Ordinal remainders of \(\psi \)-spaces
For an infinite cardinal \(\kappa\) and any almost disjoint family (ADF) \(\mathcal{A}\subset [\kappa]^{\omega}\), let \(\psi(\kappa,\mathcal{A})\) denote the space with underlying set \(\kappa\cup \mathcal{A}\) and with the topology having as a base all singletons \(\{\alpha\}\) for \(\alpha<\kappa\) and all sets of the form \(\{ A\}\cup (A\setminus F)\) where \(A\in\mathcal{A}\) and \(F\) is finite. When \(\kappa\) and the MADF \(\mathcal{M}\subset [\kappa]^{\omega}\) are understood, the space \(\psi(\kappa,\mathcal{M})\) is called a \(\psi\)-space. It is known that all \(\psi\)-spaces are locally compact, so the remainder \(\beta\psi\setminus\psi\) is closed in \(\beta\psi\), hence compact. Thus if the remainder \(\beta\psi\setminus\psi\) is homeomorphic to an ordinal, it must be a successor ordinal. In the main result of the paper under review the authors show that the cardinal \(\mathfrak{c}^+\) is the largest cardinal satisfying the following property: for every cardinal \(\omega\leq\kappa\leq\mathfrak{c}^+\), there exists a MADF \(\mathcal{M}\subset [\kappa]^{\omega}\) such that the Stone-Čech remainder of \(\psi(\kappa,\mathcal{M})\) is homeomorphic to \(\kappa+1\) with the order topology. The article is a continuation of the authors' paper [Topology Appl. 157, 1379--1394 (2010; Zbl 1196.54060)].
scientific article; zbMATH DE number 5949219
scientific article