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 ;
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.
|||J.L. Kelley, "General topology" , Springer (1975)|
See also Metric.
|[a1]||E. Čech, "Topological spaces" , Interscience (1966) pp. 532|
Pseudo-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=29453