Upper semicontinuous utilities for all upper semicontinuous total preorders

Abstract

Let $X$ be an arbitrary nonempty set. Then a topology $t$ on $X$ is said to be completely useful (or upper useful) if every upper semicontinuous total preorder $\precsim$ on the topological space $(X,t)$ can be represented by an upper semicontinuous real-valued order-preserving function (i.e., utility function). In this paper the structures of completely useful topologies on $X$ will be deeply studied and clarified. In particular, completely useful topologies will be characterized through the new notions of super-short and strongly separable topologies. Further, the incorporation of the Souslin Hypothesis and the relevance of these characterizations in mathematical utility theory will be discussed. Finally, various interrelations between the concepts of complete usefulness and other topological concepts that are of interest not only in mathematical utility theory are analyzed.