More on MAD families and \(P\)-points
The paper under review answers a question posed by the first listed author and \textit{V. I. Malykhin} [Proc. Am. Math. Soc. 124, No. 7, 2267--2273 (1996; Zbl 0849.54004)] by showing that, assuming the Continuum Hypothesis, for every maximal almost disjoint family \(\mathcal A\) there is a \(P\)-point \(\mathcal U\) such that the one-point compactification \(Fr(\mathcal A)\) (also known as the \emph{Franklin} or \emph{Alexandroff-Urysohn} space associated to \(\mathcal A\)) of the \emph{Mrówka-Isbell} space of \(\mathcal A\) is \emph{\(\mathcal U\)-Fréchet-Urysohn}, meaning that every point in the closure of a subset of the space is a \(\mathcal U\)-limit of a sequence of points of the set. Recall that a free ultrafilter \(\mathcal U\) on \(\mathbb N\) is a \emph{P-point} if for every countable family \(\mathcal X\) of elements of \(\mathcal U\) there is a \(U\in\mathcal U\) such that \(U\setminus X\) is finite for every \(X\in\mathcal X\). The \emph{Mrówka-Isbell} space associated to a maximal almost disjoint family \(\mathcal A\) of inifinite subsets of \(\mathbb N\) is the topological space which has \(\mathbb N\cup \mathcal A\) as the underlying set and \[\{\{n\}:\ n\in\mathbb N\}\cup \{A\setminus \{0, \dots, n\}: A\in \mathcal A, n\in \mathbb N\}\] as a basis for open sets. The construction is interesting and non-trivial.
scientific article; zbMATH DE number 7434147
scientific article