nhdsNE_neBot — Mathlib

English

The neighborhood filter around any x is not bottom when restricted to points ≠ x.

Русский

Границы окрестности вокруг любой точки не пусты при ограничении до точек, не совпадающих с x.

LaTeX

$$$ \forall x,\ (\mathcal{N}_{ onsuch\} (x)) \neq \bot $$$

Lean4

@[instance]
theorem nhdsNE_neBot (x : α) : NeBot (𝓝[≠] x) :=
  by
  rw [← mem_closure_iff_nhdsWithin_neBot, Metric.mem_closure_iff]
  rintro ε ε0
  rcases exists_norm_lt α ε0 with ⟨b, hb0, hbε⟩
  refine ⟨x + b, mt (Set.mem_singleton_iff.trans add_eq_left).1 <| norm_pos_iff.1 hb0, ?_⟩
  rwa [dist_comm, dist_eq_norm, add_sub_cancel_left]

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