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MAD families and \(P\)-points. (English) Zbl 1199.03028
Summary: The Katětov ordering of two maximal almost disjoint (MAD) families \(\mathcal A\) and \(\mathcal B\) is defined as follows: We say that \(\mathcal A\leq _K \mathcal B\) if there is a function \(f\: \omega \to \omega \) such that \(f^{-1}(A)\in \mathcal I(\mathcal B)\) for every \(A\in \mathcal I(\mathcal A)\). In [M. Hrušák and S. García-Ferreira, J. Symb. Log. 68, 1337–1353 (2003; Zbl 1055.03027)] a MAD family is called \(K\)-uniform if for every \(X\in \mathcal I(\mathcal A)^+\) we have that \(\mathcal A| _X\leq _K \mathcal A\). We prove that CH implies that for every \(K\)-uniform MAD family \(\mathcal A\) there is a \(P\)-point \(p\) of \(\omega ^*\) such that the set of all Rudin-Keisler predecessors of \(p\) is dense in the boundary of \(\bigcup \mathcal A^*\) as a subspace of the remainder \(\beta (\omega )\setminus \omega \).
MSC:
03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
54B99 Basic constructions in general topology
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