example of pseudometric space

Let X=2 and consider the function d:X×X to the non-negative real numbers given by


Then d(x,x)=|x1-x1|=0, d(x,y)=|x1-y1|=|y1-x1|=d(y,z) and the triangle inequalityMathworldMathworldPlanetmathPlanetmath follows from the triangle inequality on 1, so (X,d) satisfies the defining conditions of a pseudometric space.

Note, however, that this is not an example of a metric space, since we can have two distinct points that are distance 0 from each other, e.g.


Other examples:

  • Let X be a set, x0X, and let F(X) be functions XR. Then d(f,g)=|f(x0)-g(x0)| is a pseudometric on F(X) [1].

  • If X is a vector space and p is a seminorm over X, then d(x,y)=p(x-y) is a pseudometric on X.

  • The trivial pseudometric d(x,y)=0 for all x,yX is a pseudometric.


  • 1 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title example of pseudometric space
Canonical name ExampleOfPseudometricSpace
Date of creation 2013-03-22 14:40:24
Last modified on 2013-03-22 14:40:24
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Example
Classification msc 54E35
Related topic Seminorm
Related topic VectorSpace
Related topic MetricSpace
Related topic Metric
Defines trivial pseudometric