exists_isCompact_closedBall — Mathlib

English

In a weakly locally compact space, for any x there exists a radius r>0 such that the closed ball around x with radius r is compact.

Русский

В слаб Locally компактном пространстве для любой точки x существует r>0, такое что замкнутый шар вокруг x радиуса r компактно.

LaTeX

$$$\exists r>0\, IsCompact(\overline{\mathrm{closedBall}}(x,r))$$$

Lean4

theorem exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) : ∃ r, 0 < r ∧ IsCompact (closedBall x r) :=
  by
  have : ∀ᶠ r in 𝓝[>] 0, IsCompact (closedBall x r) :=
    eventually_nhdsWithin_of_eventually_nhds (eventually_isCompact_closedBall x)
  simpa only [and_comm] using (this.and self_mem_nhdsWithin).exists

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