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.

💡 Research Summary

The paper investigates the metrization problem for topological preordered spaces, i.e., spaces equipped with both a topology T and a preorder ≤. Classical metrization theorems state that a topological space is metrizable when it is second‑countable and completely regular. When a preorder is added, one needs a distance‑like function that simultaneously generates the topology and encodes the order. The authors introduce a quasi‑pseudo‑metric p: E×E →