Topological spaces for which every continuous total preorder can be represented by a continuous utility function

Abstract

Let X be an arbitrary set. Then many different topologies t are definable on X. In utility theory a topology t on X is ‘useful’, if every continuous total preorder ‘≲’ on ( X, t) can be represented by a continuous utility function. The well-known Eilenberg-Debreu theorems, for example, provide conditions for a topology t on X to be ‘useful’. The main theorem of this paper generalizes the Eilenberg-Debreu theorems. Furthermore, the characterization of all ‘useful’ topologies t on X is intimately related to the characterization problem of orderable topological spaces, which was solved for the connected case by Eilenberg.