Pseudo-uniformities (Q6060936)

In this paper a theory of pseudo-uniformity for a set \(X\) is considered and defined as a particular delta-local filter on \(X\times X\) such that its relations contain at least an element of the diagonal. Analogously to the classical theory of uniformity where the important result is the fact that each uniformity is generated by a family of pseudo-metrics, the author uses the same terminology for symmetric maps \(d : X\times X \mapsto \mathbb R^+\) satisfying the ``triangle inequality'' such that \(d(x,x)=0\) for some points \(x\in X\). Then an analogous theorem states that any delta-local filter on \(X\times X\) is defined by a family \(\mathcal P\) of pseudo-metrics with the property that for every \(x\in X\) there is \(d\in P\) such that \(d(x,x)=0\). Closely related to the former it is shown that any delta-local filter on \(X\times X\) induces a topology on \(X\). In Section 3 the author defines uniform continuity for maps with respect to delta-local filters and then the product of them is studied. In this context, the property of pseudo-uniform continuity for pseudo-metrics is considered. The main result of this treatment states that every pseudo-uniformity for \(X\) is defined by a family of pseudo-uniformly continuous pseudo-metrics on \(X\times X\). At the end it is pointed out that every topology \(\tau\) on \(X\) is defined by a family of pseudo-metrics \(d\) closely related to \(\tau\), and all topologies are pseudo-uniformizable.