Strong completions of spaces

TL;DR

This paper explores the concept of strongly complete $T_0$ spaces, where all irreducible subsets have suprema, and demonstrates that these spaces form a reflective subcategory within a broader category of $T_0$ spaces.

Contribution

It introduces the category of strongly complete $T_0$ spaces and proves it is a reflective subcategory of a larger category of $T_0$ spaces, advancing topological domain theory.

Findings

Strongly complete $T_0$ spaces have all irreducible subsets with existing suprema.

The category of strongly complete $T_0$ spaces is reflective within a certain subcategory.

This framework generalizes domain-theoretic concepts to non-Hausdorff topologies.

Abstract

A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respect to the specialization order) of a $T_0$ space are always irreducible, and (4) the topological closure of every irreducible set is again irreducible. In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2009) developed the core of domain theory directly in the context of $T_0$ spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo -- a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho (2009). The topological counterpart of a dcpo is thus this: A $T_0$ space is said to be strongly complete if the suprema of all irreducible subsets exist. In this paper, we show that the category, $\mathbf{scTop^+}$, of strongly complete $T_0$ spaces forms are reflective subcategory of a certain lluf subcategory, $\mathbf{Top^+}$, of $T_0$ spaces.