metric equivalence

Let X be a set equipped with two metrics ρ and σ. We say that ρ is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to σ (on X) if the identity mapMathworldPlanetmath on X, is a homeomorphismMathworldPlanetmath between the metric topologyMathworldPlanetmath on X induced by ρ and the metric topology on X induced by σ.

For example, if (X,ρ) is a metric space, then the function σ:X defined by


is a metric on X that is equivalent to ρ. This shows that every metric is equivalent to a boundedPlanetmathPlanetmathPlanetmath metric.

Title metric equivalence
Canonical name MetricEquivalence
Date of creation 2013-03-22 19:23:11
Last modified on 2013-03-22 19:23:11
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 54E35
Defines equivalent