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\(\kappa\)-Dowker spaces. (English) Zbl 0566.54009
Aspects of topology, Mem. H. Dowker, Lond. Math. Soc. Lect. Note Ser. 93, 175-193 (1985).
[For the entire collection see Zbl 0546.00024.]
A Dowker space is a normal space X for which \(X\times I\) is not normal, or equivalently, a normal space X such that every countable open cover of X can be shrunk. Let \(\kappa\) be an infinite regular cardinal number. The author defines the following properties of a normal space X: (1) X is \(\kappa\)-paracompact if every open cover of X of cardinality \(\leq \kappa\) has a locally finite refinement, (2) X is \(\kappa\)-\({\mathcal B}\) if every monotone open cover of X of cardinality \(\leq \kappa\) has a monotone shrinking, (3) X is \(\kappa\)-\({\mathcal D}\) if every monotone open cover of X of cardinality \(\leq \kappa\) can be shrunk, and (4) X is \(\kappa\)- shrinking if every open cover of X of cardinality \(\leq \kappa\) can be shrunk. From the above, which is known as Dowker’s Theorem, it follows that all these properties are the same for \(\kappa =\omega\). The author carefully illustrates that these notions are of importance in the complicated area of normality in products. Since \(\kappa\)-\({\mathcal D}\) is the weakest of all she defines a normal space X to be \(\kappa\)-Dowker if it is not a \(\kappa\)-\({\mathcal D}\) space. There are \(\kappa\)-Dowker spaces for every regular cardinal \(\kappa\). This interesting paper is partly a survey and partly a research paper. The author states, often with proof, almost everything that is known about Dowker spaces and their generalizations.
Reviewer: J.van Mill

MSC:
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10 Product spaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)