Sober topologies on a set

TL;DR

This paper explores the structure and properties of sober topologies within the complete lattice of all topologies on a set, revealing their relationships with T1, Alexanderoff-discrete, and Scott topologies, and providing new insights into their lattice-theoretic behavior.

Contribution

It characterizes the placement of sober topologies in the lattice of all topologies, including their joins, meets, and minimal elements, and presents new examples and structural results.

Findings

Every T1 topology is a join of sober topologies.

Every topology is a meet of sober topologies.

The set of sober topologies is directed complete.

Abstract

The collection of all topologies on a set X forms a complete lattice with respect to the inclusion order, which have been investigated by many researchers. Sobriety is one of the core and extensively studied properties in non-Hausdorff topology. This property plays a crucial role in characterizing the spectral spaces of commutative rings and topological spaces determined by their lattices of open sets. In this paper, we investigate the statute of sober topologies in the complete lattice of all topologies on a given set. The main results to be proved include: (1) every T1 topology is the join of some sober topologies; (2) every topology is the meet of some sober topologies; (3) the set of all sober topologies is directed complete; (4) every Alexanderoff - discrete topology is the meet of some sober Alexanderoff - discrete topologies; (5) the minimal sober topologies are exactly the Scott topologies of sup-complete chains; (6) an example will be constructed to show that the intersection of a decreasing sequence of Hausdorff topologies need not be sober.