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Some questions on property (a). (English) Zbl 1002.54016

The author introduces an interesting new property, called property (a), which combines aspects of open covers with aspects of dense sets. A space \(X\) is said to have property (a) provided for every open cover \({\mathcal U}\) of \(X\) and every dense subset \(D\subset X\) there exists a subset \(F\subset D\) such that \(F\) is closed in \(X\), \(F\) is a discrete subspace of \(X\), and \(\text{st}(F,{\mathcal U})=X\) (where \(\text{st}(F,{\mathcal U})= \bigcup\{U\in {\mathcal U}: U\cap F\neq \emptyset\}\)). Property (a) is motivated by the desire “to find out what is absolute countable compactness minus countable compactness”. Absolute countable compactness, introduced by the author [Topology Appl. 58, No. 1, 81-92 (1994; Zbl 0801.54021)], is equivalent to “countable compactness + property (a)”.
The main purpose of the paper is to raise questions about property (a). A theme that motivates many of the questions in this paper is an unexpected analogy between property (a) and normality. The author gives some results that illustrate this analogy. For example, the main theorem in the paper states: If \(X\) is a normal, countably paracompact space with property (a), then the space \(Y=X\times(\omega+1)\) has property (a). This may be compared with the famous theorem of C. H. Dowker [Can. J. Math. 3, 219-224 (1951; Zbl 0042.41007)] which states that if \(X\) is normal and countably paracompact, then \(Y=X\times (\omega+1)\) is normal. One of the questions asks whether it is possible to delete the assumption of normality from the author’s theorem (thus giving a closer analogy with Dowker’s theorem). Two topological games related to property (a) are also considered.
Reviewer’s remarks: Normality and property (a) are distinct even in the class of countably compact \(T_2\)-spaces; as noted in this paper, an example by the reviewer [Proc. Am. Math. Soc. 75, 339-342 (1979; Zbl 0412.54023)] provides a countably compact space with property (a) that is not normal. Recently O. I. Pavlov constructed an example of a countably compact, normal space that does not have property (a) (answering a question attributed to A. V. Arkhangel’skij in the author’s paper cited above.

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D30 Compactness
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