exists_isOpen_mem_isCompact_closure

English

In a weakly locally compact R1 space, every point has an open neighborhood with compact closure.

Русский

В слаб Locally компактном пространстве R1 любая точка имеет открытое соседство с компактным замыканием.

LaTeX

$$$\\forall x\\in X,\\ \\exists U\\ (IsOpen U \\land x \\in U \\land IsCompact (closure U))$$$

Lean4

/-- In a weakly locally compact R₁ space,
every point has an open neighborhood with compact closure. -/
theorem exists_isOpen_mem_isCompact_closure (x : X) : ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U) := by
  simpa only [singleton_subset_iff] using exists_isOpen_superset_and_isCompact_closure isCompact_singleton

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