mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : Nonempty α
⊢ PreconnectedSpace α

Tactic:
·

Goal state after:
case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : Nonempty α
⊢ PreconnectedSpace α
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mp.refine_2
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
Y : Type u
f' : Y → ι
h₁ : IsLindelof (⋃ i, (b ∘ f') i)
h₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Set Y
ht : t.Countable ∧ ⋃ i, (b ∘ f') i ⊆ ⋃ i ∈ t, (b ∘ f') i
⊢ ⋃ i ∈ f' '' t, b i ⊆ ⋃ i, (b ∘ f') i

Tactic:
·

Goal state after:
case mp.refine_2
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
Y : Type u
f' : Y → ι
h₁ : IsLindelof (⋃ i, (b ∘ f') i)
h₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Set Y
ht : t.Countable ∧ ⋃ i, (b ∘ f') i ⊆ ⋃ i ∈ t, (b ∘ f') i
⊢ ⋃ i ∈ f' '' t, b i ⊆ ⋃ i, (b ∘ f') i
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mp
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ IsLindelof U ∧ IsOpen[inst✝] U → ∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i

pp: case mpr
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ (∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i) → IsLindelof U ∧ IsOpen[inst✝] U

Tactic:
· rintro ⟨h₁, h₂⟩
    obtain ⟨Y, f, rfl, hf⟩ := hb.open_eq_iUnion h₂
    choose f' hf' using hf
    have : b ∘ f' = f := funext hf'
    subst this
    obtain ⟨t, ht⟩ :=
      h₁.elim_countable_subcover (b ∘ f') (fun i => hb.isOpen (Set.mem_range_self _)) Subset.rfl
    refine ⟨t.image f', Countable.image (ht.1) f', le_antisymm ?_ ?_⟩
    · refine Set.Subset.trans ht.2 ?_
      simp only [Set.iUnion_subset_iff]
      intro i hi
      rw [← Set.iUnion_subtype (fun x : ι => x ∈ t.image f') fun i => b i.1]
      exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, mem_image_of_mem _ hi⟩
    · apply Set.iUnion₂_subset
      rintro i hi
      obtain ⟨j, -, rfl⟩ := (mem_image ..).mp hi
      exact Set.subset_iUnion (b ∘ f') j

Goal state after:
case mpr
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ (∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i) → IsLindelof U ∧ IsOpen[inst✝] U
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mpr
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ (∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i) → IsLindelof U ∧ IsOpen[inst✝] U

Tactic:
·

Goal state after:
case mpr
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
U : Set X
⊢ (∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i) → IsLindelof U ∧ IsOpen[inst✝] U
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mpr.left
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
s : Set ι
hs : s.Countable
⊢ IsLindelof (⋃ i ∈ s, b i)

pp: case mpr.right
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
s : Set ι
hs : s.Countable
⊢ IsOpen[inst✝] (⋃ i ∈ s, b i)

Tactic:
·

Goal state after:
case mpr.left
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
s : Set ι
hs : s.Countable
⊢ IsLindelof (⋃ i ∈ s, b i)
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case mpr.right
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
s : Set ι
hs : s.Countable
⊢ IsOpen[inst✝] (⋃ i ∈ s, b i)

Tactic:
·

Goal state after:
case mpr.right
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
hb' : ∀ (i : ι), IsLindelof (b i)
s : Set ι
hs : s.Countable
⊢ IsOpen[inst✝] (⋃ i ∈ s, b i)
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht'✝ : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
t' : β
ht' : t' ∈ t
⊢ f ⁻¹' {t'} ⊆ u ∪ v

pp: case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht'✝ : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
t' : β
ht' : t' ∈ t
⊢ f ⁻¹' {t'} ∩ (u ∩ v) = ∅

Tactic:
·

Goal state after:
case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht'✝ : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
t' : β
ht' : t' ∈ t
⊢ f ⁻¹' {t'} ⊆ u ∪ v
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case pos
X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
y : Y
hf : Tendsto f (coLindelof X) (𝓝 y)
hfc : Continuous[inst✝², inst✝¹] f
l : Filter Y
hne : l.NeBot
inst✝ : CountableInterFilter l
hle : l ≤ 𝓟 (insert y (range f))
hy : ClusterPt y l
⊢ ∃ x ∈ insert y (range f), ClusterPt x l

pp: case neg
X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
y : Y
hf : Tendsto f (coLindelof X) (𝓝 y)
hfc : Continuous[inst✝², inst✝¹] f
l : Filter Y
hne : l.NeBot
inst✝ : CountableInterFilter l
hle : l ≤ 𝓟 (insert y (range f))
hy : ¬ClusterPt y l
⊢ ∃ x ∈ insert y (range f), ClusterPt x l

Tactic:
·

Goal state after:
case pos
X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
y : Y
hf : Tendsto f (coLindelof X) (𝓝 y)
hfc : Continuous[inst✝², inst✝¹] f
l : Filter Y
hne : l.NeBot
inst✝ : CountableInterFilter l
hle : l ≤ 𝓟 (insert y (range f))
hy : ClusterPt y l
⊢ ∃ x ∈ insert y (range f), ClusterPt x l
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
⊢ f ⁻¹' T₁ ⊆ f ⁻¹' t ∩ u

pp: case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
⊢ f ⁻¹' t ∩ u ⊆ f ⁻¹' T₁

Tactic:
·

Goal state after:
case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
⊢ f ⁻¹' T₁ ⊆ f ⁻¹' t ∩ u
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ s ⊆ U ∩ e

pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ t ⊆ V ∩ e

pp: case refine_3
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ Disjoint (U ∩ e) (V ∩ e)

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ s ⊆ U ∩ e
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ t ⊆ V ∩ e

pp: case refine_3
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ Disjoint (U ∩ e) (V ∩ e)

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ t ⊆ V ∩ e
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ t ⊆ V ∩ e

Tactic:
intro x hx
    exact htV ⟨subset_closure hx, fun h => hSt.notMem_of_mem_left h.1 hx⟩

Goal state after:
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_3
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ Disjoint (U ∩ e) (V ∩ e)

Tactic:
·

Goal state after:
case refine_3
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hSt : Disjoint (closure[inst✝] s) t
hsT : Disjoint s (closure[inst✝] t)
e : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ
h : NormalSpace ↑e
he : IsOpen[inst✝] e
hst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)
U : Set X
hU : IsOpen[inst✝] U
hsU : closure[inst✝] s ∩ e ⊆ U ∩ e
V : Set X
hV : IsOpen[inst✝] V
htV : closure[inst✝] t ∩ e ⊆ V ∩ e
hUV : Disjoint (Subtype.val ⁻¹' U) (Subtype.val ⁻¹' V)
⊢ Disjoint (U ∩ e) (V ∩ e)
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case a.left
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
a : α
hat : a ∈ f ⁻¹' t
hau : a ∈ u
⊢ f a ∈ t

pp: case a.right
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
a : α
hat : a ∈ f ⁻¹' t
hau : a ∈ u
⊢ f ⁻¹' {f a} ⊆ u

Tactic:
·

Goal state after:
case a.left
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
a : α
hat : a ∈ f ⁻¹' t
hau : a ∈ u
⊢ f a ∈ t
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
⊢ f ⁻¹' T₂ ⊆ f ⁻¹' t ∩ v

pp: case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
⊢ f ⁻¹' t ∩ v ⊆ f ⁻¹' T₂

Tactic:
·

Goal state after:
case a
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
⊢ f ⁻¹' T₂ ⊆ f ⁻¹' t ∩ v
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : T ∈ 𝓝 (f x)
⊢ ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T

pp: case refine_2
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T
⊢ T ∈ 𝓝 (f x)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : T ∈ 𝓝 (f x)
⊢ ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case inl
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : HereditarilyLindelofSpace X
ι : Type u_2
inst✝ : Nonempty ι
U : ι → Set X
h : ∀ (i : ι), IsOpen[inst✝²] (U i)
htc : ∅.Countable
htu : ⋃ i ∈ ∅, U i = ⋃ i, U i
⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i

pp: case inr
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : HereditarilyLindelofSpace X
ι : Type u_2
inst✝ : Nonempty ι
U : ι → Set X
h : ∀ (i : ι), IsOpen[inst✝²] (U i)
t : Set ι
htc : t.Countable
htu : ⋃ i ∈ t, U i = ⋃ i, U i
t_ne : t.Nonempty
⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i

Tactic:
·

Goal state after:
case inl
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : HereditarilyLindelofSpace X
ι : Type u_2
inst✝ : Nonempty ι
U : ι → Set X
h : ∀ (i : ι), IsOpen[inst✝²] (U i)
htc : ∅.Countable
htu : ⋃ i ∈ ∅, U i = ⋃ i, U i
⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case inr
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : HereditarilyLindelofSpace X
ι : Type u_2
inst✝ : Nonempty ι
U : ι → Set X
h : ∀ (i : ι), IsOpen[inst✝²] (U i)
t : Set ι
htc : t.Countable
htu : ⋃ i ∈ t, U i = ⋃ i, U i
t_ne : t.Nonempty
⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i

Tactic:
·

Goal state after:
case inr
X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : HereditarilyLindelofSpace X
ι : Type u_2
inst✝ : Nonempty ι
U : ι → Set X
h : ∀ (i : ι), IsOpen[inst✝²] (U i)
t : Set ι
htc : t.Countable
htu : ⋃ i ∈ t, U i = ⋃ i, U i
t_ne : t.Nonempty
⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T
⊢ T ∈ 𝓝 (f x)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : RegularSpace β
ι : Type u_5
s : ι → Set α
p : ι → Prop
x : α
h : ∀ (t : Set α), t ∈ 𝓝 x ↔ ∃ i, p i ∧ s i ⊆ t
f : α → β
hf : IsDenseInducing f
T : Set β
hT : ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ T
⊢ T ∈ 𝓝 (f x)
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case a.left
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
a : α
hat : a ∈ f ⁻¹' t
hav : a ∈ v
⊢ f a ∈ t

pp: case a.right
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
a : α
hat : a ∈ f ⁻¹' t
hav : a ∈ v
⊢ f ⁻¹' {f a} ⊆ v

Tactic:
·

Goal state after:
case a.left
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
a : α
hat : a ∈ f ⁻¹' t
hav : a ∈ v
⊢ f a ∈ t
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case a.right
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
a : α
hat : a ∈ f ⁻¹' t
hav : a ∈ v
⊢ f ⁻¹' {f a} ⊆ v

Tactic:
·

Goal state after:
case a.right
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : IsCoinducing f
t : Set β
ht : IsClosed[inst✝] t
ht' : IsConnected t
hf : Surjective f
hT : IsClosed[inst✝¹] (f ⁻¹' t)
u v : Set α
hu : IsClosed[inst✝¹] u
hv : IsClosed[inst✝¹] v
huv : f ⁻¹' t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := ⋯
T₂ : Set β := ⋯
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
a : α
hat : a ∈ f ⁻¹' t
hav : a ∈ v
⊢ f ⁻¹' {f a} ⊆ v