Theorems · Theorem · general topology

EMetric.controlled_of_isUniformInducing

∀ {α : Type u} {β : Type v} [inst : PseudoEMetricSpace α] [inst_1 : PseudoEMetricSpace β] {f : α → β},
  IsUniformInducing f →
    (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε) ∧
      ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ

If a map between pseudoemetric spaces is a uniform inducing map then the edistance between f x and f y is controlled in terms of the distance between x and y.

Defined in

Mathlib.Topology.EMetricSpace.Basic

Cited by

1 results in Mathlib

Foundations

Depth 148 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoEMetricSpacePseudoEMetricSpace