UniformEquicontinuous — Mathlib

English

Uniform equicontinuity on a subset S is defined by restricting to S in the UniformEquicontinuousOn predicate.

Русский

Уравномеренная эквиконтинуальность на подмножестве S определяется ограничением к S в предикате UniformEquicontinuousOn.

LaTeX

$$UniformEquicontinuousOn(F,S) := UniformEquicontinuousOn ((↑): H → β → α) S$$

Lean4

/-- A family `F : ι → β → α` of functions between uniform spaces is *uniformly equicontinuous* if,
for all entourages `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are
`V`-close, we have that, *for all `i : ι`*, `F i x` is `U`-close to `F i y`. -/
def UniformEquicontinuous (F : ι → β → α) : Prop :=
  ∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U

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