mathlib-initiative/mathlib-tactics huggingface dataset:
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:
]

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

Goal state before:
pp: case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
K : Set X × Set Y
hK : IsCompact K.1 ∧ IsCompact K.2
⊢ (K.1 ×ˢ K.2)ᶜ ∈ cocompact (X × Y)

Tactic:
]

Goal state after:
case a
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
K : Set X × Set Y
hK : IsCompact K.1 ∧ IsCompact K.2
⊢ (K.1 ×ˢ K.2)ᶜ ∈ cocompact (X × Y)
Module: Mathlib.Topology.Compactness.Compact

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
⊢ (cocompact X).coprod (cocompact Y) = cocompact (X × Y)

Tactic:
apply le_antisymm
  · exact sup_le (comap_cocompact_le continuous_fst) (comap_cocompact_le continuous_snd)
  · refine (hasBasis_cocompact.coprod hasBasis_cocompact).ge_iff.2 fun K hK ↦ ?_
    rw [← univ_prod, ← prod_univ, ← compl_prod_eq_union]
    exact (hK.1.prod hK.2).compl_mem_cocompact

Goal state after:
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
ht : s ∩ frontier t = s' ∩ frontier t
⊢ s ∩ t ∪ s ∩ frontier t = s ∩ t ∪ s ∩ frontier t

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
ht : s ∩ frontier t = s' ∩ frontier t
⊢ s' ∩ closure[inst✝] tᶜ = s' ∩ closure[inst✝] tᶜ

pp: α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
ht : s ∩ frontier t = s' ∩ frontier t
⊢ s' ∩ frontier tᶜ = s ∩ frontier tᶜ

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
ht : s ∩ frontier t = s' ∩ frontier t
⊢ s' ∩ frontier tᶜ = s ∩ frontier tᶜ
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
ht : s ∩ frontier t = s' ∩ frontier t
⊢ s ∩ frontier t = s' ∩ frontier t

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
s s' t : Set α
ht : s ∩ frontier t = s' ∩ frontier t
⊢ s ∩ frontier t = s' ∩ frontier t
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case ht
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ∀ a ∈ s ∩ frontier t, f a = g a

Tactic:
]

Goal state after:
case ht
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ∀ a ∈ s ∩ frontier t, f a = g a
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case hf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn f (s ∩ closure[inst✝²] t)

Tactic:
]

Goal state after:
case hf
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn f (s ∩ closure[inst✝²] t)
Module: Mathlib.Topology.Compactness.Compact

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✝
f : X✝ → Y
X : ι → Type u_2
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
⊢ IsCompact (univ.pi fun i ↦ univ)

Tactic:
]

Goal state after:
X✝ : Type u
Y : Type v
ι : Type u_1
inst✝³ : TopologicalSpace X✝
inst✝² : TopologicalSpace Y
s t : Set X✝
f : X✝ → Y
X : ι → Type u_2
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : ∀ (i : ι), CompactSpace (X i)
⊢ IsCompact (univ.pi fun i ↦ univ)
Module: Mathlib.Topology.Piecewise

Goal state before:
pp: case hg
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)

Tactic:
]

Goal state after:
case hg
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f g : α → β
s s' t : Set α
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure[inst✝²] t)
h' : ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f g (s ∩ frontier t)
⊢ ContinuousOn g (s' ∩ closure[inst✝²] tᶜ)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: Y : Type u_2
inst✝ : TopologicalSpace Y
S : Set Y
⊢ (∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥) ↔ ∀ x ∈ S, 𝓝[≠] x ⊓ 𝓟 S = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: Y : Type u_2
inst✝ : TopologicalSpace Y
s : Set Y
⊢ (∀ y ∈ s, ∃ U, IsOpen[inst✝] U ∧ U ∩ s = {y}) ↔ ∀ y ∈ s, ∃ u, IsOpen[inst✝] u ∧ u ∩ s = {y}

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ (s ∩ u).Nonempty → (closure[inst✝] s ∩ v).Nonempty → (closure[inst✝] s ∩ (u ∩ v)).Nonempty ↔
    (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty

pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] u

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ (s ∩ u).Nonempty → (closure[inst✝] s ∩ v).Nonempty → (closure[inst✝] s ∩ (u ∩ v)).Nonempty ↔
    (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty

X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] u
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ (s ∩ u).Nonempty → (s ∩ v).Nonempty → (closure[inst✝] s ∩ (u ∩ v)).Nonempty ↔
    (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty

pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] v

pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] u

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ (s ∩ u).Nonempty → (s ∩ v).Nonempty → (closure[inst✝] s ∩ (u ∩ v)).Nonempty ↔
    (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty

X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] v

X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] u
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty ↔
    (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty

pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] (u ∩ v)

pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] v

pp: X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] u

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] (u ∩ v)

X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] v

X : Type u_1
inst✝ : TopologicalSpace X
s u v : Set X
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
⊢ IsOpen[inst✝] u
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
H : s ∈ irreducibleComponents X
⊢ s = closure[inst✝] s

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
H : s ∈ irreducibleComponents X
⊢ s = closure[inst✝] s
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
⊢ (∀ t ⊆ S, ∃ u, IsOpen[inst✝] u ∧ u ∩ S = t) ↔ ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
S : Set X
⊢ (∀ t ⊆ S, ∃ u, IsOpen[inst✝] u ∧ u ∩ S = t) ↔ ∀ s ⊆ S, ∃ U, IsClosed[inst✝] U ∧ U ∩ S = s
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : X
hx : x ∈ s
hxu : x ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
⊢ (f '' s ∩ (u ∩ v)).Nonempty

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : X
hx : x ∈ s
hxu : x ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
⊢ (f '' s ∩ (u ∩ v)).Nonempty
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 ∩ S

Tactic:
]

Goal state after:
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 ∩ S
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : X
hx : x ∈ s
hxu : x ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (s ∩ v').Nonempty → (s ∩ (u' ∩ v')).Nonempty
⊢ (f '' s ∩ (u ∩ v)).Nonempty

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : X
hx : x ∈ s
hxu : x ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (s ∩ v').Nonempty → (s ∩ (u' ∩ v')).Nonempty
⊢ (f '' s ∩ (u ∩ v)).Nonempty
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 ∩ S

Tactic:
]

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

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
f : X → Y
r : X → X → Prop
inst✝ : CompactSpace X
⊢ IsCompact (range (mk r))

Tactic:
]

Goal state after:
X : Type u
Y : Type v
ι : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
f : X → Y
r : X → X → Prop
inst✝ : CompactSpace X
⊢ IsCompact (range (mk r))
Module: Mathlib.Topology.Compactness.Compact

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
f : X → Y
r : X → X → Prop
inst✝ : CompactSpace X
⊢ IsCompact (range (mk r))

Tactic:
exact isCompact_range continuous_quot_mk

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 : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : X
hx : x ∈ s
hxu : x ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (f ⁻¹' v ∩ s).Nonempty → (s ∩ (u' ∩ v')).Nonempty
⊢ (f '' s ∩ (u ∩ v)).Nonempty

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x : X
hx : x ∈ s
hxu : x ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (f ⁻¹' v ∩ s).Nonempty → (s ∩ (u' ∩ v')).Nonempty
⊢ (f '' s ∩ (u ∩ v)).Nonempty
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
sd : s ⊆ t
ht : IsDiscrete t
tc : IsClosed[inst✝] t
w✝ : Set X
rp : IsClosed[inst✝] w✝
rt : w✝ ∩ t = s
⊢ IsClosed[inst✝] (w✝ ∩ t)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
sd : s ⊆ t
ht : IsDiscrete t
tc : IsClosed[inst✝] t
w✝ : Set X
rp : IsClosed[inst✝] w✝
rt : w✝ ∩ t = s
⊢ IsClosed[inst✝] (w✝ ∩ t)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
⊢ DiscreteTopology X

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
⊢ DiscreteTopology X
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hs : IsDiscrete s
hf : IsOpenMap f
hf' : Injective f
x : X
hx : x ∈ s
⊢ 𝓝 (f x) ⊓ 𝓟 (f '' s) ≤ map f (𝓝 x) ⊓ 𝓟 (f '' s)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hs : IsDiscrete s
hf : IsOpenMap f
hf' : Injective f
x : X
hx : x ∈ s
⊢ 𝓝 (f x) ⊓ 𝓟 (f '' s) ≤ map f (𝓝 x) ⊓ 𝓟 (f '' s)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hs : IsDiscrete s
hf : IsOpenMap f
hf' : Injective f
x : X
hx : x ∈ s
⊢ map f (𝓝 x) ⊓ 𝓟 (f '' s) ≤ map f (𝓝 x) ⊓ 𝓟 (f '' s)

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x✝ : X
hx : x✝ ∈ s
hxu : x✝ ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (f ⁻¹' v ∩ s).Nonempty → (s ∩ (u' ∩ v')).Nonempty
x : X
hxs : x ∈ s
hxu' : x ∈ u'
hxv' : x ∈ v'
⊢ x ∈ f ⁻¹' u

Tactic:
]

Goal state after:
case refine_1
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x✝ : X
hx : x✝ ∈ s
hxu : x✝ ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (f ⁻¹' v ∩ s).Nonempty → (s ∩ (u' ∩ v')).Nonempty
x : X
hxs : x ∈ s
hxu' : x ∈ u'
hxv' : x ∈ v'
⊢ x ∈ f ⁻¹' u
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hs : IsDiscrete s
hf : IsOpenMap f
hs' : IsOpen[inst✝¹] s
x : X
hx : x ∈ s
⊢ 𝓝 (f x) ≤ map f (𝓝 x)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
hs : IsDiscrete s
hf : IsOpenMap f
hs' : IsOpen[inst✝¹] s
x : X
hx : x ∈ s
⊢ 𝓝 (f x) ≤ map f (𝓝 x)
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
hf' : Injective f
x : X
hx : x ∈ f ⁻¹' s
⊢ Tendsto f (𝓝[f ⁻¹' s] x) (𝓝[s] f x)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
hf' : Injective f
x : X
hx : x ∈ f ⁻¹' s
⊢ Tendsto f (𝓝[f ⁻¹' s] x) (𝓝[s] f x)
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x✝ : X
hx : x✝ ∈ s
hxu : x✝ ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (f ⁻¹' v ∩ s).Nonempty → (s ∩ (u' ∩ v')).Nonempty
x : X
hxs : x ∈ s
hxu' : x ∈ u'
hxv' : x ∈ v'
⊢ x ∈ f ⁻¹' v

Tactic:
]

Goal state after:
case refine_2
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
H : IsPreirreducible s
f : X → Y
hf : ContinuousOn f s
u v : Set Y
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
x✝ : X
hx : x✝ ∈ s
hxu : x✝ ∈ f ⁻¹' u
y : X
hy : y ∈ s
hyv : y ∈ f ⁻¹' v
u' : Set X
hu' : IsOpen[inst✝¹] u'
u'_eq : f ⁻¹' u ∩ s = u' ∩ s
v' : Set X
hv' : IsOpen[inst✝¹] v'
v'_eq : f ⁻¹' v ∩ s = v' ∩ s
this : (f ⁻¹' u ∩ s).Nonempty → (f ⁻¹' v ∩ s).Nonempty → (s ∩ (u' ∩ v')).Nonempty
x : X
hxs : x ∈ s
hxu' : x ∈ u'
hxv' : x ∈ v'
⊢ x ∈ f ⁻¹' v
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
h : ∀ (x y : X), Inseparable x y → ∃ s, x ∈ s ∧ y ∈ s ∧ T0Space ↑s
s : Set X
hs : T0Space ↑s
x y : { x // x ∈ s }
hxy : Inseparable x y
⊢ ↑x = ↑y

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
h : ∀ (x y : X), Inseparable x y → ∃ s, x ∈ s ∧ y ∈ s ∧ T0Space ↑s
s : Set X
hs : T0Space ↑s
x y : { x // x ∈ s }
hxy : Inseparable x y
⊢ ↑x = ↑y
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : PreirreducibleSpace ↑s
⊢ IsPreirreducible (Subtype.val '' univ)

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
s : Set X
inst✝ : PreirreducibleSpace ↑s
⊢ IsPreirreducible (Subtype.val '' univ)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R0Space X
x : X
ι✝ : Type u_1
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen[inst✝¹] (U i)
hxU : closure[inst✝¹] {x} ⊆ ⋃ i, U i
i : ι✝
hi : x ∈ U i
y : X
hy : y ⤳ x
⊢ y ∈ ⋃ i_1 ∈ {i}, U i_1

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R0Space X
x : X
ι✝ : Type u_1
U : ι✝ → Set X
hUo : ∀ (i : ι✝), IsOpen[inst✝¹] (U i)
hxU : closure[inst✝¹] {x} ⊆ ⋃ i, U i
i : ι✝
hi : x ∈ U i
y : X
hy : y ⤳ x
⊢ y ∈ ⋃ i_1 ∈ {i}, U i_1
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
H : ∀ (x : Y), IsDiscrete (f ⁻¹' {x})
x : X
hx : x ∈ f ⁻¹' s
h : 𝓝[f ⁻¹' s] x ≤ pure x
⊢ 𝓝[f ⁻¹' s] x ≤ pure x

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set Y
hs : IsDiscrete s
hf : ContinuousOn f (f ⁻¹' s)
H : ∀ (x : Y), IsDiscrete (f ⁻¹' {x})
x : X
hx : x ∈ f ⁻¹' s
h : 𝓝[f ⁻¹' s] x ≤ pure x
⊢ 𝓝[f ⁻¹' s] x ≤ pure x
Module: Mathlib.Topology.Irreducible

Goal state before:
pp: case refine_1.insert
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsIrreducible s
u : Set X
U : Finset (Set X)
a✝ : u ∉ U
IH : (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty
hu : ∀ u_1 ∈ insert u U, IsOpen[inst✝] u_1
hU : ∀ u_1 ∈ insert u U, (s ∩ u_1).Nonempty
⊢ (s ∩ (u ∩ ⋂₀ ↑U)).Nonempty

Tactic:
]

Goal state after:
case refine_1.insert
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsIrreducible s
u : Set X
U : Finset (Set X)
a✝ : u ∉ U
IH : (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty
hu : ∀ u_1 ∈ insert u U, IsOpen[inst✝] u_1
hU : ∀ u_1 ∈ insert u U, (s ∩ u_1).Nonempty
⊢ (s ∩ (u ∩ ⋂₀ ↑U)).Nonempty
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[s] x = 𝓝[s] x

pp: 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

Tactic:
]

Goal state after:
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: case refine_1.insert
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsIrreducible s
u : Set X
U : Finset (Set X)
a✝ : u ∉ U
IH : (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty
hu : IsOpen[inst✝] u ∧ ∀ x ∈ U, IsOpen[inst✝] x
hU : (s ∩ u).Nonempty ∧ ∀ x ∈ U, (s ∩ x).Nonempty
⊢ (s ∩ (u ∩ ⋂₀ ↑U)).Nonempty

Tactic:
]

Goal state after:
case refine_1.insert
X : Type u_1
inst✝ : TopologicalSpace X
s : Set X
h : IsIrreducible s
u : Set X
U : Finset (Set X)
a✝ : u ∉ U
IH : (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty
hu : IsOpen[inst✝] u ∧ ∀ x ∈ U, IsOpen[inst✝] x
hU : (s ∩ u).Nonempty ∧ ∀ x ∈ U, (s ∩ x).Nonempty
⊢ (s ∩ (u ∩ ⋂₀ ↑U)).Nonempty
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace Y
f : X → Y
⊢ (∀ (i : Set Y), IsCompact i → ∀ᶠ (x : X) in cofinite, f x ∈ iᶜ) ↔ ∀ (K : Set Y), IsCompact K → (f ⁻¹' K).Finite

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace Y
f : X → Y
⊢ (∀ (i : Set Y), IsCompact i → ∀ᶠ (x : X) in cofinite, f x ∈ iᶜ) ↔ ∀ (K : Set Y), IsCompact K → (f ⁻¹' K).Finite
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
hy : x ≠ y
⊢ 𝓝[≠] x ≤ 𝓝 x

Tactic:
]

Goal state after:
case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
hy : x ≠ y
⊢ 𝓝[≠] x ≤ 𝓝 x