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 utility representation. particular, show that, for completely regular spaces, topology is useful, if and only it separable, isolated chain open closed sets countable. As specific application optimization theory, characterize the representability all preorders, admit both maximal minimal element.