Quasi-metrizability of the finest quasi-proximity
Salvador Romaguera · Publicationes Mathematicae Debrecen · 2000
A characterization of the bispaces whose finest quasi-proximity is quasi-metrizable is obtained in terms of real-valued quasi-proximally continuous functions.We also prove that for a doubly Hausdorff bispace X the following are equivalent: (i) X admits a quasi-metric for which every real-valued bicontinuous function is quasiuniformly continuous; (ii) the finest quasi-proximity of X is quasi-metrizable; (iii) the finest quasi-uniformity of X is quasi-metrizable.Examples showing that double Hausdorffness of X cannot be omitted in this result are given.As an application of our methods we deduce that the fine quasi-proximity (resp.quasi-uniformity) of a T 1 topological space X is quasi-metrizable if and only if X admits a quasi-metric for which every lower semicontinuous function is quasi-proximally (resp.quasi-uniformly) continuous.We also deduce that if the finest quasi-proximity of a Hausdorff topological space X is quasi-metrizable, then its fine quasi-uniformity is quasimetrizable and, thus, X is a metrizable space with only finitely many nonisolated points.