Compactification of closed preordered spaces

TL;DR

This paper characterizes when a topological preordered space can be compactified with a Hausdorff closed preorder, constructs the maximal such compactification, and discusses the smallest one under local compactness.

Contribution

It provides a complete characterization and construction of the largest Hausdorff closed preorder compactification, linking it to Nachbin's compactification and addressing the minimal case under local compactness.

Findings

Characterization of spaces admitting Hausdorff closed preorder compactification

Construction of the largest such compactification

Analysis of the smallest compactification under local compactness

Abstract

A topological preordered space admits a Hausdorff closed preorder compactification if and only if it is Tychonoff and the preorder is represented by the family of continuous isotone functions. We construct the largest Hausdorff closed preorder compactification for these spaces and clarify its relation with Nachbin's compactification. Under local compactness the problem of the existence and identification of the smallest Hausdorff closed preorder compactification is considered.