Base-normality and product spaces
The author introduces the notion of base-normality, which is a natural generalization of base-paracompactness introduced by \textit{J. E. Porter} [Topology Appl. 128, 145--156 (2003; Zbl 1043.54010)], and studies base-normality of product spaces. A space \(X\) is base-normal if there exists a base \({\mathcal B}\) for \(X\) with \(| {\mathcal B}| =w(X)\) such that for every binary open cover \({\mathcal U}\) of \(X\), there exists a locally finite collection \({\mathcal B}'\subseteq{\mathcal B}\) which covers \(X\) and such that \(\{\overline{B}:B\in{\mathcal B}'\}\) refines \({\mathcal U}\). In particular, the following three theorems are proved: (1) For a base-normal space \(X\) and a metrizable space \(Y\), the product \(X\times Y\) is base-normal if and only if \(X\times Y\) is normal; (2) for a countable product \(X=\prod_{i\in N}X_i\) such that \(\prod_{i\leq n}X_i\) is base-normal for each \(n\in N\), \(X\) is base-normal if and only if \(X\) is normal; and (3) every \(\Sigma\)-product of metric spaces is base-normal. These theorems are applied to show that analogues to several classical theorems concerning normality of products hold for base-normality. For example, as an analogue to Dowker's theorem, it is proved that a space \(X\) is base-normal and base-countably paracompact if and only if \(X\times Y\) is base-normal for every compact metrizable space \(Y\) if and only if \(X\times[0,1]\) is base-normal. A central problem in this topic is whether every paracompact space is base-paracompact. The author reduces this to the problem whether every paracompact space is base-normal by proving that a space is base-paracompact if and only if it is paracompact and base-normal.
scientific article; zbMATH DE number 2142284
scientific article