On the structure of completely useful topologies

Abstract

Let X be an arbitrary set. Then a topology t on X is completely useful if every upper semicontinuous linear preorder on X can be represented by an upper semicontinuous order preserving  real-valued function. In this paper we characterize in ZFC (<span style="text-decoration: underline;">Z</span>ermelo-<span style="text-decoration: underline;">F</span>raenkel + Axiom of <span style="text-decoration: underline;">C</span>hoice) and ZFC+SH (ZFC + <span style="text-decoration: underline;">S</span>ouslin <span style="text-decoration: underline;">H</span>ypothesis) completely useful topologies on X. This means, in the terminology of mathematical utility theory, that we clarify the topological structure of any type of semicontinuous utility representation problem.