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

Goal state before:
pp: case inl
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝³ : TopologicalSpace α
s t✝ u v : Set α
inst✝² : TopologicalSpace β
inst✝¹ : TotallyDisconnectedSpace α
inst✝ : TotallyDisconnectedSpace β
t : Set α
ht : IsPreconnected t
x✝ : Sum.inl '' t ⊆ univ
hs : IsPreconnected (Sum.inl '' t)
⊢ (Sum.inl '' t).Subsingleton

pp: case inr
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝³ : TopologicalSpace α
s t✝ u v : Set α
inst✝² : TopologicalSpace β
inst✝¹ : TotallyDisconnectedSpace α
inst✝ : TotallyDisconnectedSpace β
t : Set β
ht : IsPreconnected t
x✝ : Sum.inr '' t ⊆ univ
hs : IsPreconnected (Sum.inr '' t)
⊢ (Sum.inr '' t).Subsingleton

Tactic:
·

Goal state after:
case inl
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝³ : TopologicalSpace α
s t✝ u v : Set α
inst✝² : TopologicalSpace β
inst✝¹ : TotallyDisconnectedSpace α
inst✝ : TotallyDisconnectedSpace β
t : Set α
ht : IsPreconnected t
x✝ : Sum.inl '' t ⊆ univ
hs : IsPreconnected (Sum.inl '' t)
⊢ (Sum.inl '' t).Subsingleton
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case inr
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝³ : TopologicalSpace α
s t✝ u v : Set α
inst✝² : TopologicalSpace β
inst✝¹ : TotallyDisconnectedSpace α
inst✝ : TotallyDisconnectedSpace β
t : Set β
ht : IsPreconnected t
x✝ : Sum.inr '' t ⊆ univ
hs : IsPreconnected (Sum.inr '' t)
⊢ (Sum.inr '' t).Subsingleton

Tactic:
·

Goal state after:
case inr
α : Type u
β : Type v
ι : Type u_1
X : ι → Type u_2
inst✝³ : TopologicalSpace α
s t✝ u v : Set α
inst✝² : TopologicalSpace β
inst✝¹ : TotallyDisconnectedSpace α
inst✝ : TotallyDisconnectedSpace β
t : Set β
ht : IsPreconnected t
x✝ : Sum.inr '' t ⊆ univ
hs : IsPreconnected (Sum.inr '' t)
⊢ (Sum.inr '' t).Subsingleton
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case inl
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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
x✝ : ∅ ⊆ univ
hs : IsPreconnected ∅
⊢ ∅.Subsingleton

pp: case inr
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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
s : Set ((i : ι) × X i)
x✝ : s ⊆ univ
hs : IsPreconnected s
h : s.Nonempty
⊢ s.Subsingleton

Tactic:
·

Goal state after:
case inl
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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
x✝ : ∅ ⊆ univ
hs : IsPreconnected ∅
⊢ ∅.Subsingleton
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case inr
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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
s : Set ((i : ι) × X i)
x✝ : s ⊆ univ
hs : IsPreconnected s
h : s.Nonempty
⊢ s.Subsingleton

Tactic:
·

Goal state after:
case inr
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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
s : Set ((i : ι) × X i)
x✝ : s ⊆ univ
hs : IsPreconnected s
h : s.Nonempty
⊢ s.Subsingleton
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case mp
α : Type u
inst✝ : TopologicalSpace α
⊢ TotallyDisconnectedSpace α → ∀ (x : α), (connectedComponent x).Subsingleton

pp: case mpr
α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), (connectedComponent x).Subsingleton) → TotallyDisconnectedSpace α

Tactic:
·

Goal state after:
case mp
α : Type u
inst✝ : TopologicalSpace α
⊢ TotallyDisconnectedSpace α → ∀ (x : α), (connectedComponent x).Subsingleton
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case mp.a
α : Type u
inst✝ : TopologicalSpace α
h : TotallyDisconnectedSpace α
x : α
⊢ connectedComponent x ⊆ univ

pp: case mp.a
α : Type u
inst✝ : TopologicalSpace α
h : TotallyDisconnectedSpace α
x : α
⊢ IsPreconnected (connectedComponent x)

Tactic:
·

Goal state after:
case mp.a
α : Type u
inst✝ : TopologicalSpace α
h : TotallyDisconnectedSpace α
x : α
⊢ connectedComponent x ⊆ univ
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case mpr.inl
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), (connectedComponent x).Subsingleton
s_sub : ∅ ⊆ univ
hs : IsPreconnected ∅
⊢ ∅.Subsingleton

pp: case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), (connectedComponent x).Subsingleton
s : Set α
s_sub : s ⊆ univ
hs : IsPreconnected s
x : α
x_in : x ∈ s
⊢ s.Subsingleton

Tactic:
·

Goal state after:
case mpr.inl
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), (connectedComponent x).Subsingleton
s_sub : ∅ ⊆ univ
hs : IsPreconnected ∅
⊢ ∅.Subsingleton
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsGδ s → ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n

pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n) → IsGδ s

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ IsGδ s → ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_1.inl
X : Type u_1
inst✝ : TopologicalSpace X
hT : ∀ t ∈ ∅, IsOpen[inst✝] t
T_count : ∅.Countable
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ ∅ = ⋂ n, f n

pp: case refine_1.inr
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT✝ : ∀ t ∈ T, IsOpen[inst✝] t
T_count : T.Countable
hT : T.Nonempty
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ T = ⋂ n, f n

Tactic:
·

Goal state after:
case refine_1.inl
X : Type u_1
inst✝ : TopologicalSpace X
hT : ∀ t ∈ ∅, IsOpen[inst✝] t
T_count : ∅.Countable
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ ∅ = ⋂ n, f n
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_1.inr
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT✝ : ∀ t ∈ T, IsOpen[inst✝] t
T_count : T.Countable
hT : T.Nonempty
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ T = ⋂ n, f n

Tactic:
·

Goal state after:
case refine_1.inr
X : Type u_1
inst✝ : TopologicalSpace X
T : Set (Set X)
hT✝ : ∀ t ∈ T, IsOpen[inst✝] t
T_count : T.Countable
hT : T.Nonempty
⊢ ∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ ⋂₀ T = ⋂ n, f n
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), (connectedComponent x).Subsingleton
s : Set α
s_sub : s ⊆ univ
hs : IsPreconnected s
x : α
x_in : x ∈ s
⊢ s.Subsingleton

Tactic:
·

Goal state after:
case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), (connectedComponent x).Subsingleton
s : Set α
s_sub : s ⊆ univ
hs : IsPreconnected s
x : α
x_in : x ∈ s
⊢ s.Subsingleton
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n) → IsGδ s

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ f, (∀ (n : ℕ), IsOpen[inst✝] (f n)) ∧ s = ⋂ n, f n) → IsGδ s
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, IsClopen s ∧ x ∈ s ∧ s ⊆ u ∪ v

Tactic:
have H1 := (hu.union hv).isClosed_compl.isCompact.inter_iInter_nonempty
      (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s) fun s => s.2.1.1

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
H1 : (∀ (u_1 : Finset { s // IsClopen s ∧ x ∈ s }), ((u ∪ v)ᶜ ∩ ⋂ i ∈ u_1, ↑i).Nonempty) → ((u ∪ v)ᶜ ∩ ⋂ i, ↑i).Nonempty
⊢ ∃ s, IsClopen s ∧ x ∈ s ∧ s ⊆ u ∪ v
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
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
t' : β
ht' : t' ∈ t
htu : f ⁻¹' {t'} ⊆ u
⊢ t' ∈ T₁ ∨ t' ∈ T₂

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
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
t' : β
ht' : t' ∈ t
htv : f ⁻¹' {t'} ⊆ v
⊢ t' ∈ T₁ ∨ t' ∈ T₂

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
T₂_v : f ⁻¹' T₂ = f ⁻¹' t ∩ v
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
t' : β
ht' : t' ∈ t
htu : f ⁻¹' {t'} ⊆ u
⊢ t' ∈ T₁ ∨ t' ∈ T₂
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
ht : t ⊆ s
hs : IsNowhereDense s
⊢ interior (closure[inst✝] t) ⊆ ∅

Tactic:
grw [ht]; rw [hs]

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

Goal state before:
pp: case refine_1
α : Type u_3
inst✝ : TopologicalSpace α
x y : α
x✝¹ : x ≠ y
x✝ : ∃ u v, IsOpen[inst✝] u ∧ IsOpen[inst✝] v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
Ux : x ∈ U
Vy : y ∈ V
f : univ ⊆ U ∪ V
disj : Disjoint U V
⊢ (fun x1 x2 ↦ ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ) x y

pp: case refine_2
α : Type u_3
inst✝ : TopologicalSpace α
x y : α
x✝¹ : x ≠ y
x✝ : (fun x1 x2 ↦ ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ) x y
U : Set α
hU : IsClopen U
Ux : x ∈ U
Ucy : y ∈ Uᶜ
⊢ ∃ u v, IsOpen[inst✝] u ∧ IsOpen[inst✝] v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case refine_1
α : Type u_3
inst✝ : TopologicalSpace α
x y : α
x✝¹ : x ≠ y
x✝ : ∃ u v, IsOpen[inst✝] u ∧ IsOpen[inst✝] v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v
U V : Set α
hU : IsOpen[inst✝] U
hV : IsOpen[inst✝] V
Ux : x ∈ U
Vy : y ∈ V
f : univ ⊆ U ∪ V
disj : Disjoint U V
⊢ (fun x1 x2 ↦ ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ) x y
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_1
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
⊢ IsClopen (⋂ U ∈ si, ↑U)

pp: case refine_2
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
⊢ x ∈ ⋂ U ∈ si, ↑U

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 ∈ si, ↑U ⊆ u ∪ v

Tactic:
·

Goal state after:
case refine_1
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
⊢ IsClopen (⋂ U ∈ si, ↑U)
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case refine_2
α : Type u_3
inst✝ : TopologicalSpace α
x y : α
x✝¹ : x ≠ y
x✝ : (fun x1 x2 ↦ ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ) x y
U : Set α
hU : IsClopen U
Ux : x ∈ U
Ucy : y ∈ Uᶜ
⊢ ∃ u v, IsOpen[inst✝] u ∧ IsOpen[inst✝] v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v

Tactic:
·

Goal state after:
case refine_2
α : Type u_3
inst✝ : TopologicalSpace α
x y : α
x✝¹ : x ≠ y
x✝ : (fun x1 x2 ↦ ∃ U, IsClopen U ∧ x1 ∈ U ∧ x2 ∈ Uᶜ) x y
U : Set α
hU : IsClopen U
Ux : x ∈ U
Ucy : y ∈ Uᶜ
⊢ ∃ u v, IsOpen[inst✝] u ∧ IsOpen[inst✝] v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v
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
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
t' : β
ht' : t' ∈ t
htv : f ⁻¹' {t'} ⊆ v
⊢ t' ∈ T₁ ∨ t' ∈ T₂

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
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
t' : β
ht' : t' ∈ t
htv : f ⁻¹' {t'} ⊆ v
⊢ t' ∈ T₁ ∨ t' ∈ T₂
Module: Mathlib.Topology.Separation.Regular

Goal state before:
pp: case refine_2
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
⊢ x ∈ ⋂ U ∈ si, ↑U

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 ∈ si, ↑U ⊆ u ∪ v

Tactic:
·

Goal state after:
case refine_2
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
⊢ x ∈ ⋂ U ∈ si, ↑U
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 ∈ si, ↑U ⊆ u ∪ v

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 ∈ si, ↑U ⊆ u ∪ v
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case inl
α : Type u
inst✝¹ : TopologicalSpace α
β : Type u_3
inst✝ : TopologicalSpace β
T : Set β
hT : IsDiscrete T
f : α → β
hne : T.Nonempty
hS : IsPreconnected ∅
hc : ContinuousOn f ∅
hTm : MapsTo f ∅ T
⊢ ∃ y ∈ T, EqOn f (const α y) ∅

pp: case inr
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
hne : T.Nonempty
x : α
hx : x ∈ S
⊢ ∃ y ∈ T, EqOn f (const α y) S

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝¹ : TopologicalSpace α
β : Type u_3
inst✝ : TopologicalSpace β
T : Set β
hT : IsDiscrete T
f : α → β
hne : T.Nonempty
hS : IsPreconnected ∅
hc : ContinuousOn f ∅
hTm : MapsTo f ∅ T
⊢ ∃ y ∈ T, EqOn f (const α y) ∅
Module: Mathlib.Topology.Connected.TotallyDisconnected

Goal state before:
pp: case inr
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
hne : T.Nonempty
x : α
hx : x ∈ S
⊢ ∃ y ∈ T, EqOn f (const α y) S

Tactic:
·

Goal state after:
case inr
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
hne : T.Nonempty
x : α
hx : x ∈ S
⊢ ∃ y ∈ T, EqOn f (const α y) S
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
⊢ S.Countable

pp: case refine_2
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S) →
    ∃ x, (∀ ⦃x_1 : Set X⦄, x_1 ∈ x → IsClosed[inst✝] x_1 ∧ IsNowhereDense x_1) ∧ (compl '' x).Countable ∧ s ⊆ ⋃₀ x

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
⊢ S.Countable
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
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 ∨ f ⁻¹' t ⊆ v

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
hT₁ : IsClosed[inst✝] T₁
hT₂ : IsClosed[inst✝] T₂
T_decomp : t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : t ⊆ T₂
⊢ f ⁻¹' t ⊆ u ∨ f ⁻¹' t ⊆ v

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
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 ∨ f ⁻¹' t ⊆ v