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Which topologies have immediate predecessors in the poset of Hausdorff topologies? (English) Zbl 1169.54002

The authors study \(\Sigma_2\), the poset of Hausdorff topologies on a set \(X\). They show that a submaximal \(H\)-closed space is upper in \(\Sigma_2\) iff it is not minimal Hausdorff, that certain dispersed non-minimal Hausdorff spaces are upper in \(\Sigma_2\) and that between two distinct first countable Hausdorff topologies there are at least \(c\) incomparable first countable topologies; further, they construct a family of \(H\)-closed, not minimal Hausdorff topologies which are not upper in \(\Sigma_2\) (also, see C. Costantini [Topol. Proc. 32, 187–225 (2008; Zbl 1163.54001]). Some questions are raised.

MSC:

54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
06A06 Partial orders, general
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54G20 Counterexamples in general topology

Citations:

Zbl 1163.54001
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