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Zbl 0883.04003
Simon, P.
A note on almost disjoint refinement.
(English)
[J] Acta Univ. Carol., Math. Phys. 37, No.2, 89-99 (1996). ISSN 0001-7140

If $D$ is a nowhere dense subset of the space $\omega^* - \beta\omega \setminus \omega$, is $D$ a $\frak c$-set? I.e., is there a pairwise disjoint family $\Cal U$ of open sets with $|\Cal U|= \frak c$ and $D \subset \bigcap_{U\in \Cal U}$cl$U$? Equivalently, if $\Cal A$ is a maximal almost disjoint family on $\omega$, is there an almost disjoint family $\Cal B$ so $\forall A \in [\omega]^\omega\setminus \Cal A$ there is $B \in \Cal B$ with $B \subset A$? (We say that $\Cal B$ is an almost disjoint refinement of $[\omega]^\omega\setminus \Cal A$.) The answer was known to be yes under the assumptions $\frak a = \frak c$ or $\frak b = \frak d$. The purpose of this paper is to show that the answer is yes if $\frak d \leq \frak a$. Here $\frak d$ and $\frak a$ are standard cardinal invariants of the reals: $\frak d$ is the size of the smallest dominating family of $\omega^{\omega}$ under $\leq^*$; $\frak a$ is the size of the smallest maximal almost disjoint family of subsets of $\omega$. Along with the main result, some interesting technical lemmas are proved, as is the following theorem:\par Theorem. The following are equivalent: (a) If $\Cal A$ is maximal almost disjoint on $\omega$, then $[\omega]^\omega\setminus\Cal A$ has an almost disjoint refinement. (b) $[\omega]^\omega$ is the union of an increasing sequence $\{\Cal I^+(\Cal C_{\alpha}):\alpha<\tau\}$ for some $\tau\leq\frak b$ with cf $\tau>\omega$, where each $\Cal C_\alpha$ is completely separable almost disjoint.\par Here $I^+(\Cal C) = \{a \subset \omega: \Cal C \cup \{a\}$ is not almost disjoint$\}$, and an almost disjoint family is completely separable iff every element of $I^+(\Cal C)$ contains some element of $\Cal C$. Completely separable almost disjoint families are, in some sense, large, so the import of the theorem is that (a) is equivalent to $[\omega]^{\omega}$ is the union of a short increasing sequence of the dual filters of large almost disjoint families''.
[J.Roitman (Lawrence)]
MSC 2000:
*03E05 Combinatorial set theory (logic)
03E35 Consistency and independence results (set theory)

Keywords: almost disjoint; $\beta\omega$; $\frak a$; $\frak d$; cardinal invariants of the reals

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