mathlib-initiative/mathlib-tactics huggingface dataset:
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
V : Set X
h : V ∈ 𝓝 x
h' : V ∩ s = {x}
⊢ ∅ = ∅

Tactic:
]

Goal state after:
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
K : Set X
inst✝ : T1Space X
hK : IsCompact K
⊢ codiscreteWithin K ≤ codiscrete X ⊓ 𝓟 K

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
K : Set X
inst✝ : T1Space X
hK : IsCompact K
⊢ codiscreteWithin K ≤ codiscrete X ⊓ 𝓟 K
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: ι : Type u_3
β : ι → Type u_4
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → TopologicalSpace (β i)
x : (i : ι) → β i
i : ι
inst✝ : ∀ (i : ι), T1Space (β i)
s : Set (β i)
hs : IsClosed[inst✝¹ i] s
⊢ IsClosed[Pi.topologicalSpace] (univ.pi (update (fun j ↦ {x j}) i s))

Tactic:
]

Goal state after:
ι : Type u_3
β : ι → Type u_4
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → TopologicalSpace (β i)
x : (i : ι) → β i
i : ι
inst✝ : ∀ (i : ι), T1Space (β i)
s : Set (β i)
hs : IsClosed[inst✝¹ i] s
⊢ IsClosed[Pi.topologicalSpace] (univ.pi (update (fun j ↦ {x j}) i s))
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsDiscrete s
ht✝ : IsDiscrete t
hsc : IsClosed[inst✝] s
ht : IsClosed[inst✝] t
⊢ IsDiscrete (⋃ b, bif b then s else t)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsDiscrete s
ht✝ : IsDiscrete t
hsc : IsClosed[inst✝] s
ht : IsClosed[inst✝] t
⊢ IsDiscrete (⋃ b, bif b then s else t)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
x y : X
⊢ ¬Inseparable x y ↔ ¬Inseparable x y

Tactic:
]

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
x y : X
h : ∃ i, (x ∈ i ∧ IsOpen[inst✝¹] i) ∧ ∃ i', (y ∈ i' ∧ IsOpen[inst✝¹] i') ∧ Disjoint i i'
⊢ ∃ u v, IsOpen[inst✝¹] u ∧ IsOpen[inst✝¹] v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
x y : X
h : ∃ i, (x ∈ i ∧ IsOpen[inst✝¹] i) ∧ ∃ i', (y ∈ i' ∧ IsOpen[inst✝¹] i') ∧ Disjoint i i'
⊢ ∃ u v, IsOpen[inst✝¹] u ∧ IsOpen[inst✝¹] v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v
Module: Mathlib.Topology.DiscreteSubset

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
S T : Set X
hs : IsDiscrete S
ht : IsDiscrete T
hs' : IsClosed[inst✝] S
ht' : IsClosed[inst✝] T
⊢ IsDiscrete (S ∪ T)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
S T : Set X
hs : IsDiscrete S
ht : IsDiscrete T
hs' : IsClosed[inst✝] S
ht' : IsClosed[inst✝] T
⊢ IsDiscrete (S ∪ T)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
K : Set X
hK : IsCompact K
U : Set X
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ i ∈ K, {y | Inseparable i y} ⊆ U

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
K : Set X
hK : IsCompact K
U : Set X
hU : IsOpen[inst✝¹] U
hKU : K ⊆ U
⊢ ∀ i ∈ K, {y | Inseparable i y} ⊆ U
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
K U V : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
h2K : K ⊆ U ∪ V
hK' : IsCompact (closure[inst✝¹] K)
⊢ closure[inst✝¹] K ⊆ U ∪ V

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
K U V : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
h2K : K ⊆ U ∪ V
hK' : IsCompact (closure[inst✝¹] K)
⊢ closure[inst✝¹] K ⊆ U ∪ V
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
K U V : Set X
hK : IsCompact K
hU : IsOpen[inst✝¹] U
hV : IsOpen[inst✝¹] V
h2K : K ⊆ U ∪ V
hK' : IsCompact (closure[inst✝¹] K)
this : SeparatedNhds (closure[inst✝¹] K \ U) (closure[inst✝¹] K \ V)
O₁ O₂ : Set X
h1O₁ : IsOpen[inst✝¹] O₁
h1O₂ : IsOpen[inst✝¹] O₂
h2O₁ : K \ U ⊆ O₁
h2O₂ : K \ V ⊆ O₂
hO : Disjoint O₁ O₂
⊢ K = K

Tactic:
]

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

Goal state before:
pp: case mp
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : ClosedIicTopology α
a : α
hd : (Iic a)ᶜ ∈ 𝓝 a
hbot : IsBot a
⊢ False

Tactic:
]

Goal state after:
case mp
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : Preorder α
inst✝ : ClosedIicTopology α
a : α
hd : (Iic a)ᶜ ∈ 𝓝 a
hbot : IsBot a
⊢ False
Module: Mathlib.Topology.Order.OrderClosed

Goal state before:
pp: α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
a : α
f : ι → α
hlt : ∀ (i : ι), f i < a
hlim : Tendsto f F (𝓝 a)
⊢ ¬∃ y, f y = a

Tactic:
]

Goal state after:
α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
a : α
f : ι → α
hlt : ∀ (i : ι), f i < a
hlim : Tendsto f F (𝓝 a)
⊢ ¬∃ y, f y = a
Module: Mathlib.Topology.Order.OrderClosed

Goal state before:
pp: α : Type u
ι : Type u_1
F : Filter ι
inst✝³ : F.NeBot
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
a : α
f : ι → α
hlt : ∀ (i : ι), f i < a
hlim : Tendsto f F (𝓝 a)
obs : a ∉ range f
⊢ Iio a = Iio a

Tactic:
]

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : ClosedIicTopology α
a : α
⊢ IsOpen[inst✝²] (Iic a)ᶜ

Tactic:
]

Goal state after:
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : ClosedIicTopology α
a : α
⊢ IsOpen[inst✝²] (Iic a)ᶜ
Module: Mathlib.Topology.Order.OrderClosed

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIicTopology α
inst✝ : PredOrder α
a : α
⊢ 𝓝[>] a = 𝓝[>] a

Tactic:
]

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIicTopology α
inst✝ : PredOrder α
a : α
⊢ 𝓝[≥] a = 𝓝[≥] a

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Clopen

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ closure[inst✝] s = s ∧ interior s = s ↔ closure[inst✝] s ⊆ interior s

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s : Set X
⊢ closure[inst✝] s = s ∧ interior s = s ↔ closure[inst✝] s ⊆ interior s
Module: Mathlib.Topology.Clopen

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ IsClopen (s ∩ (s ∩ a))

Tactic:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ IsClopen (s ∩ (s ∩ a))
Module: Mathlib.Topology.Clopen

Goal state before:
pp: case refine_1
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ s ⊆ s ∩ a ∪ s ∩ b

Tactic:
← inter_union_distrib_left

Goal state after:
case refine_1
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ s ⊆ s ∩ (a ∪ b)
Module: Mathlib.Topology.Clopen

Goal state before:
pp: case refine_1
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ s ⊆ s ∩ (a ∪ b)

Tactic:
]

Goal state after:
case refine_1
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ s ⊆ s ∩ (a ∪ b)
Module: Mathlib.Topology.Order.OrderClosed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : ClosedIicTopology α
a b : α
H : a ⋖ b
⊢ pure b = pure b

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Clopen

Goal state before:
pp: case refine_2
X : Type u
inst✝ : TopologicalSpace X
s a b : Set X
h : IsClopen s
cover : s ⊆ a ∪ b
ha : IsOpen[inst✝] a
hb : IsOpen[inst✝] b
hab : s ∩ a ∩ b = ∅
⊢ ∅ = ∅

Tactic:
]

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIicTopology α
b : α
inst✝ : PredOrder α
⊢ pure b = pure b

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Clopen

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
U : Set X
⊢ IsClopen U ↔ IsClopen U

Tactic:
]

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:
]

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
s U : Set X
⊢ Continuous[instTopologicalSpaceSubtype, _] (s.domRestrict U.boolIndicator) ↔
    Continuous[instTopologicalSpaceSubtype, _] (Subtype.val ⁻¹' U).boolIndicator
Module: Mathlib.Topology.Order.OrderClosed

Goal state before:
pp: α : Type u
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderClosedTopology α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
a : α
⊢ pure a = pure a

Tactic:
]

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 | f b ≤ g b} ∩ closure[inst✝] {b | f b ≤ g b}ᶜ ⊆ {b | f b = g b}

Tactic:
]

Goal state after:
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 | f b ≤ g b} ∩ closure[inst✝] {b | f b ≤ g b}ᶜ ⊆ {b | f b = g b}
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
⊢ frontier {b | f b < g b} ⊆ {b | f b = g b}

Tactic:
by
  simpa only [← not_lt, ← compl_ofPred, frontier_compl, eq_comm] using frontier_le_subset_eq hg hf

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

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
u v : Set α
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hs : s ⊆ u ∪ v
z : α
zs : z ∈ s
zu : z ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
xs : x ∈ s
xv : x ∈ v
t : Set α
ts : t ⊆ s
xt : x ∈ t
zt : z ∈ t
ht : IsPreconnected t
⊢ s ⊆ u ∪ v

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
s : Set α
x : α
H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t
u v : Set α
hu : IsOpen[inst✝] u
hv : IsOpen[inst✝] v
hs : s ⊆ u ∪ v
z : α
zs : z ∈ s
zu : z ∈ u
y : α
ys : y ∈ s
yv : y ∈ v
xs : x ∈ s
xv : x ∈ v
t : Set α
ts : t ⊆ s
xt : x ∈ t
zt : z ∈ t
ht : IsPreconnected t
⊢ s ⊆ u ∪ v
Module: Mathlib.Topology.Order.OrderClosed

Goal state before:
pp: case refine_1
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderClosedTopology α
f g : β → α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : (x : β) → Decidable (f x ≤ g x)
f' g' : β → γ
hf : Continuous[inst✝², inst✝⁵] f
hg : Continuous[inst✝², inst✝⁵] g
hf' : ContinuousOn f' {x | f x ≤ g x}
hg' : ContinuousOn g' {x | g x ≤ f x}
hfg : ∀ (x : β), f x = g x → f' x = g' x
⊢ ContinuousOn f' {b | f b ≤ g b}

Tactic:
]

Goal state after:
case refine_1
α : Type u
β : Type v
γ : Type w
inst✝⁵ : TopologicalSpace α
inst✝⁴ : LinearOrder α
inst✝³ : OrderClosedTopology α
f g : β → α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : (x : β) → Decidable (f x ≤ g x)
f' g' : β → γ
hf : Continuous[inst✝², inst✝⁵] f
hg : Continuous[inst✝², inst✝⁵] g
hf' : ContinuousOn f' {x | f x ≤ g x}
hg' : ContinuousOn g' {x | g x ≤ f x}
hfg : ∀ (x : β), f x = g x → f' x = g' x
⊢ ContinuousOn f' {b | f b ≤ g b}
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_3
t₁ t₂ : TopologicalSpace X
h₁ : R1Space X
h₂ : R1Space X
⊢ R1Space X

Tactic:
]

Goal state after:
X : Type u_3
t₁ t₂ : TopologicalSpace X
h₁ : R1Space X
h₂ : R1Space X
⊢ R1Space X
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)
y : Y
hys : y ∉ interior s
hxy : Inseparable (f x) y
⊢ s ∈ 𝓝 (f x)

Tactic:
]

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)
y : Y
hys : y ∉ interior s
hxy : Inseparable (f x) y
⊢ s ∈ 𝓝 (f x)
Module: Mathlib.Topology.Separation.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
⊢ ∀ (i' : Set X), IsCompact i' → ∃ i, (IsClosed[inst✝¹] i ∧ IsCompact i) ∧ iᶜ ⊆ i'ᶜ

Tactic:
]

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
⊢ ∀ (i' : Set X), IsCompact i' → ∃ i, (IsClosed[inst✝¹] i ∧ IsCompact i) ∧ iᶜ ⊆ i'ᶜ
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: case refl
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
i : ι
hi : i ∈ t
j : ι
hj : i ∈ t
⊢ IsPreconnected (s i)

Tactic:
]

Goal state after:
case refl
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
i : ι
hi : i ∈ t
j : ι
hj : i ∈ t
⊢ IsPreconnected (s i)
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: case tail
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
i : ι
hi : i ∈ t
j✝ j k : ι
a✝ : ReflTransGen R i j
hjk : R j k
ih : j ∈ t → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)
hj : k ∈ t
p : Set ι
hpt : p ⊆ t
hip : i ∈ p
hjp : j ∈ p
hp : IsPreconnected (⋃ j ∈ p, s j)
⊢ IsPreconnected (s k ∪ ⋃ x ∈ p, s x)

Tactic:
]

Goal state after:
case tail
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
i : ι
hi : i ∈ t
j✝ j k : ι
a✝ : ReflTransGen R i j
hjk : R j k
ih : j ∈ t → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)
hj : k ∈ t
p : Set ι
hpt : p ⊆ t
hip : i ∈ p
hjp : j ∈ p
hp : IsPreconnected (⋃ j ∈ p, s j)
⊢ IsPreconnected (s k ∪ ⋃ x ∈ p, s x)
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: case tail
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
i : ι
hi : i ∈ t
j✝ j k : ι
a✝ : ReflTransGen R i j
hjk : R j k
ih : j ∈ t → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)
hj : k ∈ t
p : Set ι
hpt : p ⊆ t
hip : i ∈ p
hjp : j ∈ p
hp : IsPreconnected (⋃ j ∈ p, s j)
⊢ s k ∩ s j ⊆ s k ∩ ⋃ x ∈ p, s x

Tactic:
]

Goal state after:
case tail
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
i : ι
hi : i ∈ t
j✝ j k : ι
a✝ : ReflTransGen R i j
hjk : R j k
ih : j ∈ t → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)
hj : k ∈ t
p : Set ι
hpt : p ⊆ t
hip : i ∈ p
hjp : j ∈ p
hp : IsPreconnected (⋃ j ∈ p, s j)
⊢ s k ∩ s j ⊆ s k ∩ ⋃ x ∈ p, s x
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
h : ∃ i, (IsOpen[inst✝¹] i ∧ x ∈ i ∧ IsConnected i) ∧ id i ⊆ F
⊢ connectedComponentIn F x ∈ 𝓝 x

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
h : ∃ i, (IsOpen[inst✝¹] i ∧ x ∈ i ∧ IsConnected i) ∧ id i ⊆ F
⊢ connectedComponentIn F x ∈ 𝓝 x
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen[inst✝¹] F
⊢ ∀ x_1 ∈ connectedComponentIn F x, connectedComponentIn F x ∈ 𝓝 x_1

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen[inst✝¹] F
⊢ ∀ x_1 ∈ connectedComponentIn F x, connectedComponentIn F x ∈ 𝓝 x_1
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen[inst✝¹] F
y : α
hy : y ∈ connectedComponentIn F x
⊢ connectedComponentIn F y ∈ 𝓝 y

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
F : Set α
x : α
hF : IsOpen[inst✝¹] F
y : α
hy : y ∈ connectedComponentIn F x
⊢ connectedComponentIn F y ∈ 𝓝 y
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
x : α
⊢ IsOpen[inst✝¹] (connectedComponentIn univ x)

Tactic:
]

Goal state after:
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : LocallyConnectedSpace α
x : α
⊢ IsOpen[inst✝¹] (connectedComponentIn univ x)
Module: Mathlib.Order.Filter.CountableInter

Goal state before:
pp: α : Type u_2
l : Filter α
inst✝ : CountableInterFilter l
ι : Type u_4
S : Set ι
hS : S.Countable
s : (i : ι) → i ∈ S → Set α
⊢ ⋂ x, s ↑x ⋯ ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), s i hi ∈ l

Tactic:
]

Goal state after:
α : Type u_2
l : Filter α
inst✝ : CountableInterFilter l
ι : Type u_4
S : Set ι
hS : S.Countable
s : (i : ι) → i ∈ S → Set α
⊢ ⋂ x, s ↑x ⋯ ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), s i hi ∈ l
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: case mpr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (F : Set α), IsOpen[inst✝] F → ∀ x ∈ F, IsOpen[inst✝] (connectedComponentIn F x)
⊢ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x ∈ V ∧ IsConnected V

Tactic:
]

Goal state after:
case mpr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (F : Set α), IsOpen[inst✝] F → ∀ x ∈ F, IsOpen[inst✝] (connectedComponentIn F x)
⊢ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x ∈ V ∧ IsConnected V
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: case mp
α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x ∈ V ∧ IsConnected V) →
    ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U

Tactic:
]

Goal state after:
case mp
α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen[inst✝] V ∧ x ∈ V ∧ IsConnected V) →
    ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: case mpr
α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) →
    ∀ (F : Set α), IsOpen[inst✝] F → ∀ x ∈ F, IsOpen[inst✝] (connectedComponentIn F x)

Tactic:
]

Goal state after:
case mpr
α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) →
    ∀ (F : Set α), IsOpen[inst✝] F → ∀ x ∈ F, IsOpen[inst✝] (connectedComponentIn F x)
Module: Mathlib.Order.Filter.CountableInter

Goal state before:
pp: case refine_1
ι : Sort u_1
α : Type u_2
β : Type u_3
l✝ : Filter α
inst✝ : CountableInterFilter l✝
l : Set (Set α)
hUnion : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l
hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l
S : Set (Set α)
hSc : S.Countable
hSp : S ⊆ {s | sᶜ ∈ l}
⊢ ⋃₀ (compl '' S) ∈ l

Tactic:
]

Goal state after:
case refine_1
ι : Sort u_1
α : Type u_2
β : Type u_3
l✝ : Filter α
inst✝ : CountableInterFilter l✝
l : Set (Set α)
hUnion : ∀ (S : Set (Set α)), S.Countable → (∀ s ∈ S, s ∈ l) → ⋃₀ S ∈ l
hmono : ∀ t ∈ l, ∀ s ⊆ t, s ∈ l
S : Set (Set α)
hSc : S.Countable
hSp : S ⊆ {s | sᶜ ∈ l}
⊢ ⋃₀ (compl '' S) ∈ l
Module: Mathlib.Topology.Connected.Basic

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
s : ι → Set α
H : ∀ (i : ι), IsPreconnected (s i)
K : ∀ (i j : ι), ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j
⊢ IsPreconnected (⋃ x ∈ univ, s x)

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
s : ι → Set α
H : ∀ (i : ι), IsPreconnected (s i)
K : ∀ (i j : ι), ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j
⊢ IsPreconnected (⋃ x ∈ univ, s x)
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: case mpr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U
U : Set α
hU : IsOpen[inst✝] U
x : α
x✝ : x ∈ U
y : α
hy : y ∈ connectedComponentIn U x
⊢ connectedComponentIn U y ∈ 𝓝 y

Tactic:
]

Goal state after:
case mpr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U
U : Set α
hU : IsOpen[inst✝] U
x : α
x✝ : x ∈ U
y : α
hy : y ∈ connectedComponentIn U x
⊢ connectedComponentIn U y ∈ 𝓝 y
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) ↔
    ∀ (x : α), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 x ∧ IsPreconnected s) id

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) ↔
    ∀ (x : α), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 x ∧ IsPreconnected s) id
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
b : α → ι → Set α
p : α → ι → Prop
hbasis : ∀ (x : α), (𝓝 x).HasBasis (p x) (b x)
hconnected : ∀ (x : α) (i : ι), p x i → IsPreconnected (b x i)
⊢ ∀ (x : α), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 x ∧ IsPreconnected s) id

Tactic:
]

Goal state after:
α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
b : α → ι → Set α
p : α → ι → Prop
hbasis : ∀ (x : α), (𝓝 x).HasBasis (p x) (b x)
hconnected : ∀ (x : α) (i : ι), p x i → IsPreconnected (b x i)
⊢ ∀ (x : α), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 x ∧ IsPreconnected s) id
Module: Mathlib.Topology.Connected.LocallyConnected

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : LocallyConnectedSpace α
inst✝ : TopologicalSpace β
f : β → α
h : IsOpenEmbedding f
x : β
⊢ (Filter.comap f (𝓝 (f x))).HasBasis (fun s ↦ (IsOpen[inst✝²] s ∧ f x ∈ s ∧ IsConnected s) ∧ s ⊆ range f) fun s ↦
    f ⁻¹' s

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : LocallyConnectedSpace α
inst✝ : TopologicalSpace β
f : β → α
h : IsOpenEmbedding f
x : β
⊢ (Filter.comap f (𝓝 (f x))).HasBasis (fun s ↦ (IsOpen[inst✝²] s ∧ f x ∈ s ∧ IsConnected s) ∧ s ⊆ range f) fun s ↦
    f ⁻¹' s