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Lower and upper topologies in the Hausdorff partial order on a fixed set. (English summary) Topology Appl. 154 (2007), no. 3, 619–624. Let TOP1 denote the lattice of T1 topologies on a set X, where the partial order on TOP1 is defined by inclusion, and let TOP2 be the partially ordered set of T2 topologies on the same set X, the order on TOP2 being defined also by inclusion. If τ � σ are topologies in either TOP1 or TOP2 such that from τ ⊆ µ ⊆ σ it follows that µ = τ or µ = σ, then τ is said to be a lower topology and σ an upper topology. The author considers the existence of lower and upper topologies. He observes that there are upper topologies with two distinct lower topologies, lower topologies with two distinct upper topologies, and topologies which are both lower and upper topologies. He gives a characterization of upper topologies in TOP2 and shows that there exists a countably compact, H-closed lower topology of countable tightness. The last result answers a question by O. T. Alas and R. G. Wilson [Appl. Gen. Topol. 5 (2004), no. 2, 231–242; MR2121791 (2005i:54002)], who asked whether a countably compact Hausdorff topology of countably tightness can be a lower topology. Reviewed by Ljubiˇsa Kočina

Year: 2013

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