Convexity and quasi-uniformizability of closed preordered spaces

TL;DR

This paper investigates conditions under which closed preordered spaces are convex and quasi-uniformizable, focusing on locally compact, σ-compact spaces, and applies findings to spacetime causality structures.

Contribution

It establishes that locally convex, locally compact σ-compact spaces with closed preorders are convex and quasi-uniformizable, and explores conditions for local convexity and quasi-pseudo-metrizability.

Findings

Locally convex, locally compact σ-compact spaces are convex.

Preorders generated by cones over manifolds are locally convex.

Every stably causal spacetime is quasi-uniformizable.

Abstract

In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as quasi-uniformizability, allows one to compactify the topological space and to extend its order dynamics. In this work we study locally compact $\sigma$-compact spaces endowed with a closed preorder. They are known to be normally preordered, and it is proved here that if they are locally convex, then they are convex, in the sense that the upper and lower topologies generate the topology. As a consequence, under local convexity they are quasi-uniformizable. The problem of establishing local convexity under antisymmetry is studied. It is proved that local convexity holds provided the convex hull of any compact set is compact. Furthermore, it is proved that local convexity holds whenever the preorder is compactly generated, a case which includes most examples of interest, including preorders determined by cone structures over differentiable manifolds. The work ends with some results on the problem of quasi-pseudo-metrizability. As an application, it is shown that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable.