\(\kappa\)-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.

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, LindelĂ¶f, etc.) |

54B10 | Product spaces in general topology |

54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |