nhds_comap_dist — Mathlib

English

For any a, the neighborhood filter at a equals the comap of dist(·,a) from the neighborhood of 0.

Русский

Для любого a окрестности в a совпадают с образующей тензорной несобрачности: 𝓝(a) = comap(dist(·,a)) 𝓝(0).

LaTeX

$$$\mathcal N(a) = \operatorname{comap}(\lambda t. \operatorname{dist}(t,a))(\mathcal N(0))$$$

Lean4

theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a := by
  simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap, Function.comp_def,
    dist_comm]

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