Difference between revisions of "Pseudo-metric"
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− | 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. | + | 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==== | ====References==== |
Revision as of 20:45, 5 December 2014
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 |
Pseudo-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=35374