On products of discretely generated spaces
A space \(X\) is said to be discretely generated if for any \(A\subseteq X\) and any \(x\in \overline{A}\), there is a discrete \(D\subset A\) such that \(x\in\overline{D}\). Two related notions are also considered in this paper: \(X\) is weakly discretely generated if any non-closed subset \(A\) contains a discrete subset that is not closed; and \(X\) is strongly discretely generated if for any subset \(A\), any \(x\in \overline{A}\) is contained the closure of a strongly discrete subset of \(A\) (i.e., a discrete subset whose points can be separated by a pairwise disjoint family of open sets). Two problems from [\textit{A. Dow, M. G. Tkachenko, V. V. Tkachuk} and \textit{R. G. Wilson}, Glas. Mat., III. Ser. 37, No. 1, 187--210 (2002; Zbl 1009.54005)] are considered: (1) If \(X\) is compact and discretely generated is also \(X\times X\) discretely generated? and (2) If \(X\) is weakly discretely generated is also \(X\times X\) weakly discretely generated? The following counterexamples are constructed: 1) Assuming CH a compact discretely generated space such that \(X\times X\) is not discretely generated. 2) Examples of strongly discretely generated spaces whose square is not weakly discretely generated are constructed under a number of different assumptions (CH or MA(countable) or the existence of a measurable cardinal). In addition it is proven that a product of two discretely generated spaces is discretely generated assuming that one of the factors is compact and strongly discretely generated. A corollary to this result is that the product of a countable family of compact strongly discretely generated spaces is discretely generated. Finally, it is shown that a countable Hausdorff space of weight less than the dominating number is strongly discretely generated.
scientific article; zbMATH DE number 5083209
scientific article