Almost disjoint families and topology (Q2874868)

An infinite family \(\mathcal A\), which consists of infinite subsets of the natural numbers, is denoted AD, if the intersection of any two distinct elements of \(\mathcal A\) is finite. When it is not properly included in any larger AD family, then it is denoted MAD. Structural properties are listed: Combinatorial properties, but with a strong emphasis on independence results, of AD families and their relationship with general topology as versatile sources of counterexamples, in particular, via the corresponding \(\Psi\)-spaces. As the author points out, the article contains almost no proofs. And some posted leave room for improvement. For example, it would be better to prove Proposition 2.3 as follows:NEWLINENEWLINELet an MAD family \(\mathcal A\) and a decreasing sequence \(\{X_n: n\in \omega \}\subset I^+(\mathcal A)\) be given. For each \(n\in \omega\), choose infinite sets \(Y_n \subseteq A_n\in \mathcal A\) so that \(Y_n \subseteq^* X_i\), for all \(i \in \omega\), and \(A_n\not\in \{A_m: m<n\}\). Then any set \(X\) such that \(Y_0\cup Y_1 \cup \ldots \cup Y_n \subseteq^* X \subseteq^* X_n\), for all \(n\in \omega\), is the desired one.NEWLINENEWLINEAs the author remarks in his conclusion, there are many topics on AD families that have been omitted, so the article is only intended to give complete classifications of AD and MAD families. However, such a project looks to be far from complete, since in this area one can easily put the issues that are likely to remain long unresolved due to lack of appropriate tools for proving. For example, there are still open two 40-year-old problems (in ZFC): Whether \(\omega^*\) and \(\omega^*_1\) must always be not homeomorphic. Does there always exist a completely separable MAD family. The area looks too vast and it is broken down into very specific and detached parts; there is a large bibliography indicating more detailed contributions of J. Brendle, A. Dow, M. Hrušák, S. Shelah, P. Simon and some of their co-authors.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].