Normally preordered spaces and utilities

TL;DR

This paper investigates conditions under which topological preordered spaces are normally preordered, showing that certain classes like $k_$-spaces and second countable spaces have these properties and admit utility representations.

Contribution

It establishes that $k_$-spaces with closed preorders are normally preordered and that second countable regularly preordered spaces are perfectly normally preordered with utility representations.

Findings

$k_$-spaces with closed preorders are normally preordered

Second countable regularly preordered spaces are perfectly normally preordered

Such spaces admit countable utility representations

Abstract

In applications it is useful to know whether a topological preordered space is normally preordered. It is proved that every $k_\omega$-space equipped with a closed preorder is a normally preordered space. Furthermore, it is proved that second countable regularly preordered spaces are perfectly normally preordered and admit a countable utility representation.