Compactification of closed preordered spaces

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.

💡 Research Summary

The paper investigates compactifications of topological preordered spaces that preserve both the Hausdorff topology and the closed preorder. A topological preordered space ((E,\mathscr{T},\le)) consists of a topological space ((E,\mathscr{T})) together with a reflexive, transitive, and closed relation (\le) (i.e., its graph is a closed subset of (E\times E)). The authors introduce the notion of a “closed preorder compactification”: a compact Hausdorff space (\widehat{E}) equipped with a closed preorder (\widehat{\le}) together with a continuous isotone embedding (i:E\to\widehat{E}) such that (i) is initial with respect to both the topology and the preorder.

The central result (Theorem 1) provides a necessary and sufficient condition for the existence of such a compactification. It states that a closed preorder compactification exists if and only if (i) the underlying space is Tychonoff, and (ii) the preorder can be completely represented by a family (\mathcal{F}) of continuous isotone functions (f:E\to