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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
h : ∀ x ∈ s, closure[inst✝¹] s ∉ 𝓝 x
⊢ ∀ ⦃x : X⦄, x ∈ s → closure[inst✝] (f '' s) ∉ 𝓝 (f x)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
h : ∀ x ∈ s, closure[inst✝¹] s ∉ 𝓝 x
⊢ ∀ ⦃x : X⦄, x ∈ s → closure[inst✝] (f '' s) ∉ 𝓝 (f x)
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ univ ∈ residual X

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
⊢ univ ∈ residual X
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsMeagre s
ht : IsMeagre t
⊢ sᶜ ∩ tᶜ ∈ residual X

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsMeagre s
ht : IsMeagre t
⊢ sᶜ ∩ tᶜ ∈ residual X
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
⊢ ∀ (x : ConnectedComponents α), connectedComponent x = {x}

Tactic:
]

Goal state after:
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
⊢ ∀ (x : ConnectedComponents α), connectedComponent x = {x}
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
ι' : Sort u_4
inst✝¹ : TopologicalSpace X
inst✝ : Countable ι'
f : ι' → Set X
hs : ∀ (i : ι'), IsMeagre (f i)
⊢ ⋂ i, (f i)ᶜ ∈ residual X

Tactic:
]

Goal state after:
X : Type u_1
ι' : Sort u_4
inst✝¹ : TopologicalSpace X
inst✝ : Countable ι'
f : ι' → Set X
hs : ∀ (i : ι'), IsMeagre (f i)
⊢ ⋂ i, (f i)ᶜ ∈ residual X
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
x : α
⊢ {mk x} = {mk x}

pp: case h_fibers
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
x : α
⊢ ∀ (y : ConnectedComponents α), IsConnected (mk ⁻¹' {y})

Tactic:
]

Goal state after:
case h_fibers
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
x : α
⊢ ∀ (y : ConnectedComponents α), IsConnected (mk ⁻¹' {y})
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case h_fibers
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
x y : α
⊢ IsConnected (connectedComponent y)

Tactic:
]

Goal state after:
case h_fibers
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝² : TopologicalSpace α
s t u v : Set α
inst✝¹ : TopologicalSpace β
inst✝ : TotallyDisconnectedSpace β
f : α → β
x y : α
⊢ IsConnected (connectedComponent y)
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
β : Type u_3
inst✝ : TopologicalSpace β
f : α → β
hf : IsCoinducing f
⊢ IsCoinducing (⋯.connectedComponentsMap ∘ ConnectedComponents.mk)

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
β : Type u_3
inst✝ : TopologicalSpace β
f : α → β
hf : IsCoinducing f
⊢ IsCoinducing (⋯.connectedComponentsMap ∘ ConnectedComponents.mk)
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ x,
      (∀ t ∈ compl '' x, IsOpen[inst✝] t) ∧
        (∀ t ∈ compl '' x, Dense t) ∧ (compl '' x).Countable ∧ ⋂₀ (compl '' x) ⊆ sᶜ) ↔
    ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ x,
      (∀ t ∈ compl '' x, IsOpen[inst✝] t) ∧
        (∀ t ∈ compl '' x, Dense t) ∧ (compl '' x).Countable ∧ ⋂₀ (compl '' x) ⊆ sᶜ) ↔
    ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
S : Set α
hS : IsPreconnected S
β : Type u_3
inst✝ : TopologicalSpace β
T : Set β
hT : IsDiscrete T
f : α → β
hc : ContinuousOn f S
hTm : MapsTo f S T
x y : α
hx : x ∈ S
hy : y ∈ S
F : ↑S → ↑T := MapsTo.restrict f S T hTm
this : ↑(F ⟨x, hx⟩) = ↑(F ⟨y, hy⟩)
⊢ f x = f y

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
S : Set α
hS : IsPreconnected S
β : Type u_3
inst✝ : TopologicalSpace β
T : Set β
hT : IsDiscrete T
f : α → β
hc : ContinuousOn f S
hTm : MapsTo f S T
x y : α
hx : x ∈ S
hy : y ∈ S
F : ↑S → ↑T := MapsTo.restrict f S T hTm
this : ↑(F ⟨x, hx⟩) = ↑(F ⟨y, hy⟩)
⊢ f x = f y
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
S : Set α
hS : IsPreconnected S
β : Type u_3
inst✝ : TopologicalSpace β
T : Set β
hT : DiscreteTopology ↑T
f : α → β
hc : ContinuousOn f S
hTm : MapsTo f S T
x y : α
hx : x ∈ S
hy : y ∈ S
F : ↑S → ↑T := MapsTo.restrict f S T hTm
⊢ F ⟨x, hx⟩ = F ⟨y, hy⟩

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
S : Set α
hS : IsPreconnected S
β : Type u_3
inst✝ : TopologicalSpace β
T : Set β
hT : DiscreteTopology ↑T
f : α → β
hc : ContinuousOn f S
hTm : MapsTo f S T
x y : α
hx : x ∈ S
hy : y ∈ S
F : ↑S → ↑T := MapsTo.restrict f S T hTm
⊢ F ⟨x, hx⟩ = F ⟨y, hy⟩
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: 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 := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: case refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
H1 : Disjoint (u ∪ v)ᶜ (⋂ i, ↑i) → ∃ x_1, ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ x_1, ↑i).Nonempty
si : Finset { s // IsClopen s ∧ x ∈ s }
H2 : ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ si, ↑i).Nonempty
⊢ ¬((u ∪ v)ᶜ ∩ ⋂ U ∈ si, ↑U).Nonempty

Tactic:
]

Goal state after:
case refine_3
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
H1 : Disjoint (u ∪ v)ᶜ (⋂ i, ↑i) → ∃ x_1, ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ x_1, ↑i).Nonempty
si : Finset { s // IsClopen s ∧ x ∈ s }
H2 : ¬((u ∪ v)ᶜ ∩ ⋂ i ∈ si, ↑i).Nonempty
⊢ ¬((u ∪ v)ᶜ ∩ ⋂ U ∈ si, ↑U).Nonempty
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
x✝ : ∃ x, (∀ ⦃x_1 : Set X⦄, x_1 ∈ x → IsClosed[inst✝] x_1 ∧ IsNowhereDense x_1) ∧ (compl '' x).Countable ∧ s ⊆ ⋃₀ x
S : Set (Set X)
hS : ∀ ⦃x : Set X⦄, x ∈ S → IsClosed[inst✝] x ∧ IsNowhereDense x
hc : (compl '' S).Countable
hsub : s ⊆ ⋃₀ S
⊢ (compl '' compl '' S).Countable

Tactic:
]

Goal state after:
case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
x✝ : ∃ x, (∀ ⦃x_1 : Set X⦄, x_1 ∈ x → IsClosed[inst✝] x_1 ∧ IsNowhereDense x_1) ∧ (compl '' x).Countable ∧ s ⊆ ⋃₀ x
S : Set (Set X)
hS : ∀ ⦃x : Set X⦄, x ∈ S → IsClosed[inst✝] x ∧ IsNowhereDense x
hc : (compl '' S).Countable
hsub : s ⊆ ⋃₀ S
⊢ (compl '' compl '' S).Countable
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case h
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
S : Set (Set X)
hS : ∀ t ∈ S, IsNowhereDense t
hc : S.Countable
hsub : s ⊆ ⋃₀ S
⊢ (∀ ⦃x : Set X⦄, x ∈ S → IsClosed[inst✝] (closure[inst✝] x) ∧ IsNowhereDense (closure[inst✝] x)) ∧
    (compl '' closure[inst✝] '' S).Countable ∧ s ⊆ ⋃₀ (closure[inst✝] '' S)

Tactic:
]

Goal state after:
case h
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
S : Set (Set X)
hS : ∀ t ∈ S, IsNowhereDense t
hc : S.Countable
hsub : s ⊆ ⋃₀ S
⊢ (∀ ⦃x : Set X⦄, x ∈ S → IsClosed[inst✝] (closure[inst✝] x) ∧ IsNowhereDense (closure[inst✝] x)) ∧
    (compl '' closure[inst✝] '' S).Countable ∧ s ⊆ ⋃₀ (closure[inst✝] '' S)
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case inl
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 := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ ⊆ f ⁻¹' t ∩ u ∧ f ⁻¹' t ∩ u ⊆ f ⁻¹' T₁
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₁
⊢ f ⁻¹' t ⊆ u

Tactic:
]

Goal state after:
case inl
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 := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ ⊆ f ⁻¹' t ∩ u ∧ f ⁻¹' t ∩ u ⊆ f ⁻¹' T₁
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₁
⊢ f ⁻¹' t ⊆ u
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
s : Set X
H : IsClopen s ∧ x ∈ s ∧ s ⊆ v ∪ u
H1 : IsClopen (s ∩ u)
⊢ ⋂ s, ↑s ⊆ a ∨ ⋂ s, ↑s ⊆ b

Tactic:
]

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed[inst✝²] (⋂ s, ↑s)
a b : Set X
ha : IsClosed[inst✝²] a
hb : IsClosed[inst✝²] b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen[inst✝²] u
hv : IsOpen[inst✝²] v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
s : Set X
H : IsClopen s ∧ x ∈ s ∧ s ⊆ v ∪ u
H1 : IsClopen (s ∩ u)
⊢ ⋂ s, ↑s ⊆ a ∨ ⋂ s, ↑s ⊆ b
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : BaireSpace X
Y : Type u_4
inst✝ : TopologicalSpace Y
p : Y → X
hp : IsOpenEmbedding p
f : ℕ → Set Y
hof : ∀ (n : ℕ), IsOpen[inst✝] (f n)
hdf : ∀ (n : ℕ), Dense (f n)
s : Set X := range p
c : ℕ → Set X := fun n ↦ p '' f n ∪ (closure[inst✝²] s)ᶜ
c_open : ∀ (n : ℕ), IsOpen[inst✝²] (c n)
n : ℕ
⊢ closure[inst✝²] (c n) ⊆ univ ∧ univ ⊆ closure[inst✝²] (c n)

Tactic:
]

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : BaireSpace X
Y : Type u_4
inst✝ : TopologicalSpace Y
p : Y → X
hp : IsOpenEmbedding p
f : ℕ → Set Y
hof : ∀ (n : ℕ), IsOpen[inst✝] (f n)
hdf : ∀ (n : ℕ), Dense (f n)
s : Set X := range p
c : ℕ → Set X := fun n ↦ p '' f n ∪ (closure[inst✝²] s)ᶜ
c_open : ∀ (n : ℕ), IsOpen[inst✝²] (c n)
n : ℕ
⊢ closure[inst✝²] (c n) ⊆ univ ∧ univ ⊆ closure[inst✝²] (c n)
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsNowhereDense s
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsNowhereDense s
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
h : ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ f '' s ⊆ ⋃₀ S

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
h : ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ f '' s ⊆ ⋃₀ S
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case inr
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 := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' t ∩ v ∧ f ⁻¹' t ∩ v ⊆ f ⁻¹' T₂
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₂
⊢ f ⁻¹' t ⊆ v

Tactic:
]

Goal state after:
case inr
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 := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' t ∩ u
T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' t ∩ v ∧ f ⁻¹' t ∩ v ⊆ f ⁻¹' T₂
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₂
⊢ f ⁻¹' t ⊆ v
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
T : Set (Set X)
isNowhereDense : ∀ t ∈ T, IsNowhereDense t
countable : T.Countable
cover : s ⊆ ⋃₀ T
u : Set Y
t : Set X
tT : t ∈ T
tu : f '' t = u
⊢ IsNowhereDense (f '' t)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
T : Set (Set X)
isNowhereDense : ∀ t ∈ T, IsNowhereDense t
countable : T.Countable
cover : s ⊆ ⋃₀ T
u : Set Y
t : Set X
tT : t ∈ T
tu : f '' t = u
⊢ IsNowhereDense (f '' t)
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
T : Set (Set X)
isNowhereDense : ∀ t ∈ T, IsNowhereDense t
countable : T.Countable
cover : s ⊆ ⋃₀ T
⊢ f '' s ⊆ f '' ⋃₀ T

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
T : Set (Set X)
isNowhereDense : ∀ t ∈ T, IsNowhereDense t
countable : T.Countable
cover : s ⊆ ⋃₀ T
⊢ f '' s ⊆ f '' ⋃₀ T
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
s : Set X
T : Set (Set X)
isNowhereDense : ∀ t ∈ T, IsNowhereDense t
countable : T.Countable
cover : s ⊆ ⋃₀ T
⊢ f '' ⋃₀ T ⊆ f '' ⋃₀ T

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case connected_fibers
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 : β
⊢ ∀ (t : β), IsConnected (f ⁻¹' {t})

Tactic:
· exact connected_fibers

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
hf : IsCoinducing f
h_fibers : ∀ (y : β), IsConnected (f ⁻¹' {y})
a : α
⊢ connectedComponent (f a) = connectedComponent (f a)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t u v : Set α
inst✝ : Nonempty α
⊢ Nonempty α

Tactic:
]

Goal state after:
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t u v : Set α
inst✝ : Nonempty α
⊢ Nonempty α
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ Nonempty α ↔ Nonempty α

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: α : Type u
β : Type v
ι✝ : Type u_1
X : ι✝ → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
ι : Type u_3
U : ι → Set α
hclopen : ∀ (i : ι), IsClopen (U i)
hdisj : Pairwise (Disjoint on U)
hunion : ⋃ i, U i = univ
hconn : ∀ (i : ι), IsPreconnected (U i)
x : α
i : ι
hx : x ∈ U i
⊢ connectedComponent x = connectedComponent x

Tactic:
]

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: X : Type u_1
α : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
S : Set α
f : α → Set X
ho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)
hS : S.Countable
hd : ∀ s ∈ S, Dense (f s)
⊢ Dense (⋂₀ (f '' S))

Tactic:
]

Goal state after:
X : Type u_1
α : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
S : Set α
f : α → Set X
ho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)
hS : S.Countable
hd : ∀ s ∈ S, Dense (f s)
⊢ Dense (⋂₀ (f '' S))
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
h : connectedComponent a ∩ ⋂ s, ↑s = ∅
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
h : connectedComponent a ∩ ⋂ s, ↑s = ∅
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: case refine_1
X : Type u_1
α : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
S : Set α
f : α → Set X
ho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)
hS : S.Countable
hd : ∀ s ∈ S, Dense (f s)
⊢ ∀ ⦃x : α⦄, x ∈ S → IsOpen[inst✝¹] (f x)

Tactic:
]

Goal state after:
case refine_1
X : Type u_1
α : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
S : Set α
f : α → Set X
ho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)
hS : S.Countable
hd : ∀ s ∈ S, Dense (f s)
⊢ ∀ ⦃x : α⦄, x ∈ S → IsOpen[inst✝¹] (f x)
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: case refine_2
X : Type u_1
α : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
S : Set α
f : α → Set X
ho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)
hS : S.Countable
hd : ∀ s ∈ S, Dense (f s)
⊢ ∀ ⦃x : α⦄, x ∈ S → Dense (f x)

Tactic:
]

Goal state after:
case refine_2
X : Type u_1
α : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
S : Set α
f : α → Set X
ho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)
hS : S.Countable
hd : ∀ s ∈ S, Dense (f s)
⊢ ∀ ⦃x : α⦄, x ∈ S → Dense (f x)
Module: Mathlib.Topology.Baire.Lemmas

Goal state before:
pp: case mp
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
s : Set X
⊢ (∃ S, (∀ t ∈ S, IsOpen[inst✝¹] t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s) → ∃ t ⊆ s, IsGδ t ∧ Dense t

Tactic:
]

Goal state after:
case mp
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : BaireSpace X
s : Set X
⊢ (∃ S, (∀ t ∈ S, IsOpen[inst✝¹] t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s) → ∃ t ⊆ s, IsGδ t ∧ Dense t
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
h : connectedComponent a ∩ ⋂ s, ↑s = ∅
V : Set (ConnectedComponents X)
hU : IsClopen V
ha : connectedComponent a ∩ mk ⁻¹' V = ∅
hb : connectedComponent b ⊆ mk ⁻¹' V
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
a b : X
ne : connectedComponent a ≠ connectedComponent b
h : connectedComponent a ∩ ⋂ s, ↑s = ∅
V : Set (ConnectedComponents X)
hU : IsClopen V
ha : connectedComponent a ∩ mk ⁻¹' V = ∅
hb : connectedComponent b ⊆ mk ⁻¹' V
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ mk a ∈ u ∧ mk b ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.Connected.Clopen

Goal state before:
pp: case refine_1
α : Type u
β : Type v
ι✝ : Type u_1
X : ι✝ → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
ι : Type u_3
U : ι → Set α
hclopen : ∀ (i : ι), IsClopen (U i)
hdisj : Pairwise (Disjoint on U)
hunion : ⋃ i, U i = univ
hconn : ∀ (i : ι), IsPreconnected (U i)
heq : ∀ {x : α} {i : ι} (hx : x ∈ U i), Subtype.val '' connectedComponent ⟨x, hx⟩ = connectedComponent x
x✝¹ x✝ : (i : ι) × ConnectedComponents ↑(U i)
i j : ι
x : ↑(U i)
y : ↑(U j)
hxy : connectedComponent ↑x = connectedComponent ↑y
⊢ ⟨i, mk x⟩ = ⟨j, mk y⟩

Tactic:
]

Goal state after:
case refine_1
α : Type u
β : Type v
ι✝ : Type u_1
X : ι✝ → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
ι : Type u_3
U : ι → Set α
hclopen : ∀ (i : ι), IsClopen (U i)
hdisj : Pairwise (Disjoint on U)
hunion : ⋃ i, U i = univ
hconn : ∀ (i : ι), IsPreconnected (U i)
heq : ∀ {x : α} {i : ι} (hx : x ∈ U i), Subtype.val '' connectedComponent ⟨x, hx⟩ = connectedComponent x
x✝¹ x✝ : (i : ι) × ConnectedComponents ↑(U i)
i j : ι
x : ↑(U i)
y : ↑(U j)
hxy : connectedComponent ↑x = connectedComponent ↑y
⊢ ⟨i, mk x⟩ = ⟨j, mk y⟩