Coarser compact topologies

Abstract

The concept of a quasi-king space is introduced, which is anatural generalisation of a king space. In the realm of suborderablespaces, king spaces are precisely the compact spaces, so are thequasi-king spaces. In contrast, quasi-king spaces are more flexiblein handling coarser selection topologies. The main purpose of thispaper is to show that a weakly orderable space is quasi-king if andonly if all of its coarser selection topologies are compact.