metric space

A metric space is a set X together with a real valued function d:X×X (called a metric, or sometimes a distance function) such that, for every x,y,zX,

  • d(x,y)0, with equality11This condition can be replaced with the weaker statement d(x,y)=0x=y without affecting the definition. if and only if x=y

  • d(x,y)=d(y,x)

  • d(x,z)d(x,y)+d(y,z)

For xX and ε with ε>0, the open ball around x of radius ε is the set Bε(x):={yXd(x,y)<ε}. An open set in X is a set which equals an arbitrary (possibly empty) union of open balls in X, and X together with these open sets forms a Hausdorff topological space. The topologyMathworldPlanetmath on X formed by these open sets is called the metric topology, and in fact the open sets form a basis for this topology (proof (

Similarly, the set B¯ε(x):={yXd(x,y)ε} is called a closed ball around x of radius ε. Every closed ball is a closed subset of X in the metric topology.

The prototype example of a metric space is itself, with the metric defined by d(x,y):=|x-y|. More generally, any normed vector spacePlanetmathPlanetmath has an underlying metric space structureMathworldPlanetmath; when the vector space is finite dimensional, the resulting metric space is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to Euclidean space.


  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
Title metric space
Canonical name MetricSpace
Date of creation 2013-03-22 11:53:19
Last modified on 2013-03-22 11:53:19
Owner djao (24)
Last modified by djao (24)
Numerical id 15
Author djao (24)
Entry type Definition
Classification msc 54E35
Classification msc 82-00
Classification msc 83-00
Classification msc 81-00
Related topic NeighborhoodMathworldPlanetmathPlanetmath
Related topic VectorNorm
Related topic T2Space
Related topic Ultrametric
Related topic QuasimetricSpace
Related topic NormedVectorSpace
Related topic PseudometricSpace
Defines distance metric
Defines metric
Defines distance
Defines metric topology
Defines open ball
Defines closed ball