Quasi-pseudo-metrization of topological preordered spaces

We establish that every second countable completely regularly preordered space (E,T,\leq) is quasi-pseudo-metrizable, in the sense that there is a quasi-pseudo-metric p on E for which the pseudo-metric p\veep^-1 induces T and the graph of \leq is exactly the set {(x,y): p(x,y)=0}. In the ordered case it is proved that these spaces can be characterized as being order homeomorphic to subspaces of the ordered Hilbert cube. The connection with quasi-pseudo-metrization results obtained in bitopology is clarified. In particular, strictly quasi-pseudometrizable ordered spaces are characterized as being order homeomorphic to order subspaces of the ordered Hilbert cube.Comment: Latex2e, 20 pages. v2: minor changes in the proof of theorem 2.

A quasi-pseudometrizability problem for ordered metric spaces

Includes abstract.Includes bibliographical references (leaves 83-88).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, f−1{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

The Quasimetrization Problem in the (Bi)topological Spaces

It is our main purpose in this paper to approach the quasi-pseudometrization problem in (bi)topological spaces in a way which generalizes all the well-known results on the subject naturally, and which is close to a “Bing-Nagata-Smirnov style” characterization of quasi-pseudometrizability