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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:
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