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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:
 300000 Other combinatorial set theory 3e+35 Consistency and independence results
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