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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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