Namespaces
Variants
Actions

Difference between revisions of "Pseudo-metric"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(→‎References: expand bibliodata)
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
''on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757301.png" />''
+
''on a set $X$''
  
A non-negative real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757302.png" /> defined on the set of all pairs of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757303.png" /> (that is, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757304.png" />) and subordinate to the following three restrictions, called the axioms for a pseudo-metric:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757305.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757306.png" />;
+
a) if $x = y$, then $d(x,y) = 0$;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757307.png" />;
+
b) $d(x,y) = d(y,x)$ (symmetry);
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757308.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p0757309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573011.png" /> are arbitrary elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573012.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573014.png" />. A topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573015.png" /> is determined as follows by a pseudo-metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573017.png" />: A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573018.png" /> belongs to the closure of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573020.png" />, where
+
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
 +
$$
 +
d(x,A) = \inf_{a \in A} d(x,a) \ .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075730/p07573021.png" /></td> </tr></table>
+
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 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR></table>
+
<table>
 
+
<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>
 
+
</table>
  
 
====Comments====
 
====Comments====
See also [[Metric|Metric]].
+
See also [[Metric]],  [[Quasi-metric]] and [[Symmetry on a set]].
  
 
====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 {{ZBL|0141.39401}}</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 07:22, 12 December 2016

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.

References

[1] J.L. Kelley, "General topology" , Graduate Texts in Mathematics 27 Springer (1975) ISBN 0-387-90125-6 Zbl 0306.54002

Comments

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

References

[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=14367
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