English
In a weakly locally compact R1 space, every compact set has an open neighborhood whose closure is compact.
Русский
В слаб Locally компактном R1-пространстве любому компактному множеству существует открытое окрестное множество так, что его замыкание компактно.
LaTeX
$$$\\forall X\\ (\\TopologicalSpace\,X)\\ (\\R1Space\,X)\\ (\\WeaklyLocallyCompactSpace\,X),\\ \\forall K \\subseteq X,\\ IsCompact K \\rightarrow \\exists V\\ (IsOpen V \\land K \\subseteq V \\land IsCompact (closure V))$$$
Lean4
/-- In a weakly locally compact R₁ space,
every compact set has an open neighborhood with compact closure. -/
theorem exists_isOpen_superset_and_isCompact_closure {K : Set X} (hK : IsCompact K) :
∃ V, IsOpen V ∧ K ⊆ V ∧ IsCompact (closure V) :=
by
rcases exists_compact_superset hK with ⟨K', hK', hKK'⟩
exact ⟨interior K', isOpen_interior, hKK', hK'.closure_of_subset interior_subset⟩
