| Title:
|
MAD families and $P$-points (English) |
| Author:
|
García-Ferreira, S. |
| Author:
|
Szeptycki, P. J. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
699-705 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Franklin compact space |
| Keyword:
|
$P$-point |
| Keyword:
|
$\text{FU}(p)$-space |
| Keyword:
|
maximal almost disjoint family |
| Keyword:
|
Katětov ordering |
| Keyword:
|
Rudin-Keisler ordering |
| MSC:
|
03E05 |
| MSC:
|
03E50 |
| MSC:
|
54A99 |
| MSC:
|
54B99 |
| idZBL:
|
Zbl 1199.03028 |
| idMR:
|
MR2375170 |
| . |
| Date available:
|
2009-05-05T17:05:48Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119692 |
| . |
| Reference:
|
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| Reference:
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| Reference:
|
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|
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| Reference:
|
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| Reference:
|
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| . |
