Theorems · Theorem · general topology

Metric.controlled_of_isUniformInducing

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

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

Defined in

Mathlib.Topology.MetricSpace.Pseudo.Basic

Cited by

1 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpacePseudoMetricSpace