous.homeoOfEquivCompactToT2`
cannot be generalized from `T2Space` to `T1Space`.
Let `α = OnePoint ℕ` be the one-point compactification of `ℕ`, and let `β` be the same space
`OnePoint ℕ` with the cofinite topology. Then `α` is compact, `β` is T1, and the identity map
`id : α → β` is a continuous equivalence that is not a homeomorphism.
|
@[simp] Matrix.fin_two_smul_prod (g : Matrix (Fin 2) (Fin 2) R) (v : R × R) :
g • v = (g 0 0 * v.1 + g 0 1 * v.2, g 1 0 * v.1 + g 1 1 * v.2) := by
simp [Equiv.smul_def, smul_eq_mulVec, Matrix.mulVec_eq_sum]
@[simp] lemma Matrix.GeneralLinearGroup.fin_two_smul_prod {R : Type*} [CommRing R]
(g : GL (Fin 2) R) (v : R × R) :
g • v = (g 0 0 * v.1 + g 0 1 * v.2, g 1 0 * v.1 + g 1 1 * v.2) := by
simp [Units.smul_def]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
Matrix.fin_two_smul_prod
| null |
equivProjectivization :
OnePoint K ≃ ℙ K (K × K) where
toFun p := p.elim (mk K (1, 0) (by simp)) (fun t ↦ mk K (t, 1) (by simp))
invFun p := by
refine Projectivization.lift
(fun u : {v : K × K // v ≠ 0} ↦ if u.1.2 = 0 then ∞ else ((u.1.2)⁻¹ * u.1.1)) ?_ p
rintro ⟨-, hv⟩ ⟨⟨x, y⟩, hw⟩ t rfl
have ht : t ≠ 0 := by rintro rfl; simp at hv
by_cases h₀ : y = 0 <;> simp [h₀, ht, mul_assoc]
left_inv p := by cases p <;> simp
right_inv p := by
induction p using ind with | h p hp =>
obtain ⟨x, y⟩ := p
by_cases h₀ : y = 0 <;> simp only [mk_eq_mk_iff', h₀, Projectivization.lift_mk, if_true,
if_false, OnePoint.elim_infty, OnePoint.elim_some, Prod.smul_mk, Prod.mk.injEq, smul_eq_mul,
mul_zero, and_true]
· use x⁻¹
simp_all
· exact ⟨y⁻¹, rfl, inv_mul_cancel₀ h₀⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
equivProjectivization
|
The one-point compactification of a division ring `K` is equivalent to
the projectivization `ℙ K (K × K)`.
|
equivProjectivization_apply_infinity :
equivProjectivization K ∞ = mk K ⟨1, 0⟩ (by simp) :=
rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
equivProjectivization_apply_infinity
| null |
equivProjectivization_apply_coe (t : K) :
equivProjectivization K t = mk K ⟨t, 1⟩ (by simp) :=
rfl
@[simp]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
equivProjectivization_apply_coe
| null |
equivProjectivization_symm_apply_mk (x y : K) (h : (x, y) ≠ 0) :
(equivProjectivization K).symm (mk K ⟨x, y⟩ h) = if y = 0 then ∞ else y⁻¹ * x := by
simp [equivProjectivization]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
equivProjectivization_symm_apply_mk
| null |
instGLAction : MulAction (GL (Fin 2) K) (OnePoint K) :=
(equivProjectivization K).mulAction (GL (Fin 2) K)
|
instance
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
instGLAction
|
For a field `K`, the group `GL(2, K)` acts on `OnePoint K`, via the canonical identification
with the `ℙ¹(K)` (which is given explicitly by Möbius transformations).
|
smul_infty_def {g : GL (Fin 2) K} :
g • ∞ = (equivProjectivization K).symm (.mk K (g 0 0, g 1 0) (fun h ↦ by
simpa [det_fin_two, Prod.mk_eq_zero.mp h] using g.det_ne_zero)) := by
simp [Equiv.smul_def, mulVec_eq_sum, Units.smul_def]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
smul_infty_def
| null |
smul_infty_eq_ite (g : GL (Fin 2) K) :
g • (∞ : OnePoint K) = if g 1 0 = 0 then ∞ else g 0 0 / g 1 0 := by
by_cases h : g 1 0 = 0 <;>
simp [h, div_eq_inv_mul, smul_infty_def]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
smul_infty_eq_ite
| null |
smul_infty_eq_self_iff {g : GL (Fin 2) K} :
g • (∞ : OnePoint K) = ∞ ↔ g 1 0 = 0 := by
simp [smul_infty_eq_ite]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
smul_infty_eq_self_iff
| null |
smul_some_eq_ite {g : GL (Fin 2) K} {k : K} :
g • (k : OnePoint K) =
if g 1 0 * k + g 1 1 = 0 then ∞ else (g 0 0 * k + g 0 1) / (g 1 0 * k + g 1 1) := by
simp [Equiv.smul_def, mulVec_eq_sum, div_eq_inv_mul, mul_comm, Units.smul_def]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
smul_some_eq_ite
| null |
map_smul {L : Type*} [Field L] [DecidableEq L]
(f : K →+* L) (g : GL (Fin 2) K) (c : OnePoint K) :
OnePoint.map f (g • c) = (g.map f) • (c.map f) := by
cases c with
| infty => simp [smul_infty_eq_ite, apply_ite]
| coe c => simp [smul_some_eq_ite, ← map_mul, ← map_add, apply_ite]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
map_smul
| null |
fixpointPolynomial_aeval_eq_zero_iff {c : K} {g : GL (Fin 2) K} :
g.fixpointPolynomial.aeval c = 0 ↔ g • (c : OnePoint K) = c := by
simp only [fixpointPolynomial, map_sub, map_mul, map_add, aeval_X_pow, aeval_C, aeval_X,
Algebra.algebraMap_self_apply, OnePoint.smul_some_eq_ite]
split_ifs with h
· refine ⟨fun hg ↦ (g.det_ne_zero ?_).elim, fun hg ↦ (infty_ne_coe _ hg).elim⟩
rw [det_fin_two]
grind
· rw [coe_eq_coe, div_eq_iff h]
grind
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
fixpointPolynomial_aeval_eq_zero_iff
|
The roots of `g.fixpointPolynomial` are the fixed points of `g ∈ GL(2, K)` acting on the finite
part of `OnePoint K`.
|
parabolicFixedPoint (g : GL (Fin 2) K) : OnePoint K :=
if g 1 0 = 0 then ∞ else ↑((g 0 0 - g 1 1) / (2 * g 1 0))
|
def
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
parabolicFixedPoint
|
If `g` is parabolic, this is the unique fixed point of `g` in `OnePoint K`.
|
IsParabolic.smul_eq_self_iff {g : GL (Fin 2) K} (hg : g.IsParabolic) [NeZero (2 : K)]
{c : OnePoint K} : g • c = c ↔ c = parabolicFixedPoint g := by
rcases hg with ⟨hg, hdisc⟩
rw [disc_fin_two, trace_fin_two, det_fin_two] at hdisc
cases c with
| infty => by_cases h : g 1 0 = 0 <;> simp [parabolicFixedPoint, smul_infty_eq_ite, h]
| coe c =>
suffices g 1 0 * c ^ 2 + (g 1 1 - g 0 0) * c - g 0 1 = 0 ↔ c = g.parabolicFixedPoint by
simpa [← fixpointPolynomial_aeval_eq_zero_iff, fixpointPolynomial]
by_cases hc : g 1 0 = 0
· have hd : g 1 1 = g 0 0 := by grind
suffices g 0 1 ≠ 0 by simpa [parabolicFixedPoint, hc, hd]
refine fun hb ↦ fixpointPolynomial_eq_zero_iff.not.mpr hg ?_
simp [fixpointPolynomial, hb, hc, hd]
· have : discrim (g 1 0) (g 1 1 - g 0 0) (-g 0 1) = 0 := by rw [discrim]; grind
simpa [parabolicFixedPoint, if_neg hc, sq, sub_eq_add_neg]
using quadratic_eq_zero_iff_of_discrim_eq_zero hc this c
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
IsParabolic.smul_eq_self_iff
| null |
IsParabolic.parabolicFixedPoint_pow {g : GL (Fin 2) K} (hg : IsParabolic g) [CharZero K]
{n : ℕ} (hn : n ≠ 0) :
(g ^ n).parabolicFixedPoint = g.parabolicFixedPoint := by
rw [eq_comm, ← IsParabolic.smul_eq_self_iff (hg.pow hn)]
clear hn
induction n with
| zero => simp
| succ n IH => rw [pow_succ, MulAction.mul_smul, hg.smul_eq_self_iff.mpr rfl, IH]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
IsParabolic.parabolicFixedPoint_pow
| null |
IsElliptic.smul_ne_self [LinearOrder K] [IsStrictOrderedRing K]
{g : GL (Fin 2) K} (hg : g.IsElliptic) (c : OnePoint K) :
g • c ≠ c := by
cases c with
| infty =>
rw [Ne, smul_infty_eq_self_iff]
refine fun h ↦ not_le_of_gt hg ?_
have : g.val.disc = (g 0 0 - g 1 1) ^ 2 := by
simp only [disc_fin_two, trace_fin_two, det_fin_two]
grind
rw [this]
apply sq_nonneg
| coe c =>
refine fun h ↦ not_le_of_gt hg ?_
have : g.val.disc = (2 * g 1 0 * c + (g 1 1 + -g 0 0)) ^ 2 := by
replace h : g 1 0 * (c * c) + (g 1 1 + -g 0 0) * c + -g 0 1 = 0 := by
simpa [← fixpointPolynomial_aeval_eq_zero_iff, fixpointPolynomial, sq, sub_eq_add_neg]
using h
simp only [← discrim_eq_sq_of_quadratic_eq_zero h, disc_fin_two, discrim, trace_fin_two,
det_fin_two]
grind
rw [this]
apply sq_nonneg
|
lemma
|
Topology
|
[
"Mathlib.Algebra.QuadraticDiscriminant",
"Mathlib.Data.Matrix.Action",
"Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo",
"Mathlib.LinearAlgebra.Projectivization.Action",
"Mathlib.Topology.Compactification.OnePoint.Basic"
] |
Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean
|
IsElliptic.smul_ne_self
|
Elliptic elements have no fixed points in `OnePoint K`.
|
onePointHyperplaneHomeoUnitSphere
{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
{v : E} (hv : ‖v‖ = 1) :
OnePoint (ℝ ∙ v)ᗮ ≃ₜ sphere (0 : E) 1 :=
OnePoint.equivOfIsEmbeddingOfRangeEq _ _
(isOpenEmbedding_stereographic_symm hv).toIsEmbedding (range_stereographic_symm hv)
|
def
|
Topology
|
[
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Geometry.Manifold.Instances.Sphere"
] |
Mathlib/Topology/Compactification/OnePoint/Sphere.lean
|
onePointHyperplaneHomeoUnitSphere
|
A homeomorphism from the one-point compactification of a hyperplane in Euclidean space to the
sphere.
|
onePointEquivSphereOfFinrankEq {ι V : Type*} [Fintype ι]
[AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V]
[TopologicalSpace V] [IsTopologicalAddGroup V] [ContinuousSMul ℝ V] [T2Space V]
(h : finrank ℝ V + 1 = Fintype.card ι) :
OnePoint V ≃ₜ sphere (0 : EuclideanSpace ℝ ι) 1 := by
classical
have : Nonempty ι := Fintype.card_pos_iff.mp <| by cutsat
let v : EuclideanSpace ℝ ι := .single (Classical.arbitrary ι) 1
have hv : ‖v‖ = 1 := by simp [v]
have hv₀ : v ≠ 0 := fun contra ↦ by simp [contra] at hv
have : Fact (finrank ℝ (EuclideanSpace ℝ ι) = finrank ℝ V + 1) := ⟨by simp [h]⟩
have hV : finrank ℝ V = finrank ℝ (ℝ ∙ v)ᗮ := (finrank_orthogonal_span_singleton hv₀).symm
letI e : V ≃ₜ (ℝ ∙ v)ᗮ := (FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq hV).some
exact e.onePointCongr.trans <| onePointHyperplaneHomeoUnitSphere hv
|
def
|
Topology
|
[
"Mathlib.Topology.Compactification.OnePoint.Basic",
"Mathlib.Geometry.Manifold.Instances.Sphere"
] |
Mathlib/Topology/Compactification/OnePoint/Sphere.lean
|
onePointEquivSphereOfFinrankEq
|
A homeomorphism from the one-point compactification of a finite-dimensional real vector space to
the sphere.
|
arzela_ascoli₁ [CompactSpace β] (A : Set (α →ᵇ β)) (closed : IsClosed A)
(H : Equicontinuous ((↑) : A → α → β)) : IsCompact A := by
simp_rw [Equicontinuous, Metric.equicontinuousAt_iff_pair] at H
refine TotallyBounded.isCompact_of_isClosed ?_ closed
refine totallyBounded_of_finite_discretization fun ε ε0 => ?_
rcases exists_between ε0 with ⟨ε₁, ε₁0, εε₁⟩
let ε₂ := ε₁ / 2 / 2
/- We have to find a finite discretization of `u`, i.e., finite information
that is sufficient to reconstruct `u` up to `ε`. This information will be
provided by the values of `u` on a sufficiently dense set `tα`,
slightly translated to fit in a finite `ε₂`-dense set `tβ` in the image. Such
sets exist by compactness of the source and range. Then, to check that these
data determine the function up to `ε`, one uses the control on the modulus of
continuity to extend the closeness on tα to closeness everywhere. -/
have ε₂0 : ε₂ > 0 := half_pos (half_pos ε₁0)
have : ∀ x : α, ∃ U, x ∈ U ∧ IsOpen U ∧
∀ y ∈ U, ∀ z ∈ U, ∀ {f : α →ᵇ β}, f ∈ A → dist (f y) (f z) < ε₂ := fun x =>
let ⟨U, nhdsU, hU⟩ := H x _ ε₂0
let ⟨V, VU, openV, xV⟩ := _root_.mem_nhds_iff.1 nhdsU
⟨V, xV, openV, fun y hy z hz f hf => hU y (VU hy) z (VU hz) ⟨f, hf⟩⟩
choose U hU using this
/- For all `x`, the set `hU x` is an open set containing `x` on which the elements of `A`
fluctuate by at most `ε₂`.
We extract finitely many of these sets that cover the whole space, by compactness. -/
obtain ⟨tα : Set α, _, hfin, htα : univ ⊆ ⋃ x ∈ tα, U x⟩ :=
isCompact_univ.elim_finite_subcover_image (fun x _ => (hU x).2.1) fun x _ =>
mem_biUnion (mem_univ _) (hU x).1
rcases hfin.nonempty_fintype with ⟨_⟩
obtain ⟨tβ : Set β, _, hfin, htβ : univ ⊆ ⋃y ∈ tβ, ball y ε₂⟩ :=
@finite_cover_balls_of_compact β _ _ isCompact_univ _ ε₂0
rcases hfin.nonempty_fintype with ⟨_⟩
choose F hF using fun y => show ∃ z ∈ tβ, dist y z < ε₂ by simpa using htβ (mem_univ y)
/- Associate to every function a discrete approximation, mapping each point in `tα`
to a point in `tβ` close to its true image by the function. -/
classical
refine ⟨tα → tβ, by infer_instance, fun f a => ⟨F (f.1 a), (hF (f.1 a)).1⟩, ?_⟩
rintro ⟨f, hf⟩ ⟨g, hg⟩ f_eq_g
refine lt_of_le_of_lt ((dist_le <| le_of_lt ε₁0).2 fun x => ?_) εε₁
obtain ⟨x', x'tα, hx'⟩ := mem_iUnion₂.1 (htα (mem_univ x))
calc
dist (f x) (g x) ≤ dist (f x) (f x') + dist (g x) (g x') + dist (f x') (g x') :=
dist_triangle4_right _ _ _ _
_ ≤ ε₂ + ε₂ + ε₁ / 2 := by
refine le_of_lt (add_lt_add (add_lt_add ?_ ?_) ?_)
· exact (hU x').2.2 _ hx' _ (hU x').1 hf
· exact (hU x').2.2 _ hx' _ (hU x').1 hg
· have F_f_g : F (f x') = F (g x') :=
(congr_arg (fun f : tα → tβ => (f ⟨x', x'tα⟩ : β)) f_eq_g :)
calc
dist (f x') (g x') ≤ dist (f x') (F (f x')) + dist (g x') (F (f x')) :=
dist_triangle_right _ _ _
...
|
theorem
|
Topology
|
[
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.MetricSpace.Equicontinuity"
] |
Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.lean
|
arzela_ascoli₁
|
First version, with pointwise equicontinuity and range in a compact space.
|
arzela_ascoli₂ (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β)) (closed : IsClosed A)
(in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) :
IsCompact A := by
/- This version is deduced from the previous one by restricting to the compact type in the target,
using compactness there and then lifting everything to the original space. -/
have M : LipschitzWith 1 Subtype.val := LipschitzWith.subtype_val s
let F : (α →ᵇ s) → α →ᵇ β := comp (↑) M
refine IsCompact.of_isClosed_subset ((?_ : IsCompact (F ⁻¹' A)).image (continuous_comp M)) closed
fun f hf => ?_
· haveI : CompactSpace s := isCompact_iff_compactSpace.1 hs
refine arzela_ascoli₁ _ (continuous_iff_isClosed.1 (continuous_comp M) _ closed) ?_
rw [isUniformEmbedding_subtype_val.isUniformInducing.equicontinuous_iff]
exact H.comp (A.restrictPreimage F)
· let g := codRestrict s f fun x => in_s f x hf
rw [show f = F g by ext; rfl] at hf ⊢
exact ⟨g, hf, rfl⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.MetricSpace.Equicontinuity"
] |
Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.lean
|
arzela_ascoli₂
|
Second version, with pointwise equicontinuity and range in a compact subset.
|
arzela_ascoli [T2Space β] (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β))
(in_s : ∀ (f : α →ᵇ β) (x : α), f ∈ A → f x ∈ s) (H : Equicontinuous ((↑) : A → α → β)) :
IsCompact (closure A) :=
/- This version is deduced from the previous one by checking that the closure of `A`, in
addition to being closed, still satisfies the properties of compact range and equicontinuity. -/
arzela_ascoli₂ s hs (closure A) isClosed_closure
(fun _ x hf =>
(mem_of_closed' hs.isClosed).2 fun ε ε0 =>
let ⟨g, gA, dist_fg⟩ := Metric.mem_closure_iff.1 hf ε ε0
⟨g x, in_s g x gA, lt_of_le_of_lt (dist_coe_le_dist _) dist_fg⟩)
(H.closure' continuous_coe)
|
theorem
|
Topology
|
[
"Mathlib.Topology.ContinuousMap.Bounded.Basic",
"Mathlib.Topology.MetricSpace.Equicontinuity"
] |
Mathlib/Topology/ContinuousMap/Bounded/ArzelaAscoli.lean
|
arzela_ascoli
|
Third (main) version, with pointwise equicontinuity and range in a compact subset, but
without closedness. The closure is then compact.
|
BoundedContinuousFunction (α : Type u) (β : Type v) [TopologicalSpace α]
[PseudoMetricSpace β] : Type max u v extends ContinuousMap α β where
map_bounded' : ∃ C, ∀ x y, dist (toFun x) (toFun y) ≤ C
@[inherit_doc] scoped[BoundedContinuousFunction] infixr:25 " →ᵇ " => BoundedContinuousFunction
|
structure
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
BoundedContinuousFunction
|
`α →ᵇ β` is the type of bounded continuous functions `α → β` from a topological space to a
metric space.
When possible, instead of parametrizing results over `(f : α →ᵇ β)`,
you should parametrize over `(F : Type*) [BoundedContinuousMapClass F α β] (f : F)`.
When you extend this structure, make sure to extend `BoundedContinuousMapClass`.
|
BoundedContinuousMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
[PseudoMetricSpace β] [FunLike F α β] : Prop extends ContinuousMapClass F α β where
map_bounded (f : F) : ∃ C, ∀ x y, dist (f x) (f y) ≤ C
|
class
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
BoundedContinuousMapClass
|
`BoundedContinuousMapClass F α β` states that `F` is a type of bounded continuous maps.
You should also extend this typeclass when you extend `BoundedContinuousFunction`.
|
instFunLike : FunLike (α →ᵇ β) α β where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instFunLike
| null |
instBoundedContinuousMapClass : BoundedContinuousMapClass (α →ᵇ β) α β where
map_continuous f := f.continuous_toFun
map_bounded f := f.map_bounded'
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instBoundedContinuousMapClass
| null |
instCoeTC [FunLike F α β] [BoundedContinuousMapClass F α β] : CoeTC F (α →ᵇ β) :=
⟨fun f =>
{ toFun := f
continuous_toFun := map_continuous f
map_bounded' := map_bounded f }⟩
@[simp]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instCoeTC
| null |
coe_toContinuousMap (f : α →ᵇ β) : (f.toContinuousMap : α → β) = f := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
coe_toContinuousMap
| null |
Simps.apply (h : α →ᵇ β) : α → β := h
initialize_simps_projections BoundedContinuousFunction (toFun → apply)
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
Simps.apply
|
See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections.
|
protected bounded (f : α →ᵇ β) : ∃ C, ∀ x y : α, dist (f x) (f y) ≤ C :=
f.map_bounded'
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
bounded
| null |
protected continuous (f : α →ᵇ β) : Continuous f :=
f.toContinuousMap.continuous
@[ext]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
continuous
| null |
ext (h : ∀ x, f x = g x) : f = g :=
DFunLike.ext _ _ h
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
ext
| null |
isBounded_range (f : α →ᵇ β) : IsBounded (range f) :=
isBounded_range_iff.2 f.bounded
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
isBounded_range
| null |
isBounded_image (f : α →ᵇ β) (s : Set α) : IsBounded (f '' s) :=
f.isBounded_range.subset <| image_subset_range _ _
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
isBounded_image
| null |
eq_of_empty [h : IsEmpty α] (f g : α →ᵇ β) : f = g :=
ext <| h.elim
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
eq_of_empty
| null |
mkOfBound (f : C(α, β)) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) : α →ᵇ β :=
⟨f, ⟨C, h⟩⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfBound
|
A continuous function with an explicit bound is a bounded continuous function.
|
mkOfBound_coe {f} {C} {h} : (mkOfBound f C h : α → β) = (f : α → β) := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfBound_coe
| null |
mkOfCompact [CompactSpace α] (f : C(α, β)) : α →ᵇ β :=
⟨f, isBounded_range_iff.1 (isCompact_range f.continuous).isBounded⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfCompact
|
A continuous function on a compact space is automatically a bounded continuous function.
|
mkOfCompact_apply [CompactSpace α] (f : C(α, β)) (a : α) : mkOfCompact f a = f a := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfCompact_apply
| null |
@[simps]
mkOfDiscrete [DiscreteTopology α] (f : α → β) (C : ℝ) (h : ∀ x y : α, dist (f x) (f y) ≤ C) :
α →ᵇ β :=
⟨⟨f, continuous_of_discreteTopology⟩, ⟨C, h⟩⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
mkOfDiscrete
|
If a function is bounded on a discrete space, it is automatically continuous,
and therefore gives rise to an element of the type of bounded continuous functions.
|
instDist : Dist (α →ᵇ β) :=
⟨fun f g => sInf { C | 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C }⟩
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instDist
|
The uniform distance between two bounded continuous functions.
|
dist_eq : dist f g = sInf { C | 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C } := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_eq
| null |
dist_set_exists : ∃ C, 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C := by
rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_set_exists
| null |
dist_coe_le_dist (x : α) : dist (f x) (g x) ≤ dist f g :=
le_csInf dist_set_exists fun _ hb => hb.2 x
/- This lemma will be needed in the proof of the metric space instance, but it will become
useless afterwards as it will be superseded by the general result that the distance is nonnegative
in metric spaces. -/
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_coe_le_dist
|
The pointwise distance is controlled by the distance between functions, by definition.
|
private dist_nonneg' : 0 ≤ dist f g :=
le_csInf dist_set_exists fun _ => And.left
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_nonneg'
| null |
dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C :=
⟨fun h x => le_trans (dist_coe_le_dist x) h, fun H => csInf_le ⟨0, fun _ => And.left⟩ ⟨C0, H⟩⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_le
|
The distance between two functions is controlled by the supremum of the pointwise distances.
|
dist_le_iff_of_nonempty [Nonempty α] : dist f g ≤ C ↔ ∀ x, dist (f x) (g x) ≤ C :=
⟨fun h x => le_trans (dist_coe_le_dist x) h,
fun w => (dist_le (le_trans dist_nonneg (w (Nonempty.some ‹_›)))).mpr w⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_le_iff_of_nonempty
| null |
dist_lt_of_nonempty_compact [Nonempty α] [CompactSpace α]
(w : ∀ x : α, dist (f x) (g x) < C) : dist f g < C := by
have c : Continuous fun x => dist (f x) (g x) := by fun_prop
obtain ⟨x, -, le⟩ :=
IsCompact.exists_isMaxOn isCompact_univ Set.univ_nonempty (Continuous.continuousOn c)
exact lt_of_le_of_lt (dist_le_iff_of_nonempty.mpr fun y => le trivial) (w x)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_lt_of_nonempty_compact
| null |
dist_lt_iff_of_compact [CompactSpace α] (C0 : (0 : ℝ) < C) :
dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by
fconstructor
· intro w x
exact lt_of_le_of_lt (dist_coe_le_dist x) w
· by_cases h : Nonempty α
· exact dist_lt_of_nonempty_compact
· rintro -
convert C0
apply le_antisymm _ dist_nonneg'
rw [dist_eq]
exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_lt_iff_of_compact
| null |
dist_lt_iff_of_nonempty_compact [Nonempty α] [CompactSpace α] :
dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C :=
⟨fun w x => lt_of_le_of_lt (dist_coe_le_dist x) w, dist_lt_of_nonempty_compact⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_lt_iff_of_nonempty_compact
| null |
instPseudoMetricSpace : PseudoMetricSpace (α →ᵇ β) where
dist_self f := le_antisymm ((dist_le le_rfl).2 fun x => by simp) dist_nonneg'
dist_comm f g := by simp [dist_eq, dist_comm]
dist_triangle _ _ _ := (dist_le (add_nonneg dist_nonneg' dist_nonneg')).2
fun _ => le_trans (dist_triangle _ _ _) (add_le_add (dist_coe_le_dist _) (dist_coe_le_dist _))
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instPseudoMetricSpace
|
The type of bounded continuous functions, with the uniform distance, is a pseudometric space.
|
instMetricSpace {β} [MetricSpace β] : MetricSpace (α →ᵇ β) where
eq_of_dist_eq_zero hfg := by
ext x
exact eq_of_dist_eq_zero (le_antisymm (hfg ▸ dist_coe_le_dist _) dist_nonneg)
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
instMetricSpace
|
The type of bounded continuous functions, with the uniform distance, is a metric space.
|
nndist_eq : nndist f g = sInf { C | ∀ x : α, nndist (f x) (g x) ≤ C } :=
Subtype.ext <| dist_eq.trans <| by
rw [val_eq_coe, coe_sInf, coe_image]
simp_rw [mem_setOf_eq, ← NNReal.coe_le_coe, coe_mk, exists_prop, coe_nndist]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
nndist_eq
| null |
nndist_set_exists : ∃ C, ∀ x : α, nndist (f x) (g x) ≤ C :=
Subtype.exists.mpr <| dist_set_exists.imp fun _ ⟨ha, h⟩ => ⟨ha, h⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
nndist_set_exists
| null |
nndist_coe_le_nndist (x : α) : nndist (f x) (g x) ≤ nndist f g :=
dist_coe_le_dist x
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
nndist_coe_le_nndist
| null |
dist_zero_of_empty [IsEmpty α] : dist f g = 0 := by
rw [(ext isEmptyElim : f = g), dist_self]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_zero_of_empty
|
On an empty space, bounded continuous functions are at distance 0.
|
dist_eq_iSup : dist f g = ⨆ x : α, dist (f x) (g x) := by
cases isEmpty_or_nonempty α
· rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty]
refine (dist_le_iff_of_nonempty.mpr <| le_ciSup ?_).antisymm (ciSup_le dist_coe_le_dist)
exact dist_set_exists.imp fun C hC => forall_mem_range.2 hC.2
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
dist_eq_iSup
| null |
nndist_eq_iSup : nndist f g = ⨆ x : α, nndist (f x) (g x) :=
Subtype.ext <| dist_eq_iSup.trans <| by simp_rw [val_eq_coe, coe_iSup, coe_nndist]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
nndist_eq_iSup
| null |
edist_eq_iSup : edist f g = ⨆ x, edist (f x) (g x) := by
simp_rw [edist_nndist, nndist_eq_iSup]
refine ENNReal.coe_iSup ⟨nndist f g, ?_⟩
rintro - ⟨x, hx, rfl⟩
exact nndist_coe_le_nndist x
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
edist_eq_iSup
| null |
tendsto_iff_tendstoUniformly {ι : Type*} {F : ι → α →ᵇ β} {f : α →ᵇ β} {l : Filter ι} :
Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i => F i) f l :=
Iff.intro
(fun h =>
tendstoUniformly_iff.2 fun ε ε0 =>
(Metric.tendsto_nhds.mp h ε ε0).mp
(Eventually.of_forall fun n hn x =>
lt_of_le_of_lt (dist_coe_le_dist x) (dist_comm (F n) f ▸ hn)))
fun h =>
Metric.tendsto_nhds.mpr fun _ ε_pos =>
(h _ (dist_mem_uniformity <| half_pos ε_pos)).mp
(Eventually.of_forall fun n hn =>
lt_of_le_of_lt
((dist_le (half_pos ε_pos).le).mpr fun x => dist_comm (f x) (F n x) ▸ le_of_lt (hn x))
(half_lt_self ε_pos))
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
tendsto_iff_tendstoUniformly
| null |
isInducing_coeFn : IsInducing (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) := by
rw [isInducing_iff_nhds]
refine fun f => eq_of_forall_le_iff fun l => ?_
rw [← tendsto_iff_comap, ← tendsto_id', tendsto_iff_tendstoUniformly,
UniformFun.tendsto_iff_tendstoUniformly]
simp [comp_def]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
isInducing_coeFn
|
The topology on `α →ᵇ β` is exactly the topology induced by the natural map to `α →ᵤ β`.
|
isEmbedding_coeFn : IsEmbedding (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) :=
⟨isInducing_coeFn, fun _ _ h => ext fun x => congr_fun h x⟩
variable (α) in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
isEmbedding_coeFn
| null |
@[simps! -fullyApplied]
const (b : β) : α →ᵇ β :=
⟨ContinuousMap.const α b, 0, by simp⟩
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
const
|
Constant as a continuous bounded function.
|
const_apply' (a : α) (b : β) : (const α b : α → β) a = b := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
const_apply'
| null |
@[continuity]
continuous_eval_const {x : α} : Continuous fun f : α →ᵇ β => f x :=
(continuous_apply x).comp continuous_coe
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
continuous_eval_const
|
If the target space is inhabited, so is the space of bounded continuous functions. -/
instance [Inhabited β] : Inhabited (α →ᵇ β) :=
⟨const α default⟩
theorem lipschitz_evalx (x : α) : LipschitzWith 1 fun f : α →ᵇ β => f x :=
LipschitzWith.mk_one fun _ _ => dist_coe_le_dist x
theorem uniformContinuous_coe : @UniformContinuous (α →ᵇ β) (α → β) _ _ (⇑) :=
uniformContinuous_pi.2 fun x => (lipschitz_evalx x).uniformContinuous
theorem continuous_coe : Continuous fun (f : α →ᵇ β) x => f x :=
UniformContinuous.continuous uniformContinuous_coe
/-- When `x` is fixed, `(f : α →ᵇ β) ↦ f x` is continuous.
|
@[continuity]
continuous_eval : Continuous fun p : (α →ᵇ β) × α => p.1 p.2 :=
(continuous_prod_of_continuous_lipschitzWith _ 1 fun f => f.continuous) <| lipschitz_evalx
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Indicator",
"Mathlib.Topology.Bornology.BoundedOperation",
"Mathlib.Topology.ContinuousMap.Algebra"
] |
Mathlib/Topology/ContinuousMap/Bounded/Basic.lean
|
continuous_eval
|
The evaluation map is continuous, as a joint function of `u` and `x`.
|
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