Lower and upper topologies in the Hausdorff partial order on a fixed set

Abstract

In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ ⊊ σ in which there is no Hausdorff topology μ satisfying σ ⊊ μ ⊊ τ . τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies.