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Some nowhere densely generated topological properties. (English) Zbl 0588.54023
A topological property P is said to be nowhere densely generated in a class C of topological spaces if, for every space X in C, X has property P whenever every nowhere dense closed subset of X has property P. For example, M. Katětov [Čas. Mat. Fys. 72, 101-106 (1947; Zbl 0041.515)] proved that compactness is nowhere densely generated in the class of \(T_ 1\)-spaces without isolated points. More generally, C. F. Mills and E. Wattel [Topological structures II, Proc. Symp. Amsterdam 1978, Part 2, Math. Cent. Tracts 116, 191-198 (1979; Zbl 0448.54021)] proved that for all infinite cardinals \(\kappa\) and \(\lambda\) with \(\kappa\leq \lambda\), [\(\kappa\),\(\lambda\) ]-compactness is nowhere densely generated in the class of \(T_ 1\)-spaces without isolated points. In the paper under review the author obtains a characterization of [\(\kappa\),\(\lambda\) ]-compactness from which the Mills-Wattel result follows as a corollary. Theorem: Let X be a \(T_ 1\) space, let \(\kappa\) and \(\lambda\) be infinite cardinals with \(\kappa\leq \lambda\). Then X is [\(\kappa\),\(\lambda\) ]-compact if and only if every closed screenable subset of X has cardinality \(<\kappa\), and every nowhere dense closed subset of X is [\(\kappa\),\(\lambda\) ]-compact. (A subset D of X is said to be screenable if there is a pairwise disjoint open collection \(\{V_ x: x\in D\}\) in X such that \(x\in V_ x\) for all \(x\in D.)\) The paper contains similar results for \(\kappa\)-compactness, \(\kappa\)-closed completeness, and pseudo-[\(\kappa\),\(\lambda\) ]-compactness.

MSC:
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D30 Compactness
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