# Difference between revisions of "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 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