Noetherian type in topological products
Inspired by the definition of Noetherian rings one may define any collection, \(\mathcal F\), of sets to be Noetherian if for every~\(F\in\mathcal F\) the family \(\{G\in\mathcal{F}:F\subseteq G\}\) is finite. More generally, the Noetherian type of~\(\mathcal F\) is defined to be the smallest cardinal number~\(\kappa\) such that \(\bigl|\{G\in\mathcal{F}:F\subseteq G\}\bigr|<\kappa\) for all~\(F\in\mathcal F\). The Noetherian type of a topological space~\(X\), written \(\text{Nt}(X)\), then is the minimum Noetherian type is a base for~\(X\). The authors investigate the behaviour of this cardinal function, and some variants, under the formation of products, mostly in the class of compact Hausdorff spaces. They provide two completely regular spaces~\(X\) and~\(Y\) such that the Noetherian type of their product is strictly less than the minimum of the types of~\(X\) and~\(Y\). The question whether this is possible for compact Hausdorff spaces remains open. The bulk of the paper is devoted to the Noetherian type of \(G_\delta\)-modifications (and others) of compact Hausdorff spaces, in particular of Cantor cubes. Using methods from PCF theory they show that the type of the \(G_\delta\)-modification of~\(2^{\aleph_\omega}\) is at most~\(\aleph_4\), and, more generally, the \(G_{\aleph_n}\)-modification has type at most the maximum of~\(\aleph_4\) and~\(\aleph_{n+1}\), and exactly \(\aleph_{n+1}\) if \(n\geq4\).
scientific article; zbMATH DE number 6361332
scientific article