Mild normality of finite products of subspaces of \(\omega_1\)
A topological space \(X\) is called mildly normal if every pair of disjoint regular closed sets in \(X\) can be separated by disjoint open sets. \textit{L. Kalantan} and \textit{P. J. Szeptycki} [Topology Appl. 123, 537--545 (2002; Zbl 1016.54015)] proved that the product of arbitrary many ordinals is mildly normal. \textit{L. Kalantan} and \textit{N. Kemoto} [Houston J. Math. 29, 937--947 (2003; Zbl 1155.54320)] proved that the product of two subspaces of ordinals is mildly normal, and asked if the product of arbitrary many subspaces of ordinals is mildly normal. In this paper, the authors determine when the product of finitely many subspaces of \(\omega_1\) is mildly normal. They apply the result to answer the question and show that there exist three subspaces \(A\), \(B\) and \(C\) of \(\omega_1\) such that \(A\times B\times C\) is not mildly normal. As another application, they prove that the product of finitely many subspaces of \(\omega_1\) with the pairwise non-stationary intersection is mildly normal. It, however, remains open whether the product of countable, pairwise disjoint family of stationary sets of \(\omega_1\) is mildly normal.
scientific article; zbMATH DE number 5013534