On certain generalization of normality (Q1814319)

This paper concerns certain generalizations of normality. Definitions are given but results are stated without proof. Some concepts will be defined here and some theorems stated in order to convey the general tenor of the paper. The author defines subsets \(A\) and \(B\) of a topological space \(X\) to be separated in \(X\) if there exist open subsets \(U\) and \(V\) in \(X\) such that \(A\subseteq U\) and \(B\subseteq V\) and \(U\cap V = \emptyset\). Following the paper of \textit{A. V. Arkhangelskij} and \textit{Kh. M. M. Genedi} [ibid. 1989, No. 6, 67-69 (1989; Zbl 0745.54003)], he defines a subspace \(Y\) of \(X\) as strongly normal in \(X\) if every nonintersecting subsets of \(Y\) which are closed in \(Y\) can be separated in \(X\). Let \(\tau\) be an infinite cardinal, and \(X\) a \(T_ 1\)-space. Then \(X\) is called \(\tau\)-normal if for every two nonintersecting closed subsets \(A\) and \(B\) there exist families \(\{A_ s:\;s\in S\}\) and \(\{B_ t:\;t\in T\}\) of closed sets such that \(| S|\leq \tau\), \(| T|\leq \tau\), \(A=\bigcup\{A_ s: s\in S\}\), \(B=\bigcup\{B_ t:\;t\in T\}\) and for all \(s\in S\) and \(t\in T\), \(A_ s\) and \(B_ t\) are separated in \(X\). Theorem 1: Let \(X\) be a \(T_ 2\)-space and \(X=\bigcup_{s\in S}X_ s\), where \(\{X_ s:\;s\in S\}\) is a family of compact \(T_ 2\)-spaces and \(| S | = \tau \geq \aleph_ 0\). Then \(X\) is \(\tau\)-normal. -- - Theorem 2: Let \(X\) be a \(T_ 3\)-space and the Lindelöf number of \(X=\tau>\aleph_ 0\). Then \(X\) is \(\tau\)-normal. --- Theorem 3: Let \(X\) be a space of density \(\tau>\aleph_ 0\) which contains a closed discrete subspace of cardinality \(2^ \tau\). Then \(X\) is not \(\tau\)-normal. --- The author uses the concept of strong normality in obtaining certain of his results. He also has results on \(\tau\)-paracompactness and its relation to \(t\)-normality. A number of examples showing the scope of his results are described.