A Simple Characterization of Useful Topologies in Mathematical Utility Theory

In this paper, we present a 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. We introduce the concept of weak open and closed countable chain condition (WOCCC) relative to a topology, and we then show that a useful topology always satisfies this condition. The most important result in the paper shows that a completely regular topology is useful if and only if it is separable and it satisfies WFOCCC (a stricter version of WOCCC). In this way, we generalize all the previous results concerning useful topologies. We finish the paper by presenting a simple axiomatization of useful topologies under the well-known Souslin Hypothesis.