Some Utility Theorems on Inductive Limits of Preordered Topological Spaces

One of the main problems in utility theory is to find conditions which imply the numerical representability of topological spaces on which a preorder, or partial order, is defined. This study has reached a wide development in the literature: see, for instance, the works of Eilenberg, Debreu, Fleischer and Jaffray on completely preordered topological spaces [9, 7, 10, 15], and the seminal book by Nachbin together with the papers by Mehta and Herden [22, 20, 21, 11, 12], in the framework of preordered topological spaces. In general, the existence of a continuous representation is closely related to topological conditions that he "in the neighbourhood of separability" (see [4] and [5]). Recently, some very general conditions for the existence of a continuous order preserving function have been obtained by Herden [11, 12]. The work of Herden is based on Nachbin's theory of normally preordered spaces. The objective of this paper is to consider general families of preordered (or partially ordered) spaces that admit a representation in a natural way. The idea of using a family of preference relations on a space has been outlined by Aumann [3, p.241] and others. In this paper we introduce another class of preordered spaces which admit a (utility) representation, the class of topological spaces that are inductive limits of preordered spaces. In particular, our main result proves the existence of a continuous representation of a topological space that is a suitable inductive limit of compact order-separable spaces. Then we apply this result to locally compact cr-compact preordered spaces and completely preordered topological vector spaces. Also, we extend the result to a broader class of spaces showing through an example that some elements in this class are neither compact nor order-separable. We conclude by using the idea of an inductive limit to make precise the notion of a local utility function and to prove an infinite-dimensional version of the classical Arrow-Hahn theorem on utility functions.