# Difference between revisions of "Pseudo-metric"

(Importing text file) |
m (→Comments) |
||

Line 21: | Line 21: | ||

====Comments==== | ====Comments==== | ||

− | See also [[Metric|Metric]]. | + | See also [[Metric|Metric]] and [[Quasi-metric|Quasi-metric]]. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. 532</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Čech, "Topological spaces" , Interscience (1966) pp. 532</TD></TR></table> |

## Revision as of 10:16, 20 February 2013

*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