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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.

MSC:

03E05 Other combinatorial set theory

Citations:

Zbl 0938.54004
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References:

[1] DOI: 10.1016/j.apal.2004.09.002 · Zbl 1083.03047
[2] DOI: 10.1006/jabr.2000.8396 · Zbl 0966.20001
[3] Handbook of set theory 1 pp 395– (2010)
[4] Analytic and coanalytic families of almost disjoint functions 73 pp 1158– (2008)
[5] Cofinitary groups, almost disjoint and dominating families 66 pp 1259– (2001)
[6] Set theory, on the structure of the real line (1995)
[7] Commentationes Mathematicae Universitatis Carolinae 48 pp 699– (2007)
[8] Ordering MAD families a la Katětov 68 pp 1337– (2003)
[9] Commentationes Mathematicae Universitatis Carolinae 39 pp 185– (1998)
[10] DOI: 10.1112/blms/28.2.113 · Zbl 0853.20002
[11] DOI: 10.1007/s00153-008-0104-4 · Zbl 1171.03029
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