le_nhds_of_unique_clusterPt — Mathlib

English

If X is compact, then the whole space X is compact.

Русский

Если X компактно, то все пространство X компактно.

LaTeX

$$$\text{IsCompact}(\mathrm{univ})$$$

Lean4

/-- If a filter has a unique cluster point `y` in a compact topological space,
then the filter is less than or equal to `𝓝 y`. -/
theorem le_nhds_of_unique_clusterPt [CompactSpace X] {l : Filter X} {y : X} (h : ∀ x, ClusterPt x l → x = y) :
    l ≤ 𝓝 y :=
  isCompact_univ.le_nhds_of_unique_clusterPt univ_mem fun x _ ↦ h x

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