Lower and upper topologies in the Hausdorff partial order on a fixed set |
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Authors: | Nathan Carlson |
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Affiliation: | Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., PO Box 210089, Tucson, AZ 85721-0089, USA |
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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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Keywords: | 54A10 54D25 54D35 54D40 |
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