Pseudo-uniformities

Abstract

The notions of pseudo-uniformity for a set $X$ and of pseudo-metric presented in this paper are somehow connected. Unlike the uniformity, the elements of a pseudo-uniformity do not necessarily contain the diagonal $\Delta$ of $X\times X$ but only some points (at least one point) of it. Likewise, the pseudo-metrics considered and utilized here do not vanish on the whole of $\Delta$, but only in some points of it. It will be showed how any family of pseudo-metrics defines a pseudo-uniformity. The main result achieved is the proof that every pseudo-uniformity is defined by a family of “pseudo-uniformly continuous” pseudo-metrics. In the final part of the paper, we prove that every topology is defined by a family of pseudo-metrics and, consequently, that every topology is pseudo-uniformizable: nay every topology defines a pseudo-uniformity which induces just the starting topology.