dist_mem_uniformity — Mathlib

English

The uniformity has a basis consisting of entourages { (a,b) | dist(a,b) ≤ ε } for ε > 0; equivalently, the open balls provide a neighborhood basis.

Русский

Униформность имеет базис окружностей { (a,b) | dist(a,b) ≤ ε } при ε > 0; эквивалентно, открытые шары образуют базис окрестностей.

LaTeX

$$$$ (\mathcal{U}(\alpha)).HasBasis (\varepsilon > 0) (\varepsilon \mapsto \{ (a,b) \in \alpha \times \alpha \mid \operatorname{dist}(a,b) \le \varepsilon \}). $$$$

Lean4

/-- A constant size neighborhood of the diagonal is an entourage. -/
theorem dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : {p : α × α | dist p.1 p.2 < ε} ∈ 𝓤 α :=
  mem_uniformity_dist.2 ⟨ε, ε0, fun _ _ ↦ id⟩

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