-- Mathlib/Topology/Separation/Profinite.lean
theorem nhds_basis_clopen (x : X) : (𝓝 x).HasBasis (fun s : Set X => x ∈ s ∧ IsClopen s) id :=
  ⟨fun U => by
    constructor
    · have hx : connectedComponent x = {x} :=
        totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x
      rw [connectedComponent_eq_iInter_isClopen] at hx
      intro hU
      let N := { s // IsClopen s ∧ x ∈ s }
      rsuffices ⟨⟨s, hs, hs'⟩, hs''⟩ : ∃ s : N, s.val ⊆ U
      · exact ⟨s, ⟨hs', hs⟩, hs''⟩
      haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩
      have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1
      have hdir : Directed Superset fun s : N => s.val

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
⊢ ∃ s, ↑s ⊆ U

Tactic:
have hNcl : ∀ s : N, IsClosed s.val := fun s => s.property.1.1

Result:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
⊢ ∃ s, ↑s ⊆ U

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
hdir : Directed Superset fun s => ↑s
h_nhd : ∀ y ∈ ⋂ s, ↑s, U ∈ 𝓝 y
⊢ ∃ s, ↑s ⊆ U

Tactic:
exact exists_subset_nhds_of_compactSpace hdir hNcl h_nhd

Result:
no goals

Proof state:
case mpr.intro.intro.intro.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U V : Set X
hUV : V ⊆ U
hxV : x ∈ V
V_op : IsOpen V
⊢ U ∈ 𝓝 x

Tactic:
rw [mem_nhds_iff]

Result:
case mpr.intro.intro.intro.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U V : Set X
hUV : V ⊆ U
hxV : x ∈ V
V_op : IsOpen V
⊢ ∃ t ⊆ U, IsOpen t ∧ x ∈ t

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
hdir : Directed Superset fun s => ↑s
y : X
y_in : y ∈ ⋂ s, ↑s
⊢ U ∈ 𝓝 y

Tactic:
rw [hx, mem_singleton_iff] at y_in

Result:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
hdir : Directed Superset fun s => ↑s
y : X
y_in : y = x
⊢ U ∈ 𝓝 y

Proof state:
case mp
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

Tactic:
have hx : connectedComponent x = {x} :=
  totallyDisconnectedSpace_iff_connectedComponent_singleton.mp ‹_› x

Result:
case mp
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : connectedComponent x = {x}
⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
hdir : Directed Superset fun s => ↑s
y : X
y_in : y = x
⊢ U ∈ 𝓝 y

Tactic:
rwa [y_in]

Result:
no goals

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
⊢ U ∈ 𝓝 x ↔ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

Tactic:
constructor

Result:
case mp
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
⊢ U ∈ 𝓝 x → ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

case mpr
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
⊢ (∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U) → U ∈ 𝓝 x

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
⊢ ∃ s, ↑s ⊆ U

Tactic:
haveI : Nonempty N := ⟨⟨univ, isClopen_univ, mem_univ x⟩⟩

Result:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
⊢ ∃ s, ↑s ⊆ U

Proof state:
case mk.intro.mk.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
s : Set X
hs : IsClopen s
hxs : x ∈ s
t : Set X
ht : IsClopen t
hxt : x ∈ t
⊢ ∃ z, (fun s => ↑s) ⟨s, ⋯⟩ ⊇ (fun s => ↑s) z ∧ (fun s => ↑s) ⟨t, ⋯⟩ ⊇ (fun s => ↑s) z

Tactic:
exact ⟨⟨s ∩ t, hs.inter ht, ⟨hxs, hxt⟩⟩, inter_subset_left, inter_subset_right⟩

Result:
no goals

Proof state:
case mpr.intro.intro.intro.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U V : Set X
hUV : V ⊆ U
hxV : x ∈ V
V_op : IsOpen V
⊢ ∃ t ⊆ U, IsOpen t ∧ x ∈ t

Tactic:
exact ⟨V, hUV, V_op, hxV⟩

Result:
no goals

Proof state:
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
⊢ Directed Superset fun s => ↑s

Tactic:
rintro ⟨s, hs, hxs⟩ ⟨t, ht, hxt⟩

Result:
case mk.intro.mk.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
this : Nonempty N
hNcl : ∀ (s : N), IsClosed ↑s
s : Set X
hs : IsClopen s
hxs : x ∈ s
t : Set X
ht : IsClopen t
hxt : x ∈ t
⊢ ∃ z, (fun s => ↑s) ⟨s, ⋯⟩ ⊇ (fun s => ↑s) z ∧ (fun s => ↑s) ⟨t, ⋯⟩ ⊇ (fun s => ↑s) z

Proof state:
case mp
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

Tactic:
let N := { s // IsClopen s ∧ x ∈ s }

Result:
case mp
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

Proof state:
case mp.intro.mk.intro
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : CompactSpace X
inst✝ : TotallyDisconnectedSpace X
x : X
U : Set X
hx : ⋂ s, ↑s = {x}
hU : U ∈ 𝓝 x
N : Type (max 0 u_1) := { s // IsClopen s ∧ x ∈ s }
s : Set X
hs : IsClopen s
hs' : x ∈ s
hs'' : ↑⟨s, ⋯⟩ ⊆ U
⊢ ∃ i, (x ∈ i ∧ IsClopen i) ∧ id i ⊆ U

Tactic:
exact ⟨s, ⟨hs', hs⟩, hs''⟩

Result:
no goals