Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4661516 | Topology and its Applications | 2007 | 6 Pages |
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.
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