A quasi-pseudometrizability problem for ordered metric spaces

Abstract

In this dissertation we obtain several results in the setting of ordered topological spaces related to the Hanai-Morita-Stone Theorem. The latter says that if f is a closed continuous map of a metric space X onto a topological space Y then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) For each y 2 Y, fâ1{y} has a compact boundary in X; (iii) Y is metrizable. A partial analogue of the above theorem for ordered topological spaces is herein obtained. We particularly investigate the upper and lower topologies of metrizable ordered spaces which are both C- and I-spaces in the sense of Priestley. Among other results, we show that the bitopological spaces (bispaces) consisting of the upper topology and the lower topology associated with metrizable ordered spaces which are C- and I-spaces satisfying properties like separability and local connectedness are quasi-pseudometrizable. Also, a partial order called friendly partial order is introduced and characterized. Furthermore, we show that a specified bispace associated with any uniform space endowed with this kind of partial order is quasi-uniformizable. Some interesting examples are also discussed.