Theorems · Theorem · general topology

isometry_iff_dist_eq

∀ {α : Type u} {β : Type v} [inst : PseudoMetricSpace α] [inst_1 : PseudoMetricSpace β] {f : α → β},
  Isometry f ↔ ∀ (x y : α), dist (f x) (f y) = dist x y

On pseudometric spaces, a map is an isometry if and only if it preserves distances.

Defined in

Mathlib.Topology.MetricSpace.Isometry

Cited by

3 results in Mathlib

Foundations

Depth 150 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpacePseudoMetricSpace