Theorems · Theorem · general topology

IsUniformInducing.equicontinuousWithinAt_iff

∀ {ι : Type u_1} {X : Type u_3} {α : Type u_6} {β : Type u_8} [tX : TopologicalSpace X] [uα : UniformSpace α]
  [uβ : UniformSpace β] {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β},
  IsUniformInducing u → (EquicontinuousWithinAt F S x₀ ↔ EquicontinuousWithinAt ((fun x => u ∘ x) ∘ F) S x₀)

Given u : α → β a uniform inducing map, a family 𝓕 : ι → X → α is equicontinuous at a point x₀ : X within a subset S : Set X iff the family 𝓕', obtained by composing each function of 𝓕 by u, is equicontinuous at x₀ within S.

Defined in

Mathlib.Topology.UniformSpace.Equicontinuity

Cited by

1 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceUniformSpaceUniformSpace