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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917801.png" />''
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''$X$''
  
A non-negative real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917802.png" /> defined on the set of all pairs of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917803.png" /> and satisfying the following axioms:
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A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ and satisfying the following axioms:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917804.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917805.png" />;
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1) $d(x,y) = 0$ if and only if $x = y$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917806.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917807.png" />.
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2) $d(x,y) = d(y,x)$ for any $x,y \in X$.
  
In contrast to a [[Metric|metric]] and a [[Pseudo-metric|pseudo-metric]], a symmetry need not satisfy the triangle axiom. Relative to a symmetry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917808.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917809.png" /> there is a topology defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178010.png" />: A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178011.png" /> is closed (relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178012.png" />) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178013.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178014.png" />. Here
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In contrast to a [[metric]] and a [[pseudo-metric]], a symmetry need not satisfy the triangle axiom. Relative to a symmetry $d$ on a set $X$ there is a topology defined on $X$: A set $A \subseteq X$ is closed (relative to $d$) if and only if $d(x,A) > 0$ for each $x \in X \setminus A$. Here
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$$
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d(x,A) = \inf_{a \in A} d(x,a) \ .
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$$
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The closure of a set $A$ in this topological space contains the set of all $x \in X$ for which $d(x,A) = 0$ but need not be exhausted by this set. Correspondingly, the $\epsilon$-ball around a point of $X$ may have an empty interior. A topological space is called symmetrizable if its topology is generated by the above rule from some symmetry. The class of symmetrizable spaces is much wider than the class of [[metrizable space]]s: A symmetrizable space need not be [[Paracompact space|paracompact]], [[Normal space|normal]] or [[Hausdorff space|Hausdorff]]. In addition, a symmetrizable space need not satisfy the [[first axiom of countability]].
  
<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/s/s091/s091780/s09178015.png" /></td> </tr></table>
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However, each symmetrizable space is a [[sequential space]], that is, its topology is determined by convergent sequences by the rule: A set $A$ is closed if and only if the limit of each sequence of points of $A$ that converges in $X$ belongs to $A$. For compact Hausdorff spaces symmetrizability is equivalent to metrizability.
  
The closure of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178016.png" /> in this topological space contains the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178017.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178018.png" /> but need not be exhausted by this set. Correspondingly, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178019.png" />-ball around a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178020.png" /> may have an empty interior. A topological space is called symmetrizable if its topology is generated by the above rule from some symmetry. The class of symmetrizable spaces is much wider than the class of metrizable spaces (cf. [[Metrizable space|Metrizable space]]): A symmetrizable space need not be paracompact, normal or Hausdorff. In addition, a symmetrizable space need not satisfy the first axiom of countability.
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====References====
 +
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  S.I. Nedev,  "$o$-metrizable spaces" ''Trans. Moscow Math. Soc.'' , '''24'''  (1971)  pp. 213–247  ''Trudy Moskov. Mat. Obshch.'' , '''24'''  (1971)  pp. 201–236 {{MR|0367935}} {{ZBL|0244.54016}}</TD></TR>
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</table>
  
However, each symmetrizable space is sequential, that is, its topology is determined by convergent sequences by the rule: A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178021.png" /> is closed if and only if the limit of each sequence of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178022.png" /> that converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178023.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178024.png" />. For compact Hausdorff spaces symmetrizability is equivalent to metrizability.
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.I. Nedev,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s09178025.png" />-metrizable spaces"  ''Trans. Moscow Math. Soc.'' , '''24'''  (1971)  pp. 213–247  ''Trudy Moskov. Mat. Obshch.'' , '''24'''  (1971)  pp. 201–236</TD></TR></table>
 

Latest revision as of 20:16, 11 December 2016

$X$

A non-negative real-valued function $d$ defined on the set of all pairs of elements of $X$ and satisfying the following axioms:

1) $d(x,y) = 0$ if and only if $x = y$;

2) $d(x,y) = d(y,x)$ for any $x,y \in X$.

In contrast to a metric and a pseudo-metric, a symmetry need not satisfy the triangle axiom. Relative to a symmetry $d$ on a set $X$ there is a topology defined on $X$: A set $A \subseteq X$ is closed (relative to $d$) if and only if $d(x,A) > 0$ for each $x \in X \setminus A$. Here $$ d(x,A) = \inf_{a \in A} d(x,a) \ . $$ The closure of a set $A$ in this topological space contains the set of all $x \in X$ for which $d(x,A) = 0$ but need not be exhausted by this set. Correspondingly, the $\epsilon$-ball around a point of $X$ may have an empty interior. A topological space is called symmetrizable if its topology is generated by the above rule from some symmetry. The class of symmetrizable spaces is much wider than the class of metrizable spaces: A symmetrizable space need not be paracompact, normal or Hausdorff. In addition, a symmetrizable space need not satisfy the first axiom of countability.

However, each symmetrizable space is a sequential space, that is, its topology is determined by convergent sequences by the rule: A set $A$ is closed if and only if the limit of each sequence of points of $A$ that converges in $X$ belongs to $A$. For compact Hausdorff spaces symmetrizability is equivalent to metrizability.

References

[1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[2] S.I. Nedev, "$o$-metrizable spaces" Trans. Moscow Math. Soc. , 24 (1971) pp. 213–247 Trudy Moskov. Mat. Obshch. , 24 (1971) pp. 201–236 MR0367935 Zbl 0244.54016
How to Cite This Entry:
Symmetry on a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetry_on_a_set&oldid=39975
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