Quasi-pseudo-metrization of topological preordered spaces

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.Comment: Latex2e, 20 pages. v2: minor changes in the proof of theorem 2.

Convexity and quasi-uniformizability of closed preordered spaces

“…Quasi-uniformizable topological preordered spaces are among the most well behaved topological preordered spaces. They admit completions and compactifications [7,8,45,27], and under second countability they can be shown to be quasi-pseudo-metrizable [35].…”

“…We remark that we are not claiming that the topology induced by p is the upper topology T ♯ and that induced by p −1 is the lower topology T ♭ (which would be true if we could prove strict quasi-pseudo-metrizability [35]).…”

“…Every strictly quasi-pseudo-metrizable preordered space is a quasi-pseudometrizable preordered space. Every quasi-pseudo-metrizable preordered space is a completely regularly preordered space [35,Prop. 2.3].…”

“…The topological ordered space is a completely regularly ordered space (quasi-uniformizable) by theorem 3.3 (in the k-preserving case) or theorem 4.14. Thus E is a separable quasi-pseudo-metric space by [35,Theor. 2.5].…”

Compactification of closed preordered spaces

“…We could also try a different approach by first showing that the spacetime is not only a topological preordered space, but in fact a quasi-pseudo-metric space, and then completing it with a preorder generalization of the Cauchy completion. Unfortunately, although we could prove, using the results of [20], that most interesting spacetimes are quasi-pseudo-metrizable, the completion would depend on the chosen quasi-pseudo-metric. Therefore, this strategy is not entirely viable unless we prove the existence of some natural spacetime quasi-pseudo-metric.…”