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Keywords:
Franklin compact space; $P$-point; $\text{FU}(p)$-space; maximal almost disjoint family; Katětov ordering; Rudin-Keisler ordering
Franklin compact space; $P$-point; $\text{FU}(p)$-space; maximal almost disjoint family; Katětov ordering; Rudin-Keisler ordering
Summary:
The Katětov 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 sák M., Ordering MAD families a la Katětov, 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: 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. [CH] For two $P$-points $p,q\in \omega^*$, the following are equivalent. (1) $q\leq_{\text{RK}}p$. (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.
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