Spaces with bases and \(\pi\)-basis of finite rank
An antichain is a family \({\mathcal A}\) of sets such that the inclusion \(A\subseteq B\) is false whenever A,B\(\in {\mathcal A}\) and \(A\neq B\). A family \({\mathcal F}\) has finite rank if there is an integer n such that every antichain \({\mathcal A}\subseteq {\mathcal F}\) with \(| {\mathcal A}| =n+1\) satisfies \(\cap {\mathcal A}=\emptyset\); the least such n is called the rank of \({\mathcal F}.\) By an elementary, pleasing argument, the author shows that every space with a \(\pi\)-base of finite rank has a \(\pi\)-base of rank 1. There follows a solution to a problem posed a decade ago by A. V. Arkhangel'skij: If a Hausdorff space X has a base of finite rank and no disjoint open family of cardinality \(\alpha\), then w(X)\(\leq \alpha\); if in addition X is a ccc-space and SH (the Souslin hypothesis) is assumed, then w(X)\(\leq \omega\). A consequence is the theorem of O. Ferster: Assuming SH, every regular, Hausdorff, ccc-space with a base of finite rank is metrizable.
scientific article; zbMATH DE number 3988217