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A note on almost disjoint refinement. (English) Zbl 0883.04003
If \(D\) is a nowhere dense subset of the space \(\omega^* - \beta\omega \setminus \omega\), is \(D\) a \(\mathfrak c\)-set? I.e., is there a pairwise disjoint family \(\mathcal U\) of open sets with \(|\mathcal U|= \mathfrak c\) and \(D \subset \bigcap_{U\in \mathcal U}\)cl\(U\)? Equivalently, if \(\mathcal A\) is a maximal almost disjoint family on \(\omega\), is there an almost disjoint family \(\mathcal B\) so \(\forall A \in [\omega]^\omega\setminus \mathcal A\) there is \(B \in \mathcal B\) with \(B \subset A\)? (We say that \(\mathcal B\) is an almost disjoint refinement of \([\omega]^\omega\setminus \mathcal A\).) The answer was known to be yes under the assumptions \(\mathfrak a = \mathfrak c\) or \(\mathfrak b = \mathfrak d\). The purpose of this paper is to show that the answer is yes if \(\mathfrak d \leq \mathfrak a\). Here \(\mathfrak d\) and \(\mathfrak a\) are standard cardinal invariants of the reals: \(\mathfrak d\) is the size of the smallest dominating family of \(\omega^{\omega}\) under \(\leq^*\); \(\mathfrak a\) 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:
Theorem. The following are equivalent: (a) If \(\mathcal A\) is maximal almost disjoint on \(\omega\), then \([\omega]^\omega\setminus\mathcal A\) has an almost disjoint refinement. (b) \([\omega]^\omega\) is the union of an increasing sequence \(\{\mathcal I^+(\mathcal C_{\alpha}):\alpha<\tau\}\) for some \(\tau\leq\mathfrak b\) with cf \(\tau>\omega\), where each \(\mathcal C_\alpha\) is completely separable almost disjoint.
Here \(I^+(\mathcal C) = \{a \subset \omega: \mathcal C \cup \{a\}\) is not almost disjoint\(\}\), and an almost disjoint family is completely separable iff every element of \(I^+(\mathcal C)\) contains some element of \(\mathcal C\). 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”.

MSC:
03E05 Other combinatorial set theory
03E35 Consistency and independence results
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