Nonshrinking open covers and K. Morita's third conjecture
\textit{K. Morita} conjectured [Gen. Topol. Relat. Mod. Anal. Algebra, Proc. 4th Prague Topol. Symp. 1976, Part B, 296-297 (1977; Zbl 0374.54004)] that a space \(X\) is a \(\sigma\)-locally compact, metrizable space if and only if \(X \times Y\) is normal for every normal, countably paracompact space \(Y\). Here a space is said to be \(\sigma\)-locally compact if it is the union of countably many closed locally compact subspaces. In this paper, his conjecture is proved by showing that for every regular uncountable cardinal \(\kappa\), there is a hereditarily normal, countably paracompact space \(X_{\kappa}\) with an increasing open cover \({\mathcal W} = \{ W_{\mu}: \mu \in \omega_1 \}\) such that \(\mathcal W\) does not have any refinement by \(\leq \kappa\) many closed sets. It was proved by \textit{K. Chiba, T. C. Przymusiński} and \textit{M. E. Rudin} [Topology Appl. 22, 19-32 (1986; Zbl 0575.54007)] that the existence of such spaces implies Morita's conjecture. The spaces \(X_{\kappa}\) are the first known examples of hereditarily normal \(\omega_1\)-Dowker spaces. The following conjecture (Morita's second conjecture) remains open in ZFC: A space is metrizable if and only if the product of the space with every normal Morita \(P\)-space is normal.
\(\sigma\)-locally compact metrizable space
scientific article; zbMATH DE number 1308031