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.