Topological conditions for the representation of preorders by continuous utilities

TL;DR

This paper extends Levin's theorem by removing the Hausdorff condition, showing that certain topological assumptions are equivalent to the space being Polish, thus broadening the applicability of continuous utility representations for preorders.

Contribution

It generalizes Levin's theorem by eliminating the Hausdorff requirement and establishing equivalences with Polish space assumptions in preorder representations.

Findings

Hausdorff condition can be removed from Levin's theorem

Topological assumptions are equivalent to the space being Polish

Broader class of spaces admits continuous utility representation

Abstract

We remove the Hausdorff condition from Levin's theorem on the representation of preorders by families of continuous utilities. We compare some alternative topological assumptions in a Levin's type theorem, and show that they are equivalent to a Polish space assumption.