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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf₁ : Continuous[inst✝¹, inst✝] f
hf₂ : IsOpenMap f
hf₃ : ∀ (x : Y), IsPreirreducible (f ⁻¹' {x})
hf₄ : Function.Surjective f
V : Set X
hV : V ∈ irreducibleComponents X
Z : Set Y
hZ : IsIrreducible Z
hWZ : f '' V ⊆ Z
⊢ Z ⊆ f '' V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
f : X → Y
hf₁ : Continuous[inst✝¹, inst✝] f
hf₂ : IsOpenMap f
hf₃ : ∀ (x : Y), IsPreirreducible (f ⁻¹' {x})
hf₄ : Function.Surjective f
W : ↑(irreducibleComponents Y)
⊢ (↑W ∩ range f).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
f : X → Y
hf₁ : Continuous[inst✝¹, inst✝] f
hf₂ : IsOpenMap f
hf₃ : ∀ (x : Y), IsPreirreducible (f ⁻¹' {x})
hf₄ : Function.Surjective f
W : ↑(irreducibleComponents X)
⊢ ↑((fun W ↦ ⟨f ⁻¹' ↑W, ⋯⟩) ((fun W ↦ ⟨f '' ↑W, ⋯⟩) W)) = ↑W

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
f : X → Y
hf₁ : Continuous[inst✝¹, inst✝] f
hf₂ : IsOpenMap f
hf₃ : ∀ (x : Y), IsPreirreducible (f ⁻¹' {x})
hf₄ : Function.Surjective f
W Z : ↑(irreducibleComponents Y)
⊢ { toFun := fun W ↦ ⟨f ⁻¹' ↑W, ⋯⟩, invFun := fun W ↦ ⟨f '' ↑W, ⋯⟩, left_inv := ⋯, right_inv := ⋯ } W ≤
      { toFun := fun W ↦ ⟨f ⁻¹' ↑W, ⋯⟩, invFun := fun W ↦ ⟨f '' ↑W, ⋯⟩, left_inv := ⋯, right_inv := ⋯ } Z ↔
    W ≤ Z

Tactic:
by

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

Goal state before:
pp: X✝ : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X✝
ι : Type u_4
X : ι → Type u_3
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (a : ι), T2Space (X a)
⊢ T2Space ((i : ι) × X i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsDiscrete s
hs' : IsPreirreducible s
x : X
hxs : x ∈ s
y : X
hys : y ∈ s
U : Set X
hU : IsOpen[inst✝] U
hUx : U ∩ s = {x}
V : Set X
hV : IsOpen[inst✝] V
hVy : V ∩ s = {y}
⊢ x ∈ s ∩ U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsDiscrete s
hs' : IsPreirreducible s
x : X
hxs : x ∈ s
y : X
hys : y ∈ s
U : Set X
hU : IsOpen[inst✝] U
hUx : U ∩ s = {x}
V : Set X
hV : IsOpen[inst✝] V
hVy : V ∩ s = {y}
⊢ y ∈ s ∩ V

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsDiscrete s
hs' : IsPreirreducible s
x : X
hxs : x ∈ s
y : X
hys : y ∈ s
U : Set X
hU : IsOpen[inst✝] U
hUx : U ∩ s = {x}
V : Set X
hV : IsOpen[inst✝] V
hVy : V ∩ s = {y}
z : X
hz : z ∈ s ∩ (U ∩ V)
⊢ z ∈ U ∩ s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
i✝ : T2Quotient X
x : X
j✝ : T2Quotient X
y : X
h : ¬mk x = mk y
⊢ ∃ s, T2Space (Quotient s) ∧ ⟦x⟧ ≠ ⟦y⟧

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsDiscrete s
hs' : IsPreirreducible s
x : X
hxs : x ∈ s
y : X
hys : y ∈ s
U : Set X
hU : IsOpen[inst✝] U
hUx : U ∩ s = {x}
V : Set X
hV : IsOpen[inst✝] V
hVy : V ∩ s = {y}
z : X
hz : z ∈ s ∩ (U ∩ V)
⊢ z ∈ V ∩ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsDiscrete s
hs' : IsPreirreducible s
⊢ s.Subsingleton

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
hf : Surjective f
inst✝ : PreirreducibleSpace X
⊢ IsPreirreducible univ

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
i✝ : T2Quotient X
x : X
j✝ : T2Quotient X
y : X
h : ¬mk x = mk y
s : Setoid X
hs : T2Space (Quotient s)
hsxy : ⟦x⟧ ≠ ⟦y⟧
⊢ s ∈ {s | T2Space (Quotient s)}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
hfc : Continuous[inst✝², inst✝¹] f
hf : Surjective f
inst✝ : IrreducibleSpace X
⊢ IsPreirreducible univ

Tactic:
by

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed[inst✝] {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed[inst✝] {x}) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous[inst✝, _] ⇑CofiniteTopology.of
h : Continuous[inst✝, _] ⇑CofiniteTopology.of
x : X
⊢ (⇑CofiniteTopology.of '' {x}).Finite

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
⊢ T2Space (T2Quotient X)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
U s : Set X
hU : U ∈ 𝓝[≠] x
hs : Finite ↑s
⊢ U \ s ∈ 𝓝[≠] x

Tactic:
by

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
hf : Continuous[inst✝², inst✝¹] f
⊢ ∀ (a b : X), a ≈ b → f a = f b

Tactic:
by

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed[inst✝] {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed[inst✝] {x}) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous[inst✝, _] ⇑CofiniteTopology.of
⊢ Continuous[inst✝, _] ⇑CofiniteTopology.of → ∀ (x : X), IsClosed[inst✝] {x}

Tactic:
by

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
f : X → Y
hf : Continuous[inst✝², inst✝¹] f
g : T2Quotient X → Y
hfg : g ∘ mk = f
⊢ g = lift hf

Tactic:
by

Goal state after:
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
h : ∀ i ∈ Ico n m, r i (succ i)
hnm : n ≤ m
⊢ ReflTransGen r n m

Tactic:
by

Goal state after:
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : SuccOrder α
inst✝ : IsSuccArchimedean α
r : α → α → Prop
n m : α
h : ∀ i ∈ Ico m n, r (succ i) i
hmn : m ≤ n
⊢ ReflTransGen r n m

Tactic:
by

Goal state after:
Module: Mathlib.Order.SuccPred.Relation

Goal state before:
pp: α : Type u_1
inst✝³ : LinearOrder α
inst✝² : SuccOrder α
inst✝¹ : IsSuccArchimedean α
r : α → α → Prop
n m : α
inst✝ : Std.Refl r
h1 : ∀ i ∈ Ico n m, r i (succ i)
h2 : ∀ i ∈ Ico m n, r (succ i) i
⊢ TransGen r n m

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
Z : Type u_3
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : T2Space Z
s : Set X
t : Set Y
f g : X → Y → Z
h : ∀ x ∈ s, ∀ y ∈ t, f x y = g x y
hf₁ : ∀ (x : X), Continuous[inst✝², inst✝¹] (f x)
hf₂ : ∀ (y : Y), Continuous[inst✝³, inst✝¹] fun x ↦ f x y
hg₁ : ∀ (x : X), Continuous[inst✝², inst✝¹] (g x)
hg₂ : ∀ (y : Y), Continuous[inst✝³, inst✝¹] fun x ↦ g x y
this : closure[inst✝³] s ⊆ ⋂ y ∈ closure[inst✝²] t, {x | f x y = g x y}
⊢ ∀ x ∈ closure[inst✝³] s, ∀ y ∈ closure[inst✝²] t, f x y = g x y

Tactic:
by

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed[inst✝] {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed[inst✝] {x}) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_3 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ (x : X), IsOpen[inst✝] {x}ᶜ
tfae_5_iff_6 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s
tfae_5_iff_7 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ y ∉ U
tfae_5_iff_8 : (∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)
tfae_8_iff_9 : (∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)) ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y)
tfae_1_to_4 : T1Space X → Continuous[inst✝, _] ⇑CofiniteTopology.of
tfae_4_to_2 : Continuous[inst✝, _] ⇑CofiniteTopology.of → ∀ (x : X), IsClosed[inst✝] {x}
⊢ (∀ (x : X), IsClosed[inst✝] {x}) ↔ ∀ ⦃x y : X⦄, x ⤳ y → x = y

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
s t : Set X
f g : X → Y
h : EqOn f g s
hf : ContinuousOn f t
hg : ContinuousOn g t
hst : s ⊆ t
hts : t ⊆ closure[inst✝²] s
⊢ EqOn f g t

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
s t : Set X
hs : IsCompact s
ht : IsCompact t
hst : Disjoint s t
⊢ SeparatedNhds s t

Tactic:
by

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
⊢ [T1Space X, ∀ (x : X), IsClosed[inst✝] {x}, ∀ (x : X), IsOpen[inst✝] {x}ᶜ, Continuous[inst✝, _] ⇑CofiniteTopology.of,
      ∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
      ∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen[inst✝] U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
      ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y, T0Space X ∧ R0Space X].TFAE

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
x x' : X
y : Y
hne : x' ≠ x
⊢ ContinuousAt (update f x y) x' ↔ ContinuousAt f x'

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
H1 : IsClosed[inst✝¹] s
H2 : IsCompact (closure[inst✝¹] sᶜ)
H3 : IsClosed[inst✝¹] t
H4 : Disjoint s t
⊢ SeparatedNhds s t

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
s t : Set X
hs : s.Finite
ht : t.Finite
h : Disjoint s t
⊢ SeparatedNhds s t

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
⊢ ContinuousOn (update f x y) s ↔ ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y))

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s : Set X
hs : s.Subsingleton
⊢ IsClosed[inst✝¹] s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s : Set X
hs : s.Subsingleton
⊢ (closure[inst✝¹] s).Subsingleton

Tactic:
by

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

Goal state before:
pp: X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space Y
y : Y
s : Set X
x✝ : IsClosed[inst✝²] s
h2s : s.Nonempty
⊢ IsClosed[inst✝¹] (const X y '' s)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
s : Set X
hxy : x ≠ y
⊢ 𝓝[insert y s] x = 𝓝[s] x

Tactic:
by

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

Goal state before:
pp: ι : Type u_4
X : ι → Type u_5
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), T2Space (X i)
s : Set ((i : ι) → X i)
⊢ IsCompact s ↔ IsClosed[topologicalSpace] s ∧ ∀ (i : ι), IsCompact (eval i '' s)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
s t : Set X
hu : t ⊆ insert y s
⊢ insert x s ∈ 𝓝[t] x

Tactic:
by

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

Goal state before:
pp: ι : Type u_4
X : ι → Type u_5
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), R1Space (X i)
s : Set ((i : ι) → X i)
⊢ IsCompact (closure[topologicalSpace] s) ↔ ∀ (i : ι), IsCompact (closure[inst✝¹ i] (eval i '' s))

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
x y : X
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ insert x s =ᶠ[𝓝 x] insert x t

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
⊢ (𝓝 x).ker = {x}

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
ι : Sort u_3
p : ι → Prop
s : ι → Set X
x : X
h : (𝓝 x).HasBasis p s
⊢ ⋂ i, ⋂ (_ : p i), s i = {x}

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x : X
s : Set X
⊢ {x}ᶜ ∈ 𝓝ˢ s ↔ x ∉ s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
h : ∀ x ∈ 𝓝ˢ t, x ∈ 𝓝ˢ s
x : X
hx : x ∈ s
⊢ x ∈ t

Tactic:
specialize h {x}ᶜ

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
x : X
h : {x}ᶜ ∈ 𝓝ˢ t → {x}ᶜ ∈ 𝓝ˢ s
hx : x ∈ s
⊢ x ∈ t
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
⊢ 𝓝ˢ s ≤ 𝓝ˢ t ↔ s ⊆ t

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
⊢ 𝓝ˢ s = 𝓝ˢ t ↔ s = t

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
⊢ 𝓝ˢ s = 𝓝ˢ t ↔ s = t

Tactic:
by
  simp_rw [le_antisymm_iff]
  exact and_congr nhdsSet_le_iff nhdsSet_le_iff

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s : Set X
x : X
⊢ 𝓝 x ≤ 𝓝ˢ s ↔ x ∈ s

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
x : X
f g : X → Y
hf : ContinuousAt f x
hg : ContinuousAt g x
⊢ f x ≠ g x ↔ ∀ᶠ (x : X) in 𝓝 x, f x ≠ g x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s U : Set X
⊢ s ∈ codiscreteWithin U ↔ ∀ z ∈ U, ∃ t ∈ 𝓝 z, (t ∩ (U \ s)).Finite

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T1Space X
inst✝ : ∀ (x : X), (𝓝[≠] x).NeBot
s : Set X
hs : Dense s
t : Finset X
⊢ Dense (s \ ↑t)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
x : X
f g : X → Y
hf : ContinuousAt f x
hg : ContinuousAt g x
inst✝ : (𝓝[≠] x).NeBot
hfg : f =ᶠ[𝓝[≠] x] g
hCon : ¬f x = g x
h₁ : ∀ᶠ (x : X) in 𝓝[≠] x, f x ≠ g x ∧ f x = g x
⊢ ∅ ∉ 𝓝[≠] x

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T1Space X
inst✝ : ∀ (x : X), (𝓝[≠] x).NeBot
s : Set X
hs : Dense s
t : Set X
ht : t.Finite
⊢ Dense (s \ t)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
x : X
f g : X → Y
hf : ContinuousAt f x
hg : ContinuousAt g x
inst✝ : (𝓝[≠] x).NeBot
hfg : f =ᶠ[𝓝[≠] x] g
hCon : ¬f x = g x
⊢ {x | f x ≠ g x ∧ f x = g x}.Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : ∀ (x : α), (𝓝[<] x).NeBot
s : Set α
hs : IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s
⊢ interior s = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
h : t.Subsingleton
z : X
hz : z ∈ t
⊢ univ \ t ∩ (t \ s) = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : PartialOrder α
inst✝ : TopologicalSpace β
a : α
f : α → β
⊢ ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : PartialOrder α
inst✝ : TopologicalSpace β
a : α
f : α → β
s : Set α
⊢ ContinuousWithinAt f (s ∩ Ioi a) a ↔ ContinuousWithinAt f (s ∩ Ici a) a

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space Y
x : X
f g : X → Y
hf : ContinuousAt f x
hg : ContinuousAt g x
inst✝ : (𝓝[≠] x).NeBot
⊢ f =ᶠ[𝓝[≠] x] g ↔ f =ᶠ[𝓝 x] g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
h : t.Subsingleton
⊢ s ∈ codiscreteWithin t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
a : α
⊢ 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
s : Set α
a : α
⊢ 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
S : Set X
⊢ IsPreirreducible S ↔ ∀ x ∈ S, S ⊆ closure[inst✝¹] {x}

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
S : Set X
⊢ IsPreirreducible S ↔ S.Subsingleton

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
a : α
⊢ 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
S : Set X
⊢ IsIrreducible S ↔ ∃ x, S = {x}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s : Set X
x z : X
hz : z ∈ s
⊢ univ \ {x} ∩ (s \ {x}ᶜ) = ∅

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s : Set X
x : X
⊢ {x}ᶜ ∈ codiscreteWithin s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
s : Set α
a : α
⊢ 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ici a] a = 𝓝[s] a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
a : α
⊢ 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t✝ : Set X
h : t✝.Finite
τ : X
t : Set X
hτ : τ ∉ t
h₁t : t.Finite
h₂t : tᶜ ∈ codiscreteWithin s
⊢ (insert τ t)ᶜ = {τ}ᶜ ∩ tᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
s : Set α
a : α
⊢ 𝓝[s ∩ Iic a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s] a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
h : t.Finite
⊢ tᶜ ∈ codiscreteWithin s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
a : α
⊢ 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
⊢ S ∈ codiscrete X ↔ ∀ (x : X), Disjoint (𝓝[≠] x) (𝓟 Sᶜ)

Tactic:
by

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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
a : X
p : X → Prop
hs : IsOpen[inst✝] s
⊢ (∀ᶠ (y : X) in 𝓝[s] a, ∀ᶠ (x : X) in 𝓝 y, p x) ↔ ∀ᶠ (x : X) in 𝓝[s] a, p x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
p : X
s : Set X
hs : Disjoint (𝓝[s] p) cofinite
t : Set X
h₁t : t ∈ 𝓝[s] p
h₂t : t.Finite
S : Set X := {y | y ∈ t ∩ s ∧ ¬y ⤳ p}
y : X
hy : y ∈ S
⊢ p ∈ (closure[inst✝] {y})ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
p : X
s : Set X
hs : Disjoint (𝓝[s] p) cofinite
t : Set X
h₁t : t ∈ 𝓝[s] p
h₂t : t.Finite
S : Set X := {y | y ∈ t ∩ s ∧ ¬y ⤳ p}
hS_nhds : ∀ y ∈ S, (closure[inst✝] {y})ᶜ ∈ 𝓝 p
⊢ S ⊆ t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
s : Set α
a : α
⊢ 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[s \ {a}] a

Tactic:
by

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

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T1Space X
x y : X
s : Set X
f : X → Y
⊢ ContinuousWithinAt f (insert y s) x ↔ ContinuousWithinAt f s x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.LeftRight

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : LinearOrder α
a : α
⊢ 𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a

Tactic:
by

Goal state after: