\(\kappa\)-normality and products of ordinals
Let \(Y\) be a subspace of a topological space \(X\). Following \textit{A. V. Arhangel'skii} [Topology Appl. 70, 87-99 (1996; Zbl 0848.54016)] a subset \(A\) of \(X\) is said to be concentrated on \(Y\) if \(A\subset \text{cl}_X(A\cap Y)\). \(X\) is said to be normal on \(Y\) if every two disjoint closed subsets of \(X\) concentrated on \(Y\) can be separated by disjoint open neighborhoods in \(X\). \(X\) is called densely normal if there exists a dense subspace \(Y\) of \(X\) such that \(X\) is normal on \(Y\). Every densely normal space is \(\kappa\)-normal in the sense of \textit{E. V. Shchepin} [Sib. Math. J. 13, No. 5, 820-830 (1972; Zbl 0261.54014)]. \textit{W. Just} and \textit{J. Tartir} have shown that there exist Tychonoff spaces which are densely normal but not \(\kappa\)-normal [Proc. Am. Math. Soc. 127, No. 3, 901-905 (1999; Zbl 0907.54011)]. In fact, as \textit{A. V. Arhangel'skii} has recently pointed out, the space \(X= C_p(\omega_1+ 1)\) is \(\kappa\)-normal, but no dense subspace \(Y\) of \(X\) is densely normal [Topology Appl. 123, No. 1, 27-36 (2002; Zbl 1008.54013)]. Here products of ordinals are considered and the following results are obtained. Theorem 1: Suppose that \(\alpha_i\) is an ordinal for each \(i<\omega\). Then \(\prod\{\alpha_i\mid i< \omega\}\) is densely normal. Theorem 2: Suppose that \(\alpha_i\) is an ordinal for each \(i<\lambda\). Then \(\prod\{\alpha_i\mid i<\lambda\}\) is \(\kappa\)-normal. The proof of Theorem 2 uses extensively the technique of elementary submodels.
scientific article; zbMATH DE number 1879073