Symmetry on a set

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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) $d(x,y) = 0$ 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.


[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:
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