Alas, Ofelia T.; Tkachenko, Michael G.; Wilson, Richard G. Which topologies have immediate predecessors in the poset of Hausdorff topologies? (English) Zbl 1169.54002 Houston J. Math. 35, No. 1, 149-158 (2009). 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. Reviewer: Bernhard Behrens (Göteborg) Cited in 4 Documents 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 Keywords:lattice of \(T_1\)-topologies; poset of (first countable) Hausdorff topologies; upper (lower) topology; submaximal space; minimal Hausdorff space; \(H\)-closed space; dispersed space Citations:Zbl 1163.54001 PDFBibTeX XMLCite \textit{O. T. Alas} et al., Houston J. Math. 35, No. 1, 149--158 (2009; Zbl 1169.54002) Full Text: Link