Convexity and quasi-uniformizability of closed preordered spaces (Q390817)

The author studies various conditions under which \(T_2\)-preordered topological spaces are locally convex, convex, quasi-uniformizable or quasi-pseudo-metrizable.NEWLINENEWLINEAmong other things, he proves that each locally compact \(\sigma\)-compact locally convex \(T_2\)-preordered space is a convex normally preordered space (and hence quasi-uniformizable).NEWLINENEWLINEHe also investigates two properties that under appropriate assumptions guarantee local convexity, namely the property (1) that the convex hull of every compact set is compact, orNEWLINENEWLINEthe property (2) that the preorder is ``compactly generated'' in the sense that there is a relation \(R\subseteq G(\leq)\) such that (i) for every compact set \(K\) the set \(\overline{R(K)}\) is compact and (ii) the preorder \(\leq\) is the smallest closed preorder containing \(R.\)NEWLINENEWLINEIt follows for instance that every \(T_2\)-ordered locally compact \(\sigma\)-compact space with property (1) is (locally) convex and thus quasi-uniformizable. Furthermore every locally compact \(T_2\)-ordered space satisfying property (2) is locally convex.NEWLINENEWLINEAs an application of the results the author states that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable.