Topologies for the Continuous Representability of All Continuous Total Preorders

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies
on an arbitrary set, with respect to which every continuous total preorder admits
a continuous utility representation. In particular, we show that, for completely regular
spaces, a topology is useful, if and only if it is separable, and every isolated chain of
open and closed sets is countable. As a specific application to optimization theory, we
characterize the continuous representability of all continuous total preorders, which
admit both a maximal and a minimal element.