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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
##### Keywords:
Franklin compact space
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