closure — Mathlib

English

If A ⊆ X → α is equicontinuous on S, then the closure of A is equicontinuous on S as a set of functions from S to α.

Русский

Если A ⊆ X → α эконтинуально на S, то closure(A) эконтинуально на S как множество функций.

LaTeX

$$$A\ \text{EquicontinuousOn } S \Rightarrow \overline{A}\ \text{EquicontinuousOn } S$$$

Lean4

/-- If a set of functions is equicontinuous, its closure for the product topology is also
equicontinuous. This would also be true for the coarser topology of pointwise convergence on `S`,
see `EquicontinuousOn.closure'`. -/
protected theorem closure {A : Set <| X → α} {S : Set X} (hA : A.EquicontinuousOn S) : (closure A).EquicontinuousOn S :=
  fun x hx ↦ Set.EquicontinuousWithinAt.closure (hA x hx)

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