| Title:
|
Continuous functions between Isbell-Mrówka spaces (English) |
| Author:
|
García-Ferreira, S. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
39 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
185-195 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Psi(\Sigma)$ be the Isbell-Mr'owka space associated to the $MAD$-family $\Sigma$. We show that if $G$ is a countable subgroup of the group ${\bold S}(\omega)$ of all permutations of $\omega$, then there is a $MAD$-family $\Sigma$ such that every $f \in G$ can be extended to an autohomeomorphism of $\Psi(\Sigma)$. For a $MAD$-family $\Sigma$, we set $Inv(\Sigma) = \{ f \in {\bold S}(\omega) : f[A] \in \Sigma $ for all $A \in \Sigma \}$. It is shown that for every $f \in {\bold S}(\omega)$ there is a $MAD$-family $\Sigma$ such that $f \in Inv(\Sigma)$. As a consequence of this result we have that there is a $MAD$-family $\Sigma$ such that $n+A \in \Sigma$ whenever $A \in \Sigma$ and $n < \omega$, where $n+A = \{ n+a : a \in A \}$ for $n < \omega$. We also notice that there is no $MAD$-family $\Sigma$ such that $n \cdot A \in \Sigma$ whenever $A \in \Sigma$ and $1 \leq n < \omega$, where $n \cdot A = \{ n \cdot a : a \in A \}$ for $1 \leq n < \omega$. Several open questions are listed. (English) |
| Keyword:
|
$MAD$-family |
| Keyword:
|
Isbell-Mr'owka space |
| MSC:
|
54A20 |
| MSC:
|
54A35 |
| MSC:
|
54C20 |
| idZBL:
|
Zbl 0938.54004 |
| idMR:
|
MR1623018 |
| . |
| Date available:
|
2009-01-08T18:39:58Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118997 |
| . |
| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| . |
