Mrówka maximal almost disjoint families for uncountable cardinals (Q972533)

Let \(\kappa\) be an infinite cardinal number. For any almost disjoint family \(\mathcal{A}\subset [\kappa]^{\omega}\) let \(\psi(\kappa,\mathcal{A})\) denote the space \(\kappa\cup\mathcal{A}\) 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. Such spaces were considered for \(\kappa=\omega\) and \(\mathcal{A}\) being an infinite maximal almost disjoint family by \textit{S.~Mrówka} [Fundam. Math. 94, 83--92 (1977; Zbl 0348.54017)]. Mrówka proved that there exists a maximal almost disjoint family \(\mathcal R\) such that \(|\beta \psi (\omega, \mathcal R) \setminus \psi (\omega, \mathcal R)| = 1\). (In other words there is a unique free \(z\)-ultrafilter \(p_{0}\) on the space \(\psi \).) In the paper under review, the authors extend this result to uncountable cardinals and introduce the notion of \textit{Mrówka maximal almost disjoint family \(\mathcal{M}\subset [\kappa]^{\omega}\)} on \(\kappa\) for all infinite cardinals \(\kappa\). They show the following results: ``for \(\kappa \leqslant \mathfrak c\), Mrówka's MADF \(\mathcal R\) can be used to produce a MADF \(\mathcal M \subset [\kappa]^{\omega}\) such that \(|\beta \psi (\kappa, \mathcal M) \setminus \psi (\kappa, \mathcal M)| = 1\). For \(\kappa > \mathfrak c\), and every \(\mathcal M \subset [\kappa]^{\omega}\), it is always the case that \(|\beta \psi (\kappa, \mathcal M) \setminus \psi (\kappa, \mathcal M)| \neq 1\), yet there exists a special free \(z\)-ultrafilter \(p\) on \(\psi (\kappa, \mathcal M)\) retaining some of the properties of \(p_{0}\). In particular both \(p\) and \(p_{0}\) have a clopen local base in \(\beta \psi \) (although \(\beta \psi (\kappa, \mathcal M)\) need not be zero-dimensional). A result for \(k > \mathfrak c\), that does not apply to \(p_{0}\), is that for certain \(\kappa > \mathfrak c, p\) is a P-point in \(\beta \psi \).''