Box products are often discretely generated
A space \(X\) is called discretely generated if for every \(A\subseteq X\) and every \(x\in cl_{X}A\), there exists a discrete set \(D\subseteq A\) such that \(x\in cl_{X}D\). This notion was introduced by \textit{A. Dow, M. G. Tkachenko, V. V. Tkachuk} and \textit{R. G. Wilson} [Glas. Mat. Ser. III 37 (1), 187--210 (2002; Zbl 1009.54005)]. Answering questions from that paper, the authors prove that (1) every dyadic compact discretely generated space is metrizable and (2) if there exists an \(\omega_1\)-Souslin line, then there exists a compact dicretely generated space which has no point-countable \(\pi\)-base. Concerning products, it is also proved that (3) every box product of monotonically normal spaces is discretely generated and (4) every finite product of Hausdorff spaces in which every point has a nested local base is discretely generated. Many open problems are raised.
scientific article; zbMATH DE number 5986563
scientific article