Dowker spaces revisited
It is a classical result of \textit{C. H. Dowker} [Can. J. Math. 3, 219--224 (1951; Zbl 0042.41007)] that a \(T_1\)-space \(X\) is countably paracompact and normal if and only if \(X\times[0,1]\) (or, equivalently, \(X\times(\omega+ 1)\)) is normal. The main results of this paper are (somewhat technical) analogs of Dowker's result for \(\beta\)-normality and for \(\alpha\)-normality, where a topological space \(X\) is called \(\beta\)-normal (\(\alpha\)-normal) if for every pair \(A\), \(B\) of disjoint closed subsets of \(X\) there exist open subsets \(U\) and \(V\) of \(X\) such that \(A\cap U\) is dense in \(A,B\cap V\) is dense in \(B\), and \(\text{cl\,}U\cap\text{cl\,}V= \emptyset\), (\(U\cap V=\emptyset\), respectively) (see \textit{A. V. Arhangel'skii} and \textit{L. Ludwig} [Commentat. Math. Univ. Carol. 42, No.~3, 507--519 (2001; Zbl 1053.54030)]). The authors call a topological space \(X\) \(\alpha\)-countably paracompact if for every decreasing sequence \((A_n)_{n<\omega}\) of closed subsets of \(X\) satisfying \(\bigcap_{n<\omega} A_n=\emptyset\) there exists a sequence \((U_n)_{n<\omega}\) of open subsets of \(X\) such that \(U_n\cap A_n\) is dense in \(A_n\) for each \(n<\omega\) and \(\bigcap_{n< \omega}\text{cl\,} U_n=\emptyset\). It follows from their main results that if \(X\times(\omega+ 1)\) is \(\beta\)-normal, then \(X\) is \(\beta\)-normal and \(\alpha\)-countably paracompact. Whether the converse is true is an open problem. Additionally, it is shown that there exists a completely regular \(T_1\)-space that is first countable, collectionwise Hausdorff, \(\alpha\)-countably paracompact, and \(\alpha\)-normal but not \(\beta\)-normal. Moreover, there exists a Moore space that is collectionwise Hausdorff and \(\alpha\)-normal but not \(\beta\)-normal.
scientific article; zbMATH DE number 5792241