On mappings from products into developable spaces
A topological space is subnormal, respectively D-normal, if every pair of disjoint closed subsets can be separated by \(G_{\delta}\)-sets, respectively closed \(G_{\delta}\)-sets. This interesting paper addresses the problem as to whether there exists, for every non-compact Hausdorff space X, a cardinal \(\kappa\) such that \(^{\kappa}X\) is not subnormal, and gives the partial answer that \(^{\kappa}X\) is not D-normal whenever \(\kappa \geq \max (\omega_ 1,w(X))\), where w(X) is the weight of X. The proof involves the use of a characterization of continuous mappings from a product into a developable \(T_ 1\)-space, depending on countably many coordinates, and the whole leans on work of \textit{N. Noble} [Trans. Am. Math. Soc. 160, 169-183 (1971; Zbl 0233.54004)] and \textit{J. Keesling} [Adv. Math. 9, 90-92 (1972; Zbl 0243.54003)]. Some open problems are noted.
mapping depending on countably many coordinates
scientific article; zbMATH DE number 4028375
scientific article