Invariance properties of almost disjoint families. (English) Zbl 1317.03032

The paper’s main result concerns families of \(\mathcal{A}\) of subsets of \(\omega\) that are almost disjoint and are maximal with respect to this property, the MAD families. Under the assumption that the cardinal invariants \(\mathfrak{t}\) and \(\mathfrak{c}\) are equal, the authors prove that there exists a \(\mathrm{MAD}\) family maximal in the Katětov order. An important role in this investigation is played by the cofinitary subgroups of \(\mathrm{Sym}(\omega)\): those subgroups in which the non-identity elements have only finitely many fixed points.
To each AD family \(\mathcal{A}\) one can associate the subgroup \(\mathrm{Inv}(\mathcal{A})\) of \(\mathrm{Sym}(\omega)\) consisting of those permutations that preserve \(\mathcal{A}\). Giving \(\mathrm{Sym}(\omega)\) the subspace topology of \(\omega^{\omega}\), \(\mathrm{Sym}(\omega)\) can be thought of as a topological group.
In this context, the authors consider three questions posed earlier by S. García-Ferreira [Commentat. Math. Univ. Carol. 39, No. 1, 185–195 (1998; Zbl 0938.54004)]: (1) For any countable \(F\subseteq \mathrm{Sym}(\omega)\), is there a MAD family \(\mathcal{A}\) such that \(F\subseteq\mathrm{Inv}(\mathcal{A})\)? (2) Is there a MAD family \(\mathcal{A}\) such that \(\mathrm{Inv}(\mathcal{A})\) is a closed subspace? (3) Is there a MAD family \(\mathcal{A}\) such that \(\mathrm{Inv}(\mathcal{A})\) is a dense subspace?
They show that the answer to (1) is negative, while both (2) and (3) have positive answers. In the course of establishing these results, the authors construct a countable cofinitary group that is dense in \(\mathrm{Sym}(\omega)\).
The article ends with a list of open questions.


03E05 Other combinatorial set theory


Zbl 0938.54004
Full Text: DOI Euclid Link


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