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On the product of a compact space with an absolutely countably compact space. (English) Zbl 0917.54023

Coplakova, Eva (ed.) et al., Papers on general topology and applications. Proceedings of the 10th summer conference, Amsterdam, Netherlands, August 15–18, 1994. New York, NY: New York Academy of Sciences. Ann. N. Y. Acad. Sci. 788, 203-208 (1996).
It is known that a space \(X\) is countably compact if and only if for every open cover \(\mathcal U\), there is a finite subset \(F\) of \(X\) such that \(St(F,\mathcal U)=X\). A space \(X\) is absolutely countably compact (acc) provided that for every open cover \(\mathcal U\) and dense subset \(D\), there is a finite subset \(F\) of \(D\) such that \(St(F,\mathcal U)=X\). Unlike countable compactness, the stronger acc property of \(X\) is not preserved by taking a product of \(X\) with a compact space. In this paper, sufficient conditions are given for such a product with a compact space to be acc. The main result is that if \(Y\) is a compact sequential \(T_2\)-space and \(X\) is an acc \(T_3\)-space then \(X\times Y\) is acc. It also follows from a related result that every countably compact GO-space is acc.
For the entire collection see [Zbl 0903.00049].

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10 Product spaces in general topology
54D55 Sequential spaces
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