pseudometric space

A pseudometric space is a set X together with a non-negative real-valued function d:X×X (called a pseudometric) such that, for every x,y,zX,

  • d(x,x)=0.

  • d(x,y)=d(y,x)

  • d(x,z)d(x,y)+d(y,z)

In other words, a pseudometric space is a generalizationPlanetmathPlanetmath of a metric space in which we allow the possibility that d(x,y)=0 for distinct values of x and y.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
Title pseudometric space
Canonical name PseudometricSpace
Date of creation 2013-03-22 14:40:18
Last modified on 2013-03-22 14:40:18
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 54E35
Synonym pesudo-metric space
Related topic MetricSpace
Related topic QuasimetricSpace
Related topic NormedVectorSpace
Related topic Seminorm
Defines pseudometric
Defines pseudo-metricPlanetmathPlanetmath