Paracompactness and the Lindelöf property in countable products
Following \textit{A. Hohti} and \textit{Y. Ziqui} [Czech. Math. J. 49, No.~3, 569--583 (1999; Zbl 1003.54006)] a topological space \(X\) is called Čech-scattered if, for every nonempty closed subset \(A\) of \(X\), there is a point \(x\in A\) which has a Čech-complete neighborhood in \(A\). Obviously, every scattered topological space is C-scattered in the sense of \textit{R. Telgarsky} [Fundam. Math. 73, 59--74 (1974; Zbl 0226.54018)], an every C-scattered space is Čech-scattered. Therefore the main result of this interesting paper is a nice generalization of a well-known theorem of \textit{K. Alster} [ibid. 127, 241--246 (1987; Zbl 0641.54005)]. Theorem 1. If \(Y\) is a perfect paracompact Hausdorff space and \(\{X_n \mid n < \omega\}\) is a countable collection of Čech-scattered paracompact Hausdorff spaces, then the product \(Y\times \prod_{n<\omega} X_n\) is paracompact. It is pointed out that the next theorem can be proved in a similar way. Theorem 2. If \(Y\) is a hereditarily Lindelöf space and \(\{X_n \mid n <\omega\}\) is a countable collection of Čech-scattered Lindelöf spaces, then the product \(Y\times \prod_{n<\omega} X_n\) is Lindelöf. Note that Theorem 2 generalizes another well-known theorem of \textit{K. Alster} [ibid. 114, 173--181 (1981; Zbl 0369.54005)].
scientific article; zbMATH DE number 2136454
scientific article