Bispaces admitting only bicomplete or only totally bounded quasi-metrics (Q1851418)

The authors extend the following result: a (pseudo) metrizable topological space admits only complete (pseudo) metrics if and only if it is compact. In section 2 the above result is studied for quasi-(pseudo) metrizable bispaces. An example shows that it is not true that quasi-metrizable bispaces \((X,S,T)\) admit only bicomplete quasi-metrics if and only if \((X, S\vee T)\) is compact. In section 3, the authors discuss the problem when quasi-pseudometrizable bispaces admit totally bounded quasi-pseudometrics.