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Lindelöf spaces which are indestructible, productive, or \(D\). (English) Zbl 1234.54032

This interesting study may be considered as a continuation of a recent paper of F. D. Tall and B. Tsaban concerning productively Lindelöf spaces [“On productively Lindelöf spaces”, Topology Appl. 158, No. 11, 1239–1248 (2011; Zbl 1229.54032)], where a topological space \(X\) is called productively Lindelöf if \(X\times Y\) is Lindelöf for every Lindelöf space \(Y\). It is shown that the following assertions are equivalent:
(i) every completely metrizable productively Lindelöf space is \(\sigma\)-compact; (ii) every completely metrizable productively Lindelöf space is Alster; (iii) every completely metrizable productively Lindelöf space is Menger; (iv) there exists a Lindelöf space \(X\) such that \(X\times\mathbb{P}\) is not Lindelöf, where \(\mathbb{P}\) is the space of irrationals.
The axiom \({\mathfrak d}=\aleph_1\) implies that every metrizable productively Lindelöf space is Hurewicz (and therefore Menger). Moreover, the continuum hypothesis implies that every separable productively Lindelöf \(T_3\)-space is a \(d\)-space in the sense of E. K. van Douwen and W. F. Pfeffer [“Some properties of the Sorgenfrey line and related spaces”, Pac. J. Math. 81, 371–377 (1979; Zbl 0409.54011)], and that every first countable productively Lindelöf \(T_3\)-space is an indestructible \(d\)-space, i.e., a \(d\)-space in every extension obtained by countably closed forcing. The authors call a topological space indestructibly Lindelöf (respectively, indestructibly productively Lindelöf) if it is Lindelöf (respectively, productively Lindelöf) in every extension obtained by countably closed forcing. It is shown that a metrizable space is indestructibly productively Lindelöf if and only if it is \(\sigma\)-compact. The various relationships between these and many other classes of Lindelöf spaces are summarized in a very useful diagram.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D99 Fairly general properties of topological spaces
03E35 Consistency and independence results
54G20 Counterexamples in general topology
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References:

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