Some remarks concerning the Lindelöf property of the product of a Lindelöf space with the irrationals (Q1203812)

Assuming (CH), the reviewer showed in Bull. Am. Math. Soc. 69, 375-376 (1963; Zbl 0114.389) that there exists a regular Lindelöf space \(X\) such that \(X\times P\), where \(P\) denotes the irrationals, is not Lindelöf (or even normal). It has since been shown that such a space exists under assumptions weaker than (CH), but it remains unknown whether one exists in ZFC. The author makes new contributions to this problem, primarily by studying its relationship to certain questions about a partial order on \(P\) introduced by F. Rothberger.