uniformity_basis_dist_inv_nat_succ

English

There is a basis of the uniformity indexed by natural numbers n, with entourages { (a,b) | dist(a,b) < 1/(n+1) }.

Русский

Существует база униформности, индексируемая по Нат-числам n: entourages { (a,b) | dist(a,b) < 1/(n+1) }.

LaTeX

$$$$ (\mathcal{U}(\alpha)) \HasBasis (n: \mathbb{N}) \bigl( n \mapsto \{ (a,b) \in \alpha \times \alpha \mid \operatorname{dist}(a,b) < \tfrac{1}{n+1} \} \bigr) $$$$

Lean4

theorem uniformity_basis_dist_inv_nat_succ :
    (𝓤 α).HasBasis (fun _ => True) fun n : ℕ => {p : α × α | dist p.1 p.2 < 1 / (↑n + 1)} :=
  Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun _ε ε0 =>
    (exists_nat_one_div_lt ε0).imp fun _n hn => ⟨trivial, le_of_lt hn⟩

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