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 subordinate to the following three restrictions, 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" , Springer (1975)

Comments

See also Metric, Quasi-metric and Symmetry on a set.

References

[a1]	E. Čech, "Topological spaces" , Interscience (1966) pp. 532