Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions

Under the additional assumption of complete regularity, we furnish a simple characterization of all the topologies such that every continuous total preorder is representable by a continuous utility function. In particular, we prove that a completely regular topology satisfies such property if, and only if, it is separable and every linearly ordered collection of clopen sets is countable. Since it is not restrictive to refer to completely regular topologies when dealing with this kind of problem, this is, as far as we are concerned, the simplest characterization of this sort available in the literature. All the famous utility representation theorems are corollaries of our result.