Topological conditions for the representation of preorders by continuous utilities

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.

💡 Research Summary

The paper revisits Levin’s theorem on the representation of preorders by families of continuous utility functions and removes the traditional Hausdorff requirement on the underlying topological space. A preorder ((E,T,\le)) is a reflexive, transitive relation on a topological space ((E,T)). A utility function is an isotone map (f:E\to