Compact productivity of Lindelöf-type properties
All hypothesized spaces are Hausdorff. For a space \(X\) and topological property \(P\), the space \(X\) is said to be \textit{almost cellular-}\(P\) (resp., \textit{strongly almost cellular}-\(P\)) if whenever \(\mathcal{U}\) is a pairwise disjoint family of nonempty open sets of \(X\) there is a subspace (resp., closed subspace) of \(X\) which has property \(P\) and meets \(\vert \mathcal{U}\vert \) elements of \(\mathcal{U}\). \par A few of the many very nice results the authors establish, some of which they use to answer previously published questions or raise new ones, are the following. A generalized Isbell-Mrówka \(\Psi\)-space extension of the discrete space \(\omega_1\) (Example 2.6) is a Tychonoff strongly almost cellular-Lindelöf space, but is neither cellular-Lindelöf nor weakly Lindelöf. The Tychonoff space due to \textit{M. Bell} et al. [Pac. J. Math. 79, 37--45 (1979; Zbl 0367.54003)], shown by them to be weakly Lindelöf but not Lindelöf, is also not almost cellular-Lindelöf. A feebly Lindelöf space with a rank 3 diagonal has cardinality at most \(\mathfrak c\). Thus, no first countable almost cellular-Lindelöf space has cardinality \(>\mathfrak c\). For a space \(X\), \(X\times K\) is almost cellular-Lindelöf for each compact space \(K\) iff \(X\times A_\kappa\) is almost cellular-Lindelöf for each infinite cardinal \(\kappa\) (where \(A_\kappa\) denotes the one-point compactification of a discrete space of cardinality \(\kappa\)). For a space \(X\), \(X\times K\) is almost cellular-Lindelöf (resp., almost strongly cellular-Lindelöf) whenever \(K\) is compact iff for every set \(\mathcal{U}\) of nonempty open subsets of \(X\), there is a Lindelöf subspace (resp., a closed Lindelöf subspace) of \(X\) which meets \(\vert \mathcal{U}\vert \)-many elements of \(\mathcal{U}\). Applying this characterization theorem to the space in Example 2.6, they present a proof providing a negative answer to the natural question: If \(X\) is a space such that \(X\times K\) is almost cellular-Lindelöf for every compact space \(K\), must \(X\) have a dense Lindelöf subspace?
scientific article; zbMATH DE number 7614061
scientific article