/-- In a compact T₂ space, the connected component of a point equals the intersection of all
its clopen neighbourhoods. -/
theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x : X) :
    connectedComponent x = ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s := by
  apply Subset.antisymm connectedComponent_subset_iInter_isClopen
  -- Reduce to showing that the clopen intersection is connected.
  refine' IsPreconnected.subset_connectedComponent _ (mem_iInter.2 fun s => s.2.2)
  -- We do this by showing that any disjoint cover by two closed sets implies
  -- that one of these closed sets must contain our whole thing.
  -- To reduce to the case where the cover is disjoint on all of `X` we need that `s` is closed
  have hs : @IsClosed X _ (⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, s) :=
    isClosed_iInter fun s => s.2.1.2
  rw [isPreconnected_iff_subset_of_fully_disjoint_closed hs]
  intro a b ha hb hab ab_disj
  -- Since our space is normal, we get two larger disjoint open sets containing the disjoint
  -- closed sets. If we can show that our intersection is a subset of any of these we can then
  -- "descend" this to show that it is a subset of either a or b.
  rcases normal_separation ha hb ab_disj with ⟨u, v, hu, hv, hau, hbv, huv⟩
  obtain ⟨s, H⟩ : ∃ s : Set X, IsClopen s ∧ x ∈ s ∧ s ⊆ u ∪ v
  /- Now we find a clopen set `s` around `x`, contained in `u ∪ v`. We utilize the fact that
  `X \ u ∪ v` will be compact, so there must be some finite intersection of clopen neighbourhoods of
  `X` disjoint to it, but a finite intersection of clopen sets is clopen so we let this be our
  `s`. -/
  · have H1 := (hu.union hv).isClosed_compl.isCompact.inter_iInter_nonempty
      (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s) fun s => s.2.1.2
    rw [← not_disjoint_iff_nonempty_inter, imp_not_comm, not_forall] at H1
    cases' H1 (disjoint_compl_left_iff_subset.2 <| hab.trans <| union_subset_union hau hbv)
      with si H2
    refine' ⟨⋂ U ∈ si, Subtype.val U, _, _, _⟩
    · exact isClopen_biInter_finset fun s _ => s.2.1
    · exact mem_iInter₂.2 fun s _ => s.2.2
    · rwa [← disjoint_compl_left_iff_subset, disjoint_iff_inter_eq_empty,
        ← not_nonempty_iff_eq_empty]
  -- So, we get a disjoint decomposition `s = s ∩ u ∪ s ∩ v` of clopen sets. The intersection of all
  -- clopen neighbourhoods will then lie in whichever of u or v x lies in and hence will be a subset
  -- of either a or b.
  · have H1 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hu hv huv
    rw [union_comm] at H
    have H2 := isClopen_inter_of_disjoint_cover_clopen H.1 H.2.2 hv hu huv.symm
    by_cases hxu : x ∈ u <;> [left; right]
    -- The x ∈ u case.
    · suffices ⋂ s : { s : Set X // IsClopen s ∧ x ∈ s }, ↑s ⊆ u
        from Disjoint.left_le_of_le_sup_right hab (huv.mono this hbv)
      · apply Subset.trans _ (inter_subset_right s u)
        

Proof state:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : CompactSpace X
x : X
hs : IsClosed (⋂ s, ↑s)
a b : Set X
ha : IsClosed a
hb : IsClosed b
hab : ⋂ s, ↑s ⊆ a ∪ b
ab_disj : Disjoint a b
u v : Set X
hu : IsOpen u
hv : IsOpen v
hau : a ⊆ u
hbv : b ⊆ v
huv : Disjoint u v
s : Set X
H : IsClopen s ∧ x ∈ s ∧ s ⊆ v ∪ u
H1 : IsClopen (s ∩ u)
H2 : IsClopen (s ∩ v)
hxu : x ∈ u
⊢ ⋂ s, ↑s ⊆ s ∩ u

Next tactic:
exact iInter_subset (fun s : { s : Set X // IsClopen s ∧ x ∈ s } => s.1)
          ⟨s ∩ u, H1, mem_inter H.2.1 hxu⟩