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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
t : Set ↑s
⊢ Dense t ↔ s ⊆ closure[inst✝] ↑t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Z
g : Y → Z
hf : IsOpenEmbedding f
hg : IsOpenEmbedding g
h : Injective (Sum.elim f g)
⊢ IsOpenEmbedding (Sum.elim f g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.NhdsWithin

Goal state before:
pp: α : Type u_5
β : Type u_6
t : TopologicalSpace β
f : α → β
s : Set α
⊢ 𝓝ˢ s = comap f (𝓝ˢ (f '' s))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
x : α
⊢ ContinuousWithinAt f {x}ᶜ x ↔ ContinuousAt f x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
x : α
h : ¬AccPt x (𝓟 {x}ᶜ)
⊢ ContinuousAt f x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.NhdsWithin

Goal state before:
pp: α : Type u_5
β : Type u_6
t : TopologicalSpace β
f : α → β
s : Set α
⊢ map f (𝓝ˢ s) = 𝓝ˢ[range f] (f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hst : s ⊆ t
⊢ DenseRange (inclusion hst) ↔ t ⊆ closure[inst✝] s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
x : α
h : ¬AccPt x ⊤
hh : AccPt x (𝓟 {x}ᶜ)
⊢ 𝓟 {x}ᶜ ≤ ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Topology.NhdsWithin

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
t : Set ↑s
⊢ map Subtype.val (𝓝ˢ t) = 𝓝ˢ[s] (Subtype.val '' t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.NhdsWithin

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s : Set α
t u : Set ↑s
⊢ u ∈ 𝓝ˢ t ↔ Subtype.val '' u ∈ 𝓝ˢ[s] (Subtype.val '' t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
x : ↑s
⊢ map Subtype.val (𝓝 x) = 𝓝[s] ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
x : α
h : s ∈ 𝓝 x
⊢ ContinuousWithinAt f s x ↔ ContinuousAt f x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X ⊕ Y → Z
⊢ IsClosedMap f ↔ (IsClosedMap fun a ↦ f (inl a)) ∧ IsClosedMap fun b ↦ f (inr b)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Z
g : Y → Z
⊢ IsClosedMap (Sum.elim f g) ↔ IsClosedMap f ∧ IsClosedMap g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
s : Set α
h : Continuous[inst✝¹, inst✝] f
⊢ ContinuousOn f s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
p : X → Prop
x : X
hx : p x
h : ∀ᶠ (x : X) in 𝓝 x, p x
⊢ range Subtype.val ∈ 𝓝 ↑⟨x, hx⟩

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set (Set α)
hs : t = generateFrom s
⊢ IsTopologicalBasis ((fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s})

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t✝ : TopologicalSpace α
s : Set (Set α)
hsg : t✝ = generateFrom s
hsi : FiniteInter s
t : Set α
ht : t ∈ s
⊢ {t} ∈ {f | f.Finite ∧ f ⊆ s} ∧ (fun f ↦ ⋂₀ f) {t} = t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
Y : Type u_5
p : X → Prop
l : Filter Y
f : Y → Subtype p
a : X
ha : p a
⊢ Tendsto f l (𝓝 ⟨a, ha⟩) ↔ Tendsto (fun x ↦ ↑(f x)) l (𝓝 ↑⟨a, ha⟩)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Z
g : Y → Z
hf : IsClosedEmbedding f
hg : IsClosedEmbedding g
h : Injective (Sum.elim f g)
⊢ IsClosedEmbedding (Sum.elim f g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f g : α → β
s s₁ : Set α
h : ContinuousOn f s
h' : EqOn g f s₁
h₁ : s₁ ⊆ s
⊢ ContinuousOn g s₁

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set (Set α)
hsg : t = generateFrom s
hsi : FiniteInter s
⊢ IsTopologicalBasis s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ (Homeomorph.refl X).sumCongr (Homeomorph.refl Y) = Homeomorph.refl (X ⊕ Y)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f g : α → β
s : Set α
x : α
h : f =ᶠ[𝓝[s] x] g
hx : f x = g x
⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
r : Set (Set α)
hsg : t = generateFrom r
hsi : ∀ ⦃s : Set α⦄, s ∈ r → ∀ ⦃t : Set α⦄, t ∈ r → s ∩ t ∈ r
⊢ t = generateFrom (insert univ r)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
s : Set α
x : α
g : β → γ
t : Set β
y : β
hg : ContinuousWithinAt g t y
hf : ContinuousWithinAt f s x
h : MapsTo f s t
hy : f x = y
⊢ ContinuousWithinAt (g ∘ f) s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set (Set α)
h_nhds : ∀ (a : α), (𝓝 a).HasBasis (fun t ↦ t ∈ s ∧ a ∈ t) id
t₁ : Set α
ht₁ : t₁ ∈ s
t₂ : Set α
ht₂ : t₂ ∈ s
x : α
hx : x ∈ t₁ ∩ t₂
⊢ ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
s : Set α
x : α
g : β → γ
t : Set β
y : β
hg : ContinuousWithinAt g t y
hf : ContinuousWithinAt f s x
hy : f x = y
⊢ ContinuousWithinAt (g ∘ f) (s ∩ f ⁻¹' t) x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set (Set α)
h_nhds : ∀ (a : α), (𝓝 a).HasBasis (fun t ↦ t ∈ s ∧ a ∈ t) id
x : α
⊢ 𝓝 x = 𝓝 x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
X' : Type u_5
Y' : Type u_6
inst✝³ : TopologicalSpace X'
inst✝² : TopologicalSpace Y'
X'' : Type u_7
Y'' : Type u_8
inst✝¹ : TopologicalSpace X''
inst✝ : TopologicalSpace Y''
h₁ : X ≃ₜ X'
h₂ : Y ≃ₜ Y'
h₃ : X' ≃ₜ X''
h₄ : Y' ≃ₜ Y''
⊢ (h₁.sumCongr h₂).trans (h₃.sumCongr h₄) = (h₁.trans h₃).sumCongr (h₂.trans h₄)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
s : Set α
x : α
g : β → γ
t : Set β
y : β
hg : ContinuousWithinAt g t y
hf : ContinuousWithinAt f s x
h : f ⁻¹' t ∈ 𝓝[s] x
hy : f x = y
⊢ ContinuousWithinAt (g ∘ f) s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : ∀ (a : X), f a ∈ s
⊢ Continuous[inst✝¹, instTopologicalSpaceSubtype] (codRestrict f s hs) ↔ Continuous[inst✝¹, inst✝] f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
⊢ Continuous[instTopologicalSpaceSum, instTopologicalSpaceSum] (Sum.elim inl (inr ∘ inl))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
⊢ Continuous[inst✝, instTopologicalSpaceSum] (inr ∘ inr)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
⊢ Continuous[inst✝², instTopologicalSpaceSum] (inl ∘ inl)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
s : Set α
x : α
g : β → γ
t : Set β
y : β
hg : ContinuousWithinAt g t y
hf : ContinuousWithinAt f s x
hs : t ∈ 𝓝[f '' s] y
hy : f x = y
⊢ ContinuousWithinAt (g ∘ f) s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
⊢ Continuous[instTopologicalSpaceSum, instTopologicalSpaceSum] (Sum.elim (inl ∘ inr) inr)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set (Set α)
h_open : ∀ u ∈ s, IsOpen[t] u
h_nhds : ∀ (a : α) (u : Set α), a ∈ u → IsOpen[t] u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u
a : α
⊢ ∀ (i : Set α), a ∈ i ∧ IsOpen[t] i → ∃ i', (i' ∈ s ∧ a ∈ i') ∧ id i' ⊆ i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
s : Set α
x : α
g : β → γ
y : β
hg : ContinuousAt g y
hf : ContinuousWithinAt f s x
hy : f x = y
⊢ ContinuousWithinAt (g ∘ f) s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsOpenEmbedding f
s : Set X
t : Set Y
H : MapsTo f s t
hs : IsOpen[inst✝¹] s
⊢ IsOpen[instTopologicalSpaceSubtype] (range (MapsTo.restrict f s t H))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
g : β → γ
f : α → β
s : Set β
hg : ContinuousOn g s
hf : Continuous[inst✝², inst✝¹] f
hs : ∀ (x : α), f x ∈ s
⊢ Continuous[inst✝², inst✝] (g ∘ f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : β × γ → δ
g : α → β
h : α → γ
x : α
s : Set α
y : β × γ
hf : ContinuousAt f y
hg : ContinuousWithinAt g s x
hh : ContinuousWithinAt h s x
e : (g x, h x) = y
⊢ ContinuousWithinAt (fun x ↦ f (g x, h x)) s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hst : s ⊆ t
hs : IsOpen[instTopologicalSpaceSubtype] (t ↓∩ s)
⊢ IsOpen[instTopologicalSpaceSubtype] (range (inclusion hst))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hst : s ⊆ t
hs : IsClosed[instTopologicalSpaceSubtype] (t ↓∩ s)
⊢ IsClosed[instTopologicalSpaceSubtype] (range (inclusion hst))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u_5
Y : Type u_6
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s : Set Y
inst✝ : DiscreteTopology ↑s
f : X → Y
hc : Continuous[inst✝², inst✝¹] f
hinj : Injective f
⊢ MapsTo f (f ⁻¹' s) s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology α
f : α × β → γ
s : Set (α × β)
x : α × β
⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt (fun x_1 ↦ f (x.1, x_1)) {b | (x.1, b) ∈ s} x.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsCoinducing f
s : Set Y
hs : IsOpen[inst✝] s
⊢ IsCoinducing (s.restrictPreimage f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
a : α
s : Set α
b : Set (Set α)
hb : IsTopologicalBasis b
⊢ s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set α
b : Set (Set α)
hb : IsTopologicalBasis b
⊢ IsOpen[t] s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
⊢ IsClosed[instTopologicalSpaceSubtype] (s ↓∩ t) ↔ s ∩ closure[inst✝] (s ∩ t) ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
ht : IsOpen[inst✝] t
⊢ frontier (s ∩ t) ∩ t = frontier s ∩ t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
b : Set (Set α)
hb : IsTopologicalBasis b
a : α
s : Set α
⊢ (∃ t ∈ b, a ∈ t ∧ t ⊆ s) ↔ ∃ i, (i ∈ b ∧ a ∈ i) ∧ i ⊆ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology β
f : α × β → γ
s : Set (α × β)
x : α × β
⊢ ContinuousWithinAt f s x ↔ ContinuousWithinAt (fun x_1 ↦ f (x_1, x.2)) {a | (a, x.2) ∈ s} x.1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
t : Set ↑s
ht : IsOpen[instTopologicalSpaceSubtype] t
⊢ ∃ c, IsOpen[inst✝] c ∧ Subtype.val '' t = c ∩ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set α
b : Set (Set α)
hb : IsTopologicalBasis b
hs : s ∈ b
⊢ IsOpen[t] s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set (Set α)
h : IsTopologicalBasis s
⊢ ∀ u ∈ insert ∅ s, IsOpen[t] u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology α
f : α × β → γ
x : α × β
⊢ ContinuousAt f x ↔ ContinuousAt (fun x_1 ↦ f (x.1, x_1)) x.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology β
f : α × β → γ
x : α × β
⊢ ContinuousAt f x ↔ ContinuousAt (fun x_1 ↦ f (x_1, x.2)) x.1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
B B' : Set (Set α)
hB : IsTopologicalBasis B
h_open : ∀ u ∈ B', IsOpen[t] u
h : ∀ u ∈ B, ∀ x ∈ u, ∃ v ∈ B', x ∈ v ∧ v ⊆ u
⊢ IsTopologicalBasis B'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
B : Set (Set α)
hB : IsTopologicalBasis B
u : Set α
ou : IsOpen[t] u
⊢ u = ⋃ i, ↑i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology α
f : α × β → γ
s : Set (α × β)
⊢ ContinuousOn f s ↔ ∀ (a : α), ContinuousOn (fun x ↦ f (a, x)) {b | (a, b) ∈ s}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
B : Set (Set α)
P : Set α → Prop
hB : IsTopologicalBasis B
basis : ∀ b ∈ B, P b
sUnion : ∀ (S : Set (Set α)), (∀ s ∈ S, P s) → P (⋃₀ S)
s : Set α
hs : IsOpen[t] s
⊢ P s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t✝ : TopologicalSpace α
B : Set (Set α)
s t : Set α
hB : IsTopologicalBasis B
hs : IsOpen[t✝] s
h : ∀ U ∈ B, U ⊆ s → U ⊆ t
⊢ s ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology β
f : α × β → γ
s : Set (α × β)
⊢ ContinuousOn f s ↔ ∀ (b : β), ContinuousOn (fun x ↦ f (x, b)) {a | (a, b) ∈ s}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : Dense s
u : Set ↑s
huo : IsOpen[instTopologicalSpaceSubtype] u
hud : Dense u
⊢ ∃ v, IsOpen[inst✝] v ∧ Dense v ∧ Subtype.val ⁻¹' v = u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology α
f : α × β → γ
⊢ Continuous[instTopologicalSpaceProd, inst✝¹] f ↔ ∀ (a : α), Continuous[inst✝², inst✝¹] fun x ↦ f (a, x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
t : Set ↑s
ht : IsClosed[instTopologicalSpaceSubtype] t
⊢ ∃ c, IsClosed[inst✝] c ∧ Subtype.val '' t = c ∩ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t✝ : TopologicalSpace α
B : Set (Set α)
s t : Set α
hB : IsTopologicalBasis B
hs : IsOpen[t✝] s
ht : IsOpen[t✝] t
h : ∀ U ∈ B, U ⊆ s ↔ U ⊆ t
⊢ s = t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hs : IsOpen[inst✝] s
h : IsOpen[instTopologicalSpaceSubtype] (s ↓∩ t)
⊢ IsOpen[inst✝] (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
b : Set (Set α)
hb : IsTopologicalBasis b
s : Set α
a : α
⊢ (∀ (i : Set α), i ∈ b ∧ a ∈ i → (i ∩ s).Nonempty) ↔ ∀ o ∈ b, a ∈ o → (o ∩ s).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology β
f : α × β → γ
⊢ Continuous[instTopologicalSpaceProd, inst✝¹] f ↔ ∀ (b : β), Continuous[inst✝³, inst✝¹] fun x ↦ f (x, b)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hs : IsClosed[inst✝] s
h : IsClosed[instTopologicalSpaceSubtype] (s ↓∩ t)
⊢ IsClosed[inst✝] (s ∩ t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: X : Type u
ι : Type u_5
A : ι → Type u_6
inst✝ : TopologicalSpace X
T : (i : ι) → TopologicalSpace (A i)
f : X → (i : ι) → A i
⊢ Continuous[inst✝, Pi.topologicalSpace] f ↔ ∀ (i : ι), Continuous[inst✝, T i] fun a ↦ f a i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
b : Set (Set α)
hb : IsTopologicalBasis b
s : Set α
⊢ Dense s ↔ ∀ o ∈ b, o.Nonempty → (o ∩ s).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: ι : Type u_5
A : ι → Type u_6
T : (i : ι) → TopologicalSpace (A i)
a : (i : ι) → A i
⊢ 𝓝 a = Filter.pi fun i ↦ 𝓝 (a i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions.SumProd

Goal state before:
pp: X : Type u
Y : Type v
Z : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Z
g : Y → Z
hf : IsInducing f
hg : IsInducing g
hFg : Disjoint (closure[inst✝] (range f)) (range g)
hfG : Disjoint (range f) (closure[inst✝] (range g))
⊢ IsInducing (Sum.elim f g)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology α
f : α × β → γ
⊢ IsOpenMap f ↔ ∀ (a : α), IsOpenMap fun x ↦ f (a, x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
β : Type u_1
t : TopologicalSpace α
inst✝ : TopologicalSpace β
B : Set (Set α)
hB : IsTopologicalBasis B
f : α → β
⊢ IsOpenMap f ↔ ∀ s ∈ B, IsOpen[inst✝] (f '' s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
⊢ ∀ u ∈ {U | IsOpen[t] U}, IsOpen[t] u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: ι : Type u_5
A : ι → Type u_6
B : ι → Type u_7
T : (i : ι) → TopologicalSpace (A i)
inst✝ : (i : ι) → TopologicalSpace (B i)
f : (i : ι) → A i → B i
hfo : ∀ (i : ι), IsOpenMap (f i)
hsurj : ∀ᶠ (i : ι) in cofinite, Surjective (f i)
⊢ IsOpenMap (Pi.map f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen[t] u → ∃ v ∈ {U | IsOpen[t] U}, a ∈ v ∧ v ⊆ u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
β : Type u_1
t : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
T : Set (Set β)
hf : IsInducing f
h : IsTopologicalBasis T
a : α
⊢ (𝓝 a).HasBasis (fun t ↦ t ∈ preimage f '' T ∧ a ∈ t) id

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: Y : Type v
ι : Type u_5
A : ι → Type u_6
T : (i : ι) → TopologicalSpace (A i)
f : Y → (i : ι) → A i
g : (i : ι) → A i
u : Filter Y
⊢ Tendsto f u (𝓝 g) ↔ ∀ (x : ι), Tendsto (fun i ↦ f i x) u (𝓝 (g x))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: ι : Type u_5
A : ι → Type u_6
B : ι → Type u_7
T : (i : ι) → TopologicalSpace (A i)
inst✝ : (i : ι) → TopologicalSpace (B i)
f : (i : ι) → A i → B i
hf : ∀ (i : ι), IsInducing (f i)
⊢ IsInducing (Pi.map f)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: ι : Type u_5
A : ι → Type u_6
T : (i : ι) → TopologicalSpace (A i)
ι' : Type u_9
φ : ι' → ι
⊢ induced (fun f j ↦ f (φ j)) topologicalSpace = ⨅ i', induced (eval (φ i')) (T (φ i'))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: ι : Type u_5
A : ι → Type u_6
T : (i : ι) → TopologicalSpace (A i)
S : Set ι
⊢ induced S.domRestrict topologicalSpace = ⨅ i ∈ S, induced (eval i) (T i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousOn

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : DiscreteTopology β
f : α × β → γ
⊢ IsOpenMap f ↔ ∀ (b : β), IsOpenMap fun x ↦ f (x, b)

Tactic:
by

Goal state after: