chapter  7
45 Pages

The Quasi-Metrization Problem

WithFletcher Peter, William F. Lindgren

As we observed in Chapter 1, a widely known result in the study of quasi-uniform spaces is the theorem of A. H. Frink that a T1 space admits a quasi-uniformity with a countable base if and only if it is quasi-metrizable. We begin this section by considering the problem of finding necessary and sufficient topological conditions for a topological space to admit a quasi-metric. Unlike the corresponding metrization problem, no completely satisfactory solution to the quasi-metrization problem has yet been found; nonetheless there is a purely topological characterization of the non- Archimedeanly quasi-metrizable spaces.