Box products and Hurewicz spaces
Normality and paracompactness of products \(X \times \square^\omega (\omega + 1)\) for Lindelöf spaces \(X\) are considered. The box product \(\square^\omega (\omega + 1)\) is the cartesian product of countably many copies of the convergent sequence \(\omega + 1\), whose topology is generated by the base consisting of sets of the form \(\prod_{n \in \omega} U_n\) where each \(U_n\) is open in \(\omega + 1\). A space is called a Hurewicz space if for every sequence \(\langle {\mathcal U}_n \rangle_{n \in \omega}\) of open covers of \(X\), there is a sequence \(\langle {\mathcal V}_n \rangle_{n \in \omega}\) such that each \({\mathcal V}_n\) is a finite subfamily of \({\mathcal U}_n\) and \(X = \bigcup_{n \in \omega} {\mathcal V}_n\). Every \(\sigma\)-compact space is a Hurewicz space, and every Hurewicz space is Lindelöf. The author shows the following: (a) It is consistent that \(X \times \square^\omega (\omega + 1)\) is paracompact. (b) If \(X\) is a separable Lindelöf space which is not a Hurewicz space, then \(X \times \square^\omega (\omega + 1)\) is not normal. (c) Martin's axiom implies the existence of a Hurewicz subspace \(X\) of the real line such that \(X \times \square^\omega (\omega + 1)\) is paracompact but \(X^2 \times \square^\omega (\omega + 1)\) is not normal.
scientific article; zbMATH DE number 792528
scientific article