Convergence in probabilistic semimetric spaces
A probabilistic semimetric space (S,F) is a set S together with a function F defined on \(S\times S\) with values in the space \(\Delta^+\), which is a space of real-valued functions, satisfying some weak assumptions resembling those for a metric except for the triangular inequality. Since \(\Delta^+\) carries a metric which turns it into a (complete) metric space, S carries a natural Cauchy structure \({\mathcal C}\), namely the one generated by the set of all filters \(\phi\) such that F(\(\phi\) \(\times \phi)\) is a Cauchy filter in \(\Delta^+.\) The author shows that the additional assumption that F is Cauchy- continuous has quite a few interesting consequences, e.g.: (S,\({\mathcal C})\) is regular and \({\mathcal C}\) induces a pretopology on S and if (S,\({\mathcal C})\) is totally bounded then \({\mathcal C}\) is uniformizable. Furthermore the questions about the existence of completions and compactifications have very satisfactory solutions. Various examples show that the assumption of the Cauchy continuity of F cannot be dropped.
scientific article; zbMATH DE number 4129604