closure — Mathlib

English

If a set A of functions is equicontinuous at x0, then its closure (in the product topology) is equicontinuous at x0.

Русский

Если множество функций A экквинтоентно в x0, то его замыкание по произведной топологии экквинтоентно в x0.

LaTeX

$$$\\text{A EquicontinuousAt } x_0 \\Rightarrow \\overline{A} \\text{ EquicontinuousAt } x_0.$$$

Lean4

/-- If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
also equicontinuous at `x₀`. -/
protected theorem closure {A : Set (X → α)} {x₀ : X} (hA : A.EquicontinuousAt x₀) : (closure A).EquicontinuousAt x₀ :=
  hA.closure' (u := id) continuous_id

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