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Difference between revisions of "Pseudo-metric"

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This topology is [[Completely-regular space|completely regular]] but is not necessarily [[Hausdorff space|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 space|completely regular]] but is not necessarily [[Hausdorff space|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====
 
<table>
 
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR>
 
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====Comments====
 
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====References====
 
====References====
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "Topological spaces" , Interscience  (1966)  pp. 532</TD></TR>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Graduate Texts in Mathematics '''27''' Springer  (1975) {{ISBN|0-387-90125-6}} {{ZBL|0306.54002}}</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,  "Topological spaces" , Interscience  (1966)  pp. 532 {{ZBL|0141.39401}}</TD></TR>
 
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Latest revision as of 11:48, 23 November 2023

on a set $X$

A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ (that is, on $X \times X$) and satisfying the following three conditions, called the axioms for a pseudo-metric:

a) if $x = y$, then $d(x,y) = 0$;

b) $d(x,y) = d(y,x)$ (symmetry);

c) $d(x,y) \le d(x,z) + d(z,y)$ (triangle inequality), where $x,y,z$ are arbitrary elements of $X$.

It is not required that $d(x,y) = 0$ implies $x=y$. A topology on $X$ is determined by a pseudo-metric $d$ on $X$ as follows: A point $x$ belongs to the closure of a set $A \subseteq X$ if $d(x,A) = 0$, where $$ d(x,A) = \inf_{a \in A} d(x,a) \ . $$

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.

Comments

See also Metric, Quasi-metric and Symmetry on a set.

References

[1] J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002
[a1] E. Čech, "Topological spaces" , Interscience (1966) pp. 532 Zbl 0141.39401
How to Cite This Entry:
Pseudo-metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-metric&oldid=39978
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article