S. Garc\'{\i }a-Ferreira, P.J. Szeptycki
MAD families and $P$-points

Comment.Math.Univ.Carolin. 48,4 (2007) 699-705.

Abstract:The {Kat\v etov ordering} of two maximal almost disjoint (MAD) families $\Cal A$ and $\Cal B$ is defined as follows: We say that $\Cal A\leq _K \Cal B$ if there is a function $f: \omega \to \omega $ such that $f^{-1}(A)\in \Cal I(\Cal B)$ for every $A\in \Cal I(\Cal A)$. In [Garcia-Ferreira S., Hru\v s\'ak M., {Ordering MAD families a la Kat\v etov}, J. Symbolic Logic {68} (2003), 1337--1353] a MAD family is called $K$-{uniform} if for every $X\in \Cal I(\Cal A)^+$, we have that $\Cal A|_X\leq _K \Cal A$. We prove that CH implies that for every $K$-uniform MAD family $\Cal A$ 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 $\bigcup \Cal A^*$ as a subspace of the remainder $\beta (\omega )\setminus \omega $. This result has a nicer topological interpretation: \par The symbol $\Cal F(\Cal A)$ will denote the Franklin compact space associated to a MAD family $\Cal A$. Given an ultrafilter $p\in \beta (\omega )\setminus \omega $, we say that a space $X$ is a {$\text {FU}(p)$-space} if for every $A\subseteq X$ and $x\in cl_X(A)$ there is a sequence $(x_n)_{n < \omega }$ in $A$ such that $x = p$-$\lim _{n \to \infty }x_n$ (that is, for every neigborhood $V$ of $x$, we have that $\{n < \omega : x_n \in V\}\in p$). {[CH]} For every $K$-uniform MAD family $\Cal A$ there is a $P$-point $p$ of $\omega ^*$ such that $\Cal F(\Cal A)$ is a $\text {FU}(p)$-{space}. We also establish the following. \par {[CH]} For two $P$-points $p,q\in \omega ^*$, the following are equivalent. \roster \item "(1)" $q\leq _{\text {RK}}p$. \item "(2)" For every $MAD$ family $\Cal A$, the space $\Cal F(\Cal A)$ is a $\text {FU}(p)$-space whenever it is a $\text {FU}(q)$-space. \endroster

Keywords: Franklin compact space, $P$-point, $\text {FU}(p)$-space, maximal almost disjoint family, Kat\v etov ordering, Rudin-Keisler ordering
AMS Subject Classification: Primary 03E05; Secondary 54B99