disjoint_nhds_nhds — Mathlib

English

In a Hausdorff space, neighborhoods of distinct points are disjoint, and conversely disjoint neighborhoods imply inequality.

Русский

В нормальном пространстве НХДС точек различны и наоборот, если их окрестности не пересекаются, точки различны.

LaTeX

$$$ [T2Space X] \iff Disjoint(\mathcal{N}(x), \mathcal{N}(y)) \iff x \neq y $$$

Lean4

@[simp]
theorem disjoint_nhds_nhds [T2Space X] {x y : X} : Disjoint (𝓝 x) (𝓝 y) ↔ x ≠ y :=
  ⟨fun hd he => by simp [he, nhds_neBot.ne] at hd, (t2Space_iff_disjoint_nhds.mp ‹_› ·)⟩

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