disjoint_nhds_nhds_iff_not_specializes

English

For a pair of points x,y in an R1-space, the neighborhoods are disjoint if and only if x does not specialize to y.

Русский

Для пары точек x,y в пространстве R1 окрестности дисjunkты тогда и только тогда, когда x не специализируется на y.

LaTeX

$$$ \\forall X\\ [TopologicalSpace X] [R1Space X],\\; Disjoint(\\mathcal{N}_x, \\mathcal{N}_y) \\iff \\neg x \\triangleright y. $$$

Lean4

theorem disjoint_nhds_nhds_iff_not_specializes : Disjoint (𝓝 x) (𝓝 y) ↔ ¬x ⤳ y :=
  ⟨fun hd hspec ↦ hspec.not_disjoint hd, (specializes_or_disjoint_nhds _ _).resolve_left⟩

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