Coarser Compact Topologies

TL;DR

This paper introduces quasi-king spaces as a generalization of king spaces, showing their equivalence to compactness in coarser selection topologies within weakly orderable spaces.

Contribution

It defines quasi-king spaces and establishes their characterization via coarser selection topologies in weakly orderable spaces.

Findings

Quasi-king spaces are equivalent to compactness in coarser selection topologies.

In suborderable spaces, king and quasi-king spaces coincide with compact spaces.

The paper provides a new perspective on the structure of weakly orderable spaces.

Abstract

It is introduced the concept of a quasi-king space, which is a natural generalisation of a king space. In the realm of suborderable spaces, king spaces are precisely the compact spaces, so are the quasi-king spaces. In contrast, quasi-king spaces are more flexible in handling coarser selection topologies. The main purpose of this paper is to show that a weakly orderable space is quasi-king if and only if all of its coarser selection topologies are compact.