A regular non-weakly discretely generated \(P\)-space
A space is discretely generated if the topology is generated by its discrete subsets in the sense that if \(x\in\overline A\) then \(x\in\overline D\) for some (relatively) discrete subset \(D\) of \(A\). It is \textit{weakly} discretely generated if every non-closed set \(A\) contains a (relatively) discrete subset \(D\) such that \(\overline D\setminus A\neq\emptyset\).\par After surveying what is known about these properties the authors construct, under the assumption that \(2^{\aleph_0}=\aleph_1\) and \(2^{\aleph_1}=\aleph_2\), a regular \(P\)-space that is not weakly discretely generated. This is related to the result that compact Hausdorff spaces are weakly discretely generated; since Lindelöf \(P\)-spaces behave much like compact Hausdorff spaces the question whether they are also weakly discretely generated is a natural one. The present example is the first regular \(P\)-space that is not weakly discretely generated; but it is not Lindelöf. Thus the original question remains open as well as the question for a regular example in ZFC.
scientific article; zbMATH DE number 7491525