Quasi-pseudo-metrization of topological preordered spaces

We establish that every second countable completely regularly preordered space ( E , T , ⩽ ) is quasi-pseudo-metrizable, in the sense that there is a quasi-pseudo-metric p on E for which the pseudo-metric p ∨ p − 1 induces 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.