View source for Pseudo-metric

''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 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.

====Comments====
See also [[Metric]],  [[Quasi-metric]] and [[Symmetry on a set]].

====References====
<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>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Čech,   "Topological spaces" , Interscience  (1966)  pp. 532 {{ZBL|0141.39401}}</TD></TR>
</table>

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