quasimetric space

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

  • d(x,y)0 with equality if and only if x=y.

  • d(x,z)d(x,y)+d(y,z)

In other words, a quasimetric space is a generalizationPlanetmathPlanetmath of a metric space in which we drop the requirement that, for two points x and y, the “distance” between x and y is the same as the “distance” between y and x (i.e. the symmetry axiom of metric spaces).

Some properties:

  • If (X,d) is a quasimetric space, we can form a metric space (X,d) where d is defined for all x,yX by

  • Every metric space is trivially a quasimetric space.

  • A quasimetric that is (i.e. d(x,y)=d(y,x) for all x,yX is a metric.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
  • 2 Z. Shen, Lectures of Finsler geometry, World Sientific, 2001.
Title quasimetric space
Canonical name QuasimetricSpace
Date of creation 2013-03-22 14:40:21
Last modified on 2013-03-22 14:40:21
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 54E35
Synonym quasi-metric space
Related topic PseudometricSpace
Related topic MetricSpace
Related topic GeneralizationOfAPseudometric
Defines quasimetric
Defines quasi-metric