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Some thoughts on countable Lindelöf products. (English) Zbl 1426.54012

It is well-known that the Cartesian product of two Lindelöf spaces is not necessarily a Lindelöf space. Even more, under the assumption of the Continuum Hypothesis (CH) E. A. Michael [Compos. Math. 23, 199–214 (1971; Zbl 0216.44304)] showed the following:
(a) There exists a first countable space \(X\) such that \(X^n\) is Lindelöf for every \(n<\omega\) but \(X^\omega\) is not normal;
(b) For every \(n<\omega\) there exists a first countable space \(X=X(n)\) such that \(X^n\) is Lindelöf but \(X^{n+1}\) is not normal;
(c) For every \(n<\omega\) there exists a first countable space \(X=X(n)\) such that \(X^n\) is Lindelöf and \(X^{n+1}\) is paracompact but \(X^{n+1}\) is not Lindelöf.
Later K. Alster and P. Zenor [in: Set-theor. Topol., Vol. dedic. to M. K. Moore, 1–10 (1977; Zbl 0372.54016)] again assuming CH showed that for evey \(n<\omega\) there exists a first countable separable space \(X=X(n)\) such that \(X^n\) is Lindelöf and \(X^{n+1}\) is (collectionwise) normal, but \(X^{n+1}\) is not paracompact.
In [Fundam. Math. 105, 87–104 (1980; Zbl 0438.54021)], T. C. Przymusiński showed that the assumption of the Continuum Hypothesis can be omitted in the above four theorems.
On the other hand, there are non-compact examples of spaces \(X\), usually called powerfully Lindelöf, for which \(X^\omega\) is Lindelöf. Even more, as the author of the paper under review points out, there are classes of spaces such that the product of each of its countable subcollections is Lindelöf. The author calls such collections of topological spaces powerfully Lindelöf.
In this well-written paper the author studies powerfully Lindelöf collections of topological spaces, gives an overview of many results related to such collections, and generalizes some of them. The reader can find in this paper several techniques used by different authors to prove countable products Lindelöf as well as some open questions.

MSC:

54B10 Product spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54G10 \(P\)-spaces
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