Quasi-pseudo-metrization of topological preordered spaces

Abstract

We establish that every second countable completely regularly preordered space $(E,\mathcal{T},⩽)$ is quasi-pseudo-metrizable, in the sense that there is a quasi-pseudo-metric p on E for which the pseudo-metric $p\vee p^{-1}$ induces $\mathcal{T}$ and the graph of ⩽ 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-pseudo-metrizable ordered spaces are characterized as being order homeomorphic to order subspaces of the ordered Hilbert cube.