Quasi-pseudo-metrization of topological preordered spaces

TL;DR

This paper proves that second countable completely regularly preordered spaces are quasi-pseudo-metrizable and characterizes ordered spaces as subspaces of the ordered Hilbert cube, connecting to results in bitopology.

Contribution

It establishes quasi-pseudo-metrization for a broad class of preordered spaces and characterizes ordered spaces via the ordered Hilbert cube, linking to bitopological results.

Findings

Second countable completely regularly preordered spaces are quasi-pseudo-metrizable.

Ordered spaces can be characterized as order homeomorphic to subspaces of the ordered Hilbert cube.

Strictly quasi-pseudometrizable ordered spaces are order homeomorphic to subspaces of the ordered Hilbert cube.

Abstract

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.