View source for Symmetry on a set

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091780/s0917801.png" />''

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:

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" />;

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" />.

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

<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>

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.

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.

====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>