# Pseudo-metric

*on a set *

A non-negative real-valued function defined on the set of all pairs of elements of (that is, on ) and subordinate to the following three restrictions, called the axioms for a pseudo-metric:

a) if , then ;

b) ;

c) , where , and are arbitrary elements of .

It is not required that implies . A topology on is determined as follows by a pseudo-metric on : A point belongs to the closure of a set if , where

This topology is completely regular but is not necessarily Hausdorff: singleton sets can be non-closed. Every completely-regular topology can be given by a collection of pseudo-metrics as the lattice union of the corresponding pseudo-metric topologies. Analogously, families of pseudo-metrics can be used in defining, describing and investigating uniform structures.

#### References

[1] | J.L. Kelley, "General topology" , Springer (1975) |

#### Comments

See also Metric and Quasi-metric.

#### References

[a1] | E. Čech, "Topological spaces" , Interscience (1966) pp. 532 |

**How to Cite This Entry:**

Pseudo-metric.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=29453