Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or \(D\)
A space \(X\) is \(D\) (introduced by \textit{E. K. van Dowen} and \textit{W. F. Pfeffer} [Pac. J. Math. 81, 371--377 (1979; Zbl 0409.54011)]) if for every neighbourhood assignment \(\{\mathbf V_x\}_{x \in X}\), that is, each \(\mathbf V_x\) is an open set containing \(x\), there is a closed discrete \(Y \subseteq X\) such that \(\{\mathbf V_x\}_{x \in Y}\) covers \(X\). A space \(X\) is productively Lindelöf (as defined by \textit{M. Barr, J. F. Kenison} and \textit{R. Raphael} [Sci. Math. Jpn. 65, No. 3, 319--332 (2007; Zbl 1146.54013)]), if \(X \times Y\) is Lindelöf for any Lindelöf space \(Y\). A space is Alster [\textit{K. Alster}, Topology Appl. 44, No.1-3, 19--25 (1992; Zbl 0766.54017)] if every cover by \(G_\delta\) sets that covers each compact set finitely includes a countable subcover. A space is Rothberger (or Menger) if for each sequence \(\{\mathbf U_n\}_{n<\omega}\) of open covers (such that each finite union of elements of \(\mathbf U_n\) is a member of \(\mathbf U_n\)), there are \(\mathbf U_n\) in \(\mathbf U_n, n< \omega\) such that \(\{\mathbf U_{n}:n< \omega\}\) is an open cover. A \(\gamma\)-cover of a space is a countably infinite open cover such that each point is in all but finitely many members of the cover. A space is Hurewicz if given a sequence \(\{\mathbf U_n\}_{n< \omega}\) of \(\gamma\)-covers, there is for each \(n\), a finite \(\mathbf V_n \subseteq \mathbf U_n\) such that either \(\{\cup \mathbf V_n : n \varepsilon\omega\}\) is a \(\gamma\)-cover, or else, for some \(n, \cup \mathbf V_n\) is a cover. In this note which is a continuation of an earlier paper by \textit{L. F. Aurichi} and the author [Topology Appl. 159, No. 1, 331--340 (2012; Zbl 1234.54032)], various relationships in Lindelöf spaces among the above properties are discussed (in the presence of CH, MA, Axiom of Projective Determinacy and so on).
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