Normal covers of infinite products and normality of \(\sigma\) -products
This paper studies normal covers of a product of infinitely many \(\beta\)-spaces and the normality of a \(\Sigma\)-product of \(\beta\)-spaces. The notion of \(\beta\)-spaces was introduced by \textit{R. E. Hodel} [Pac. J. Math. 38, 641--652 (1971; Zbl 0219.54024)], and it is known that all \(\Sigma\)-spaces and all semi-stratifiable spaces are \(\beta\)-spaces. First, let \(X=\prod_{\lambda\in\Lambda}X_\lambda\) be an infinite product space, and define \(X_\theta=\prod_{\lambda\in\theta}X_\lambda\) for each \(\theta\subseteq\Lambda\). A subset \(A\) of \(X\) is called a \textit{cylinder} if there is a finite set \(\theta\subseteq\Lambda\) and \(T\subseteq X_\theta\) such that \(A=\pi_\theta^{-1}[T]\), where \(\pi_\theta:X\to X_\theta\) is the projection. A subset \(A\) of \(X\) is called a \textit{rectangle} if there is a finite set \(\theta\subseteq\Lambda\) and \(T_\lambda\subseteq X_\lambda\) \((\lambda\in\theta)\) such that \(A=\bigcap_{\lambda\in\theta}\pi_\lambda^{-1}[T_\lambda]\), where \(\pi_\lambda:X\to X_\lambda\) is the projection. The author proves a technical theorem which implies that (1) if each \(X_\lambda\) is a \(\beta\)-space and if \(X_\theta\) is paracompact and has countable tightness for every finite set \(\theta\subseteq\Lambda\), then an open cover \({\mathcal O}\) of \(X\) is normal if and only if \({\mathcal O}\) has a locally finite, \(\sigma\)-discrete, cylindrical cozero refinement which has a cylindrical zero-set shrinking. The following conditions (a)--(e) on a cover \({\mathcal O}\) of \(X\) are considered: (a) \({\mathcal O}\) is normal; (b) \({\mathcal O}\) has a \(\sigma\)-locally finite rectangular cozero refinement; (c) \({\mathcal O}\) has a \(\sigma\)-discrete rectangular cozero refinement; (d) \({\mathcal O}\) has a locally finite rectangular cozero refinement; (e) \({\mathcal O}\) has a locally finite, \(\sigma\)-discrete, rectangular cozero refinement which has a rectangular zero-set shrinking. As an application of (1), the author proves the following theorems (2) and (3). (2) If each \(X_\lambda\) is a paracompact \(\Sigma\)-space and if \(X_\theta\) has countable tightness for every finite set \(\theta\subseteq\Lambda\), then (a)--(e) are equivalent for every open cover \({\mathcal O}\) of \(X\). (3) If each \(X_\lambda\) is a paracompact \(M\)-space, then (a)--(e) are equivalent for every open cover \({\mathcal O}\) of \(X\). Next, let \(\Sigma\) be the \(\Sigma\)-product of spaces \(X_\lambda\) \((\lambda\in\Lambda)\). The author proves the following theorems (4) and (5): (4) If each \(X_\lambda\) is a \(\beta\)-space and if \(X_\theta\) is paracompact and has countable tightness for every finite set \(\theta\subseteq\Lambda\), then \(\Sigma\) is collectionwise normal. (5) If each \(X_\lambda\) is a \(\beta\)-space and if \(X_\theta\) is base-paracompact and has countable tightness for every finite set \(\theta\subseteq\Lambda\), then \(\Sigma\) is base-normal. (4) is an affirmative answer to Question 4 of the author's paper [Tsukuba J. Math. 13, 83--98 (1989; Zbl 0697.54006)]. (5) implies that a \(\Sigma\)-product of paracompact \(M\)-spaces is base-normal if and only if it is normal, which is an affirmative answer to Question 5.8 of \textit{K. Yamazaki} [Topology Proc. 29, 389--403 (2005; Zbl 1162.54318)].
scientific article; zbMATH DE number 5080157
scientific article