The normality of \(\Sigma\)-products and the perfect \(\kappa\)-normality of Cartesian products
The author first proves that, if \(\Sigma\) is a \(\Sigma\)-product of paracompact \(\Sigma\)-spaces, then \(\Sigma\) is collectionwise normal if and only if it is normal. This generalizes a theorem of \textit{A. P. Kombarov} [Sov. Math. Dokl. 19, 403-407; Dokl. Akad. Nauk SSSR 239, 775- 778 (1978; Zbl 0397.54013)]. Next it is proved that, if X is a Cartesian product of paracompact \(\sigma\)-spaces and if each finite subproduct of X has countable tightness, then X is perfectly \(\kappa\)-normal (that is, if for each disjoint open sets \(V_ 1\) and \(V_ 2\) in X there exist disjoint cozero-sets \(U_ 1\) and \(U_ 2\) in X such that \(V_ k\subset U_ k\), \(k=1,2)\). This generalizes a theorem of \textit{V. E. Shchepin} [Sov. Math. Dokl. 17, 152-155; Dokl. Akad. Nauk SSSR 226, 527-529 (1976; Zbl 0338.54022)]. A result about the countable paracompactness of \(\Sigma\)-products is also obtained.
countable paracompactness of \(\Sigma \) -products
scientific article; zbMATH DE number 3886718