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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : IsEmpty α
⊢ IsPreconnected univ

Tactic:
by

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

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

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

Tactic:
· exact ⟨by rw [univ_eq_empty_iff.mpr hα]; exact isPreconnected_empty⟩

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.Connected.Basic

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : Nonempty α
⊢ IsPreconnected univ

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ PreconnectedSpace α ↔ ∀ (x : α), connectedComponent x = univ

Tactic:
by

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

Goal state before:
pp: 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 : PreconnectedSpace α
inst✝ : PreconnectedSpace β
⊢ IsPreconnected univ

Tactic:
by

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

Goal state before:
pp: 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
⊢ Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[inst✝] t)

Tactic:
by

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

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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), PreconnectedSpace (X i)
⊢ IsPreconnected univ

Tactic:
by

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

Goal state before:
pp: X : Type u_3
inst✝ : TopologicalSpace X
u v : Set X
hu : IsClopen (v ↓∩ u)
huv₁ : u ⊆ v
⊢ u = ⋃ x ∈ u, connectedComponentIn v x

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
s : Set α
h : IsPreconnected s
⊢ IsPreconnected univ

Tactic:
by

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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
s : Set α
h : PreconnectedSpace ↑s
⊢ IsPreconnected s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
s : Set α
a : α
di : IsDenseInducing i
hs : s ∈ 𝓝 a
⊢ closure[inst✝] (i '' s) ∈ 𝓝 (i a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
di : IsDenseInducing i
s : Set α
⊢ Dense (i '' s) ↔ Dense s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
i : α → β
inst✝ : T2Space β
di : IsDenseInducing i
hd : Dense (range i)ᶜ
s : Set α
hs : IsCompact s
⊢ interior s = ∅

Tactic:
by

Goal state after:
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:
· exact hs.isLindelof_biUnion fun i _ => hb' i

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.Compactness.Lindelof

Goal state before:
pp: 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

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
i : α → β
inst✝ : TopologicalSpace δ
f : γ → α
g : γ → δ
h : δ → β
d : δ
a : α
di : IsDenseInducing i
H : Tendsto h (𝓝 d) (𝓝 (i a))
comm : h ∘ g = i ∘ f
⊢ Tendsto f (comap g (𝓝 d)) (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: 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
⊢ l ≤ 𝓟 (insert y (range f)) → ∃ x ∈ insert y (range f), ClusterPt x l

Tactic:
hle

Goal state after:
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))
⊢ ∃ x ∈ insert y (range f), ClusterPt x l
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
di : IsDenseInducing i
b : β
s : Set β
hs : s ∈ 𝓝 b
⊢ ∃ a, i a ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
i : α → β
inst✝ : TopologicalSpace γ
di : IsDenseInducing i
f : α → γ
a : α
hf : ContinuousAt f a
⊢ Tendsto f (𝓝 a) (𝓝 (di.extend f (i a)))

Tactic:
by

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
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}
⊢ ∀ t' ∈ t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: 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))
s : Set Y
hsy : s ∈ 𝓝 y
t : Set Y
htl : t ∈ l
hd : Disjoint s t
K : Set X
hKc : IsLindelof K
hKs : Kᶜ ⊆ f ⁻¹' s
⊢ f '' K ∈ l

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
f : α → γ
di : IsDenseInducing i
hf : ∀ (b : β), ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)
a : α
⊢ di.extend f (i a) = f a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T2Space γ
b : β
f : α → γ
g : β → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : α) in comap i (𝓝 b), g (i x) = f x
hg : ContinuousAt g b
⊢ di.extend f b = g b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
y : Y
hf : Tendsto f (coLindelof X) (𝓝 y)
hfc : Continuous[inst✝¹, inst✝] f
⊢ IsLindelof (insert y (range f))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this✝ : ∀ (b : β), (comap i (𝓝 b)).NeBot
this : ∀ V' ∈ 𝓝 (φ b), IsClosed[inst✝¹] V' → φ ⁻¹' V' ∈ 𝓝 b
⊢ ContinuousAt φ b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ (coclosedLindelof X).HasBasis (fun s ↦ IsClosed[inst✝] s ∧ IsLindelof s) compl

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ coclosedLindelof X ↔ ∃ t, IsClosed[inst✝] t ∧ IsLindelof t ∧ tᶜ ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ s ∈ coclosedLindelof X ↔ ∃ t, IsClosed[inst✝] t ∧ IsLindelof t ∧ sᶜ ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝³ : TopologicalSpace X
inst✝² : LindelofSpace X
f : Filter X
inst✝¹ : f.NeBot
inst✝ : CountableInterFilter f
⊢ ∃ x, ClusterPt x f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: case right
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : LindelofSpace X
U : X → Set X
hU : ∀ (x : X), U x ∈ 𝓝 x
t : Set X
tc : t.Countable
s : univ ⊆ ⋃ x ∈ t, U x
⊢ ⋃ x ∈ t, U x = univ

Tactic:
apply top_unique s

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : LindelofSpace X
U : X → Set X
hU : ∀ (x : X), U x ∈ 𝓝 x
⊢ ∃ t, t.Countable ∧ ⋃ x ∈ t, U x = univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
h :
  ∀ {ι : Type u} (t : ι → Set X),
    (∀ (i : ι), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ ⋂ i ∈ u, t i = ∅
ι✝ : Type u
t : ι✝ → Set X
⊢ (∀ (i : ι✝), IsClosed[inst✝] (t i)) → univ ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ univ ∩ ⋂ i ∈ u, t i = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
⊢ IsLindelof s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsSigmaCompact s
⊢ IsLindelof s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : NonLindelofSpace X
⊢ (coLindelof X).NeBot

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed[inst✝¹] V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
⊢ V₁ ∈ 𝓝 b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : Continuous[inst✝¹, inst✝] f
⊢ comap f (coLindelof Y) ≤ coLindelof X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : LindelofSpace X
f : X → Y
hf : Continuous[inst✝², inst✝¹] f
⊢ IsLindelof (range f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hf : IsInducing f
⊢ IsLindelof s ↔ IsLindelof (f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed[inst✝¹] V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
⊢ ∃ V₂ ∈ 𝓝 b, IsOpen[inst✝²] V₂ ∧ ∀ x ∈ i ⁻¹' V₂, f x ∈ V'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsInducing f
hf' : IsClosed[inst✝] (range f)
K : Set Y
hK : IsLindelof K
⊢ IsLindelof (f ⁻¹' K)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ IsLindelof s ↔ IsLindelof univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
φ : β → γ := di.extend f
this✝ : ∀ (b : β), (comap i (𝓝 b)).NeBot
V' : Set γ
V'_in : V' ∈ 𝓝 (φ b)
V'_closed : IsClosed[inst✝¹] V'
V₁ : Set β := {x | Tendsto f (comap i (𝓝 x)) (𝓝 (φ x))}
V₁_in : V₁ ∈ 𝓝 b
V₂ : Set β
V₂_in : V₂ ∈ 𝓝 b
V₂_op : IsOpen[inst✝²] V₂
hV₂ : ∀ x ∈ i ⁻¹' V₂, f x ∈ V'
this : ∀ x ∈ V₁ ∩ V₂, φ x ∈ V'
⊢ φ ⁻¹' V' ∈ 𝓝 b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
h : LindelofSpace Y
f : X → Y
hf : IsClosedEmbedding f
⊢ IsLindelof univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
ι : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
s t : Set X
inst✝¹ : LindelofSpace X
inst✝ : LindelofSpace Y
⊢ IsLindelof univ

Tactic:
by

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
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
⊢ f ⁻¹' T₁ = f ⁻¹' t ∩ u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X✝ : Type u
Y : Type v
ι : Type u_1
inst✝⁴ : TopologicalSpace X✝
inst✝³ : TopologicalSpace Y
s t : Set X✝
X : ι → Type u_2
inst✝² : Countable ι
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), LindelofSpace (X i)
⊢ IsLindelof univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
r : X → X → Prop
inst✝ : LindelofSpace X
⊢ IsLindelof univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : LindelofSpace X
hf : Continuous[inst✝², inst✝¹] f
hsur : Function.Surjective f
⊢ IsLindelof univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : HereditarilyLindelofSpace X
s : Set X
⊢ IsLindelof s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
H : ∀ (s : Set X), IsOpen[inst✝] s → IsLindelof s
⊢ HereditarilyLindelofSpace X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
i : α → β
inst✝¹ : TopologicalSpace γ
inst✝ : T3Space γ
b : β
f : α → γ
di : IsDenseInducing i
hf : ∀ᶠ (x : β) in 𝓝 b, ∃ c, Tendsto f (comap i (𝓝 x)) (𝓝 c)
⊢ ContinuousAt (di.extend f) b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
c : Continuous[inst✝¹, inst✝] i
dense : ∀ (x : β), x ∈ closure[inst✝] (range i)
H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ (b : α), i b ∈ t → b ∈ s
a : α
⊢ comap i (𝓝 (i a)) ≤ 𝓝 a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
e : α → β
de : IsDenseEmbedding e
p : α → Prop
x✝ : { x // p x }
x : α
hx : p x
⊢ 𝓝 ⟨x, hx⟩ = comap (subtypeEmb p e) (𝓝 (subtypeEmb p e ⟨x, hx⟩))

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
⊢ CompletelyNormalSpace X ↔ ∀ (s : Set X), IsOpen[inst✝] s → NormalSpace ↑s

Tactic:
by

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 : CompletelyNormalSpace X
inst✝ : R0Space X
⊢ ∀ (x : SeparationQuotient X), IsClosed {x}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
e : α → β
de : IsDenseEmbedding e
p : α → Prop
⊢ closure[instTopologicalSpaceSubtype] (range (subtypeEmb p e)) = univ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace β
e : α → β
p : β → Prop
he : DenseRange e
hp : IsClosed[inst✝] {x | p x}
h : ∀ (a : α), p (e a)
⊢ ∀ (b : β), p b

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t✝ : Set X
inst✝ : SecondCountableTopology X
t : Set X
x✝¹ : t ⊆ univ
x✝ : Filter X
⊢ ∀ [x✝.NeBot] [CountableInterFilter x✝], x✝ ≤ 𝓟 t → ∃ x ∈ t, ClusterPt x x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: 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 : (𝓝 x).HasBasis p s
f : α → β
hf : IsDenseInducing f
⊢ (𝓝 (f x)).HasBasis p fun i ↦ closure[inst✝¹] (f '' s i)

Tactic:
rw [Filter.hasBasis_iff] at h ⊢

Goal state after:
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), t ∈ 𝓝 (f x) ↔ ∃ i, p i ∧ closure[inst✝¹] (f '' s i) ⊆ t
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : HereditarilyLindelofSpace X
ι : Type u_2
U : ι → Set X
h : ∀ (i : ι), IsOpen[inst✝¹] (U i)
⊢ ∃ t, t.Countable ∧ ⋃ i ∈ t, U i = ⋃ i, U i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: 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)
T' : Set β
hT₁ : T' ∈ 𝓝 (f x)
hT₂ : IsClosed[inst✝¹] T'
hT₃ : T' ⊆ T
⊢ f ⁻¹' T' ∈ 𝓝 x

Tactic:
by

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 : CompletelyNormalSpace X
inst✝ : R0Space X
s t : Set (SeparationQuotient X)
hd₁ : Disjoint (closure s) t
hd₂ : Disjoint s (closure t)
⊢ Disjoint (𝓝ˢ s) (𝓝ˢ t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: 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
i : ι
hi : p i
hi' : closure[inst✝¹] (f '' s i) ⊆ T
this : closure[inst✝¹] (f '' s i) ∈ 𝓝 (f x)
⊢ T ∈ 𝓝 (f x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: 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)
⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DenseEmbedding

Goal state before:
pp: 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 : (𝓝 x).HasBasis p s
f : α → β
hf : IsDenseInducing f
⊢ (𝓝 (f x)).HasBasis p fun i ↦ closure[inst✝¹] (f '' s i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : HereditarilyLindelofSpace X
ι : Type u_2
C : ι → Set X
h : ∀ (i : ι), IsClosed[inst✝¹] (C i)
⊢ ∃ t, t.Countable ∧ ⋂ i ∈ t, C i = ⋂ i, C i

Tactic:
by

Goal state after:
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✝³ : TopologicalSpace α
s t u v : Set α
inst✝² : TopologicalSpace β
inst✝¹ : TotallyDisconnectedSpace α
inst✝ : TotallyDisconnectedSpace β
⊢ TotallyDisconnectedSpace (α ⊕ β)

Tactic:
by

Goal state after:
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 : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), TotallyDisconnectedSpace (X i)
⊢ TotallyDisconnectedSpace ((i : ι) × X i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Compactness.Lindelof

Goal state before:
pp: X : Type u
inst✝² : TopologicalSpace X
inst✝¹ : HereditarilyLindelofSpace X
ι : Type u_2
inst✝ : Nonempty ι
C : ι → Set X
h : ∀ (i : ι), IsClosed[inst✝²] (C i)
⊢ ∃ k, ⋂ n, C (k n) = ⋂ i, C i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsOpen[inst✝] s
⊢ ∀ t ∈ {s}, IsOpen[inst✝] t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
ι : Type u_3
inst✝ : TopologicalSpace X
I : Set ι
hI : I.Countable
f : ι → Set X
hf : ∀ i ∈ I, IsOpen[inst✝] (f i)
⊢ ∀ t ∈ f '' I, IsOpen[inst✝] t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.GDelta.Basic

Goal state before:
pp: X : Type u_1
ι : Type u_3
inst✝ : TopologicalSpace X
I : Set ι
hI : I.Countable
f : ι → Set X
hf : ∀ i ∈ I, IsOpen[inst✝] (f i)
⊢ ⋂ i ∈ I, f i = ⋂₀ (f '' I)

Tactic:
by

Goal state after: