mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ Tendsto f (map (fun a ↦ (a, x.2)) (𝓝 x.1 ⊓ 𝓟 ((fun a ↦ (a, x.2)) ⁻¹' s))) (𝓝 (f (x.1, x.2))) ↔
    Tendsto (fun x_1 ↦ f (x_1, x.2)) (𝓝 x.1 ⊓ 𝓟 {a | (a, x.2) ∈ s}) (𝓝 (f (x.1, x.2)))

Tactic:
rfl

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)
⊢ (∀ (a : α) (b : β), (a, b) ∈ s → ContinuousWithinAt (fun x ↦ f (a, x)) {b | (a, b) ∈ s} b) ↔
    ∀ (a : α), ∀ x ∈ {b | (a, b) ∈ s}, ContinuousWithinAt (fun x ↦ f (a, x)) {b | (a, b) ∈ s} x

Tactic:
rfl

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
⊢ (∀ (a : α) (b : β), 𝓝 (f (a, b)) ≤ map (f ∘ Prod.mk a) (𝓝 b)) ↔
    ∀ (a : α) (x : β), 𝓝 (f (a, x)) ≤ map (fun x ↦ f (a, x)) (𝓝 x)

Tactic:
rfl

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
⊢ (∀ (b : β) (a : α), 𝓝 (f (a, b)) ≤ map (f ∘ fun a ↦ (a, b)) (𝓝 a)) ↔
    ∀ (b : β) (x : α), 𝓝 (f (x, b)) ≤ map (fun x ↦ f (x, b)) (𝓝 x)

Tactic:
rfl

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
g : β → γ
hg : IsInducing g
s : Set α
x : α
⊢ Tendsto (g ∘ f) (𝓝[s] x) (𝓝 (g (f x))) ↔ Tendsto (g ∘ f) (𝓝[s] x) (𝓝 ((g ∘ f) x))

Tactic:
rfl

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
g : β → γ
h : IsQuotientMap f
s : Set β
hs : IsOpen[inst✝¹] s
⊢ Continuous[instTopologicalSpaceSubtype, inst✝] (s.domRestrict g ∘ s.restrictPreimage f) ↔
    Continuous[instTopologicalSpaceSubtype, inst✝] ((f ⁻¹' s).domRestrict (g ∘ f))

Tactic:
rfl

Goal state after:
Module: Mathlib.Order.SetAccumulate

Goal state before:
pp: α : Type u_1
s : ℕ → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
n m : ℕ
hm : accumulate s n ⊆ s m
k : ℕ
hk : s m ⊆ s k ∧ s (n + 1) ⊆ s k
⊢ accumulate s n ⊆ s k ∧ s (n + 1) ⊆ s k

Tactic:
grind

Goal state after:
Module: Mathlib.Order.SetAccumulate

Goal state before:
pp: α : Type u_1
s : ℕ → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
h : ∀ (n : ℕ), s n ≠ univ
n m : ℕ
hm : accumulate s n ⊆ s m
⊢ accumulate s n ≠ univ

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
inst✝² : TopologicalSpace β
inst✝¹ : SeparableSpace α
inst✝ : SeparableSpace β
s : Set α
hsc : s.Countable
hsd : Dense s
t : Set β
htc : t.Countable
htd : Dense t
⊢ Dense (Sum.inl '' s ∪ Sum.inr '' t)

Tactic:
dense_iff_closure_eq,

Goal state after:
α : Type u
β : Type u_1
t✝ : TopologicalSpace α
B : Set (Set α)
s✝ : Set α
inst✝² : TopologicalSpace β
inst✝¹ : SeparableSpace α
inst✝ : SeparableSpace β
s : Set α
hsc : s.Countable
hsd : Dense s
t : Set β
htc : t.Countable
htd : Dense t
⊢ closure[instTopologicalSpaceSum] (Sum.inl '' s ∪ Sum.inr '' t) = univ
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set α
hs : Dense s
⊢ IsSeparable s ↔ SeparableSpace α

Tactic:
IsSeparable,

Goal state after:
α : Type u
t : TopologicalSpace α
s : Set α
hs : Dense s
⊢ (∃ c, c.Countable ∧ s ⊆ closure[t] c) ↔ SeparableSpace α
Module: Mathlib.Topology.Bases

Goal state before:
pp: α : Type u
t : TopologicalSpace α
s : Set α
hs : Dense s
⊢ (∃ c, c.Countable ∧ s ⊆ closure[t] c) ↔ ∃ s, s.Countable ∧ Dense s

Tactic:
dense_iff_closure_eq,

Goal state after:
α : Type u
t : TopologicalSpace α
s : Set α
hs : Dense s
⊢ (∃ c, c.Countable ∧ s ⊆ closure[t] c) ↔ ∃ s, s.Countable ∧ closure[t] s = univ
Module: Mathlib.Order.Filter.Ultrafilter.Defs

Goal state before:
pp: α : Type u
β : Type v
γ : Type u_1
f✝ g : Ultrafilter α
s t : Set α
p q : α → Prop
f : Filter α
h : ∀ (s : Set α), sᶜ ∉ f ↔ s ∈ f
hf : f = ⊥
⊢ False

Tactic:
by simp [hf] at h

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)
X : Type u_9
f : X → (i : ι) → A i
⊢ ⨅ i, induced ((fun f ↦ f i) ∘ f) (T i) = ⨅ i, induced (fun x ↦ f x i) inferInstance

Tactic:
Function.comp_def

Goal state after:
Module: Mathlib.Topology.Constructions

Goal state before:
pp: ι : Type u_5
σ : ι → Type u_7
inst✝ : (i : ι) → TopologicalSpace (σ i)
s : Set (Sigma σ)
⊢ (∀ (i : ι), IsOpen[coinduced (Sigma.mk i) (inst✝ i)] s) ↔ ∀ (i : ι), IsOpen[inst✝ i] (Sigma.mk i ⁻¹' s)

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: case refine_2.h₁
α : Type u
t✝ : TopologicalSpace α
inst✝ : SecondCountableTopology α
t : Set (Set α)
ht : IsTopologicalBasis t
u : Set α
hu : IsOpen[t✝] u
a : α → Set α
hat : ∀ x ∈ u, a x ∈ t
xa : ∀ x ∈ u, x ∈ a x
au : ∀ x ∈ u, a x ⊆ u
T : Set ↑u
T_count : T.Countable
hT : ⋃ i ∈ T, a ↑i = ⋃ i, a ↑i
x : α
hx : x ∈ u
⊢ ∃ i ∈ u, x ∈ a i

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: case refine_2.h₂
α : Type u
t✝ : TopologicalSpace α
inst✝ : SecondCountableTopology α
t : Set (Set α)
ht : IsTopologicalBasis t
u : Set α
hu : IsOpen[t✝] u
a : α → Set α
hat : ∀ x ∈ u, a x ∈ t
xa : ∀ x ∈ u, x ∈ a x
au : ∀ x ∈ u, a x ⊆ u
T : Set ↑u
T_count : T.Countable
hT : ⋃ i ∈ T, a ↑i = ⋃ i, a ↑i
⊢ ∀ i ∈ u, a i ⊆ u

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: case h_open
α : Type u
t✝ : TopologicalSpace α
inst✝ : SecondCountableTopology α
t : Set (Set α)
ht : IsTopologicalBasis t
s : Set α → Set (Set α)
hst : ∀ u ∈ countableBasis α, s u ⊆ t
s_count : ∀ u ∈ countableBasis α, (s u).Countable
hs : ∀ u ∈ countableBasis α, u = ⋃ a ∈ s u, a
this : ∀ s ∈ t, IsOpen[t✝] s
⊢ ∀ (u x : Set α), x ∈ countableBasis α → u ∈ s x → IsOpen[t✝] u

Tactic:
grind

Goal state after:
Module: Mathlib.Order.Filter.SmallSets

Goal state before:
pp: α : Type u_1
f : Filter α
⊢ f.lift' powerset = ⨅ b ∈ f.sets, 𝓟 (𝒫 b)

Tactic:
Filter.lift',

Goal state after:
α : Type u_1
f : Filter α
⊢ f.lift (𝓟 ∘ powerset) = ⨅ b ∈ f.sets, 𝓟 (𝒫 b)
Module: Mathlib.Order.Filter.SmallSets

Goal state before:
pp: α : Type u_1
L : Filter (Set α)
l : Filter α
⊢ L.bind 𝓟 ≤ l ↔ l.sets ⊆ powerset ⁻¹' L.sets

Tactic:
rfl

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocallyCompactSpace X
K : Set X
hK : IsCompact K
⊢ ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ r ∈ 𝓝ˢ K, IsCompact r ∧ r ⊆ s) → ∃ r ∈ 𝓝ˢ K, IsCompact r ∧ r ⊆ t

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Bases

Goal state before:
pp: case a.h
α : Type u_1
ts : TopologicalSpace α
inst✝ : SecondCountableTopology α
t : Set (Set α)
ht : ts = generateFrom t
t' : Set (Set α) := (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ t}
this : IsTopologicalBasis t'
s' : Set (Set α)
s't' : s' ⊆ t'
s'_count : s'.Countable
hs' : IsTopologicalBasis s'
f : Set α → Set (Set α)
f_fin : ∀ u ∈ s', (f u).Finite
ft : ∀ u ∈ s', f u ⊆ t
hf : ∀ u ∈ s', ⋂₀ f u = u
u : Set α
hu : u ∈ s'
i : Set α
hi : i ∈ f u
⊢ ∃ i_1 ∈ s', i ∈ f i_1

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.LocallyFinite

Goal state before:
pp: ι : Type u_1
X : Type u_4
inst✝ : TopologicalSpace X
f : Option ι → Set X
⊢ LocallyFinite ((f ∘ ⇑(Equiv.optionEquivSumPUnit ι).symm) ∘ Sum.inl) ↔ LocallyFinite (f ∘ some)

Tactic:
rfl

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
f : Filter X
hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f
⊢ sᶜ ∈ f

Tactic:
by
  contrapose! hf
  simp only [notMem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢
  exact @hs _ hf inf_le_right

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

Goal state before:
pp: X : Type u_1
ι : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : SigmaCompactSpace X
f : ι → Set X
hf : LocallyFinite f
hne : ∀ (i : ι), (f i).Nonempty
⊢ univ.Countable

Tactic:
by
  have := fun n => hf.finite_nonempty_inter_compact (isCompact_compactCovering X n)
  refine (countable_iUnion fun n => (this n).countable).mono fun i _ => ?_
  rcases hne i with ⟨x, hx⟩
  rcases iUnion_eq_univ_iff.1 (iUnion_compactCovering X) x with ⟨n, hn⟩
  exact mem_iUnion.2 ⟨n, x, hx, hn⟩

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

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : DiscreteTopology X
hs : IsCompact s
⊢ s.Finite

Tactic:
have : ∀ x : X, ({x} : Set X) ∈ 𝓝 x := by simp [nhds_discrete]
  rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, _, hst⟩
  simp only [← t.set_biUnion_coe, biUnion_of_singleton] at hst
  exact t.finite_toSet.subset hst

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

Goal state before:
pp: X : Type u
T : TopologicalSpace X
S : Set (Set X)
hTS : T = generateFrom S
s : Set X
h : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q
F : Ultrafilter X
hsF : s ∈ F
hF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x
U : X → Set X
hxU : ∀ x ∈ s, x ∈ U x
hSU : ∀ x ∈ s, U x ∈ S
hUF : ∀ x ∈ s, U x ∉ F
x : X
hx : x ∈ s
⊢ U x ∈ U '' s

Tactic:
grind

Goal state after:
Module: Mathlib.Order.UpperLower.CompleteLattice

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
s : UpperSet α
b : β
⊢ b ∈ (map f.symm).symm s ↔ f.symm b ∈ s

Tactic:
rfl

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

Goal state before:
pp: X : Type u
T : TopologicalSpace X
𝔅 : Set (Set X)
hT : T = generateFrom 𝔅
h𝔅 : ∀ s ∈ 𝔅, sᶜ ∈ 𝔅
h : ∀ P ⊆ 𝔅, (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) → (⋂₀ P).Nonempty
⊢ CompactSpace X

Tactic:
refine compactSpace_generateFrom hT fun P hP𝔅 hP ↦ ?_

Goal state after:
X : Type u
T : TopologicalSpace X
𝔅 : Set (Set X)
hT : T = generateFrom 𝔅
h𝔅 : ∀ s ∈ 𝔅, sᶜ ∈ 𝔅
h : ∀ P ⊆ 𝔅, (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) → (⋂₀ P).Nonempty
P : Set (Set X)
hP𝔅 : P ⊆ 𝔅
hP : ⋃₀ P = univ
⊢ ∃ Q ⊆ P, Q.Finite ∧ ⋃₀ Q = univ
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
⊢ (∃ s, IsOpen[inst✝] s ∧ y ∈ s ∧ x ∉ s) ↔ ∃ s, IsOpen[inst✝] s ∧ y ∈ s ∧ x ∉ s

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x y : X
⊢ (∃ s, IsClosed[inst✝] s ∧ x ∈ s ∧ y ∉ s) ↔ ∃ s, IsClosed[inst✝] s ∧ x ∈ s ∧ y ∉ s

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
⊢ (∀ ⦃a : X⦄ ⦃b : Y⦄, b ∈ closure[inst✝] {f a} → ∃ a' ∈ closure[inst✝¹] {a}, f a' = b) ↔
    ∀ (x : X), closure[inst✝] {f x} ⊆ f '' closure[inst✝¹] {x}

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsClosed[inst✝] s
⊢ mk ⁻¹' mk '' s = s

Tactic:
refine Subset.antisymm ?_ (subset_preimage_image _ _)
  rintro x ⟨y, hys, hxy⟩
  exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
Y : Type u_2
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf) ∘ Prod.map mk mk) (𝓝 (x, y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
Y : Type u_2
α : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y → α
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
x : X
y : Y
s : Set (SeparationQuotient X × SeparationQuotient Y)
l : Filter α
⊢ Tendsto (uncurry (lift₂ f hf)) (map (Prod.map mk mk) (𝓝 (x, y) ⊓ 𝓟 (Prod.map mk mk ⁻¹' s))) l ↔
    Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.Inseparable

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y → Z
hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d
s : Set (SeparationQuotient X × SeparationQuotient Y)
⊢ (∀ (a : X) (b : Y), (mk a, mk b) ∈ s → ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (a, b)) ↔
    ∀ (a : X) (b : Y), (a, b) ∈ Prod.map mk mk ⁻¹' s → ContinuousWithinAt (uncurry f) (Prod.map mk mk ⁻¹' s) (a, b)

Tactic:
rfl

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: Y : Type u_2
inst✝ : TopologicalSpace Y
S : Set Y
⊢ (∀ a ∈ S, ∃ t, IsOpen[inst✝] t ∧ ∀ a_1 ∈ S, a_1 ∈ t ↔ a_1 = a) ↔
    ∀ y ∈ S, ∃ U, IsOpen[inst✝] U ∧ ∀ (x : Y), x ∈ U ∧ x ∈ S ↔ x = y

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ t ⊆ S, ∃ u, IsOpen[inst✝] u ∧ u ∩ S = t
s : Set X
sS : s ⊆ S
U : Set X
Uo : IsOpen[inst✝] U
Us : U ∩ S = sᶜ ∩ S
⊢ Uᶜ ∩ S = s

Tactic:
by rw [left_eq_inter.mpr sS]; simp_all [Set.ext_iff]

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
h : ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
s : Set X
sS : s ⊆ S
U : Set X
Uo : IsClosed[inst✝] U
Us : U ∩ S = sᶜ ∩ S
⊢ Uᶜ ∩ S = s

Tactic:
by rw [left_eq_inter.mpr sS]; simp_all [Set.ext_iff]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∃ x, ↑s = {x}

Tactic:
refine minimal_nonempty_open_eq_singleton ho hne ?_

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T0Space X
s : Finset X
ihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen[inst✝¹] ↑t → ∃ x ∈ ↑t, IsOpen[inst✝¹] {x}
hne : (↑s).Nonempty
ho : IsOpen[inst✝¹] ↑s
ht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen[inst✝¹] ↑t
⊢ ∀ t ⊆ ↑s, t.Nonempty → IsOpen[inst✝¹] t → t = ↑s
Module: Mathlib.Topology.Separation.Basic

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

Tactic:
nhdsWithin_inter_of_mem

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
h : x ≠ y
s : Set X
⊢ 𝓝[s] x = 𝓝[s] x

X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
h : x ≠ y
s : Set X
⊢ {y}ᶜ ∈ 𝓝[s] x
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_3
inst✝ : TopologicalSpace X
⊢ (∀ (U : Set X), IsOpen[inst✝] U → U.Nonempty → closure[inst✝] U = univ) ↔
    ∀ ⦃U : Set X⦄, IsOpen[inst✝] U → U.Nonempty → ∀ (x : X), x ∈ closure[inst✝] U

Tactic:
grind

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T2Space X
x : X
t : Set X
H1 : IsCompact t
H2 : x ∉ t
⊢ Disjoint (𝓝ˢ t) (𝓝 x)

Tactic:
simpa using SeparatedNhds.disjoint_nhdsSet <| .of_isCompact_isCompact_isClosed H1
    isCompact_singleton isClosed_singleton <| disjoint_singleton_right.mpr H2

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case h
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Y → X
h : IsOpenMap f
s : Set X
hne : (f ⁻¹' s).Nonempty
hs : IsPreirreducible s
hs' : IsClosed[inst✝¹] s
⊢ s ∩ range f ⊆ f '' f ⁻¹' s

Tactic:
grind

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:
intro h x

Goal state after:
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
⊢ IsClosed[inst✝] {x}
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:
grind

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:
grind

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:
grind

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:
grind

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
  rw [← h.ker, ker_nhds]

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:
rw [← nhdsSet_singleton, nhdsSet_le_iff, singleton_subset_iff]

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
h : ∀ᶠ (y : X) in 𝓝[s] a, ∀ᶠ (x : X) in 𝓝 y, p x
a✝ : X
hy : ∀ᶠ (x : X) in 𝓝 a✝, p x
⊢ ∀ᶠ (x : X) in 𝓝[s] a✝, p x

Tactic:
exact eventually_nhdsWithin_of_eventually_nhds hy

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:
grind

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

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
hs : IsDiscrete s
x : X
hx : x ∈ s
U : Set X
hU_nhds : U ∈ 𝓝 x
hU_inter : U ∩ s = {x}
t : Set X
ht_sub : t ⊆ U
ht_open : IsOpen[inst✝] t
ht_x : x ∈ t
⊢ ∃ U, IsOpen[inst✝] U ∧ U ∩ s = {x}

Tactic:
grind

Goal state after:
Module: Mathlib.Topology.Clopen

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s U : Set X
⊢ Continuous[instTopologicalSpaceSubtype, _] (s.domRestrict U.boolIndicator) ↔
    Continuous[instTopologicalSpaceSubtype, _] (Subtype.val ⁻¹' U).boolIndicator

Tactic:
rfl

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

Goal state before:
pp: α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderClosedTopology α
f g : β → α
inst✝ : TopologicalSpace β
hf : Continuous[inst✝, inst✝³] f
hg : Continuous[inst✝, inst✝³] g
b : β
hb₁ : b ∈ {b | f b ≤ g b}
hb₂ : b ∈ closure[inst✝] {b | f b ≤ g b}ᶜ
⊢ {b | ¬f b ≤ g b} = {b | f b ≤ g b}ᶜ

Tactic:
rfl

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✝ : R1Space Y
f : X → Y
x : X
K : Set X
s : Set Y
hf : Continuous[inst✝², inst✝¹] f
hs : s ∈ 𝓝 (f x)
hKc : IsCompact K
hKx : K ∈ 𝓝 x
hc : IsCompact (f '' K \ interior s)
⊢ SeparatedNhds {f x} (f '' K \ interior s)

Tactic:
simp_rw [separatedNhds_iff_disjoint, nhdsSet_singleton, hc.disjoint_nhdsSet_right,
      disjoint_nhds_nhds_iff_not_inseparable]

Goal state after:
X : Type u_3
Y : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : R1Space Y
f : X → Y
x : X
K : Set X
s : Set Y
hf : Continuous[inst✝², inst✝¹] f
hs : s ∈ 𝓝 (f x)
hKc : IsCompact K
hKx : K ∈ 𝓝 x
hc : IsCompact (f '' K \ interior s)
⊢ ∀ x_1 ∈ f '' K \ interior s, ¬Inseparable (f x) x_1
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
t : Set ι
s : ι → Set α
H : ∀ i ∈ t, IsPreconnected (s i)
K : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ t) i j
R : ι → ι → Prop := fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ t
P : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen R i j → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)
x : α
hx : x ∈ ⋃ n ∈ t, s n
y : α
hy : y ∈ ⋃ n ∈ t, s n
i : ι
hi : i ∈ t
hxi : x ∈ s i
⊢ ∃ t_1 ⊆ ⋃ n ∈ t, s n, x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1

Tactic:
obtain ⟨j : ι, hj : j ∈ t, hyj : y ∈ s j⟩ := mem_iUnion₂.1 hy

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
t : Set ι
s : ι → Set α
H : ∀ i ∈ t, IsPreconnected (s i)
K : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ t) i j
R : ι → ι → Prop := fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ t
P : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen R i j → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)
x : α
hx : x ∈ ⋃ n ∈ t, s n
y : α
hy : y ∈ ⋃ n ∈ t, s n
i : ι
hi : i ∈ t
hxi : x ∈ s i
j : ι
hj : j ∈ t
hyj : y ∈ s j
⊢ ∃ t_1 ⊆ ⋃ n ∈ t, s n, x ∈ t_1 ∧ y ∈ t_1 ∧ IsPreconnected t_1
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ LocallyConnectedSpace α ↔ ∀ (F : Set α), IsOpen[inst✝] F → ∀ x ∈ F, IsOpen[inst✝] (connectedComponentIn F x)

Tactic:
by
  constructor
  · intro h
    exact fun F hF x _ => hF.connectedComponentIn
  · intro h
    rw [locallyConnectedSpace_iff_subsets_isOpen_isConnected]
    refine fun x U hU =>
        ⟨connectedComponentIn (interior U) x,
          (connectedComponentIn_subset _ _).trans interior_subset, h _ isOpen_interior x ?_,
          mem_connectedComponentIn ?_, isConnected_connectedComponentIn_iff.mpr ?_⟩ <;>
      exact mem_interior_iff_mem_nhds.mpr hU

Goal state after: