MAD families and $P$-points

García-Ferreira, Salvador, and Szeptycki, Paul J.. "MAD families and $P$-points." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 699-705. <http://eudml.org/doc/250207>.

@article{García2007,
abstract = {The Katětov ordering of two maximal almost disjoint (MAD) families $\mathcal \{A\}$ and $\mathcal \{B\}$ is defined as follows: We say that $\mathcal \{A\}\le _K \mathcal \{B\}$ if there is a function $f: \omega \rightarrow \omega $ such that $f^\{-1\}(A)\in \mathcal \{I\}(\mathcal \{B\})$ for every $A\in \mathcal \{I\}(\mathcal \{A\})$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in \mathcal \{I\}(\mathcal \{A\})^+$, we have that $\mathcal \{A\}|_X\le _K \mathcal \{A\}$. 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 $. This result has a nicer topological interpretation: The symbol $\mathcal \{F\}(\mathcal \{A\})$ will denote the Franklin compact space associated to a MAD family $\mathcal \{A\}$. 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 $x = p$-$\lim _\{n \rightarrow \infty \}x_n$ (that is, for every neigborhood $V$ of $x$, we have that $\lbrace n < \omega : x_n \in V\rbrace \in p$). [CH] For every $K$-uniform MAD family $\mathcal \{A\}$ there is a $P$-point $p$ of $\omega ^*$ such that $\mathcal \{F\}(\mathcal \{A\})$ 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) $q\le _\{\text\{RK\}\}p$. (2) For every $MAD$ family $\mathcal \{A\}$, the space $\mathcal \{F\}(\mathcal \{A\})$ is a $\text\{FU\}(p)$-space whenever it is a $\text\{FU\}(q)$-space.},
author = {García-Ferreira, Salvador, Szeptycki, Paul J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Franklin compact space; $P$-point; $\text\{FU\}(p)$-space; maximal almost disjoint family; Katětov ordering; Rudin-Keisler ordering; Franklin compact space},
language = {eng},
number = {4},
pages = {699-705},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {MAD families and $P$-points},
url = {http://eudml.org/doc/250207},
volume = {48},
year = {2007},
}

TY - JOUR
AU - García-Ferreira, Salvador
AU - Szeptycki, Paul J.
TI - MAD families and $P$-points
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 4
SP - 699
EP - 705
AB - The Katětov ordering of two maximal almost disjoint (MAD) families $\mathcal {A}$ and $\mathcal {B}$ is defined as follows: We say that $\mathcal {A}\le _K \mathcal {B}$ if there is a function $f: \omega \rightarrow \omega $ such that $f^{-1}(A)\in \mathcal {I}(\mathcal {B})$ for every $A\in \mathcal {I}(\mathcal {A})$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in \mathcal {I}(\mathcal {A})^+$, we have that $\mathcal {A}|_X\le _K \mathcal {A}$. 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 $. This result has a nicer topological interpretation: The symbol $\mathcal {F}(\mathcal {A})$ will denote the Franklin compact space associated to a MAD family $\mathcal {A}$. 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 $x = p$-$\lim _{n \rightarrow \infty }x_n$ (that is, for every neigborhood $V$ of $x$, we have that $\lbrace n < \omega : x_n \in V\rbrace \in p$). [CH] For every $K$-uniform MAD family $\mathcal {A}$ there is a $P$-point $p$ of $\omega ^*$ such that $\mathcal {F}(\mathcal {A})$ 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) $q\le _{\text{RK}}p$. (2) For every $MAD$ family $\mathcal {A}$, the space $\mathcal {F}(\mathcal {A})$ is a $\text{FU}(p)$-space whenever it is a $\text{FU}(q)$-space.
LA - eng
KW - Franklin compact space; $P$-point; $\text{FU}(p)$-space; maximal almost disjoint family; Katětov ordering; Rudin-Keisler ordering; Franklin compact space
UR - http://eudml.org/doc/250207
ER -