Continuous functions between Isbell-Mrówka spaces
García-Ferreira, Salvador
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998), p. 185-195 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 1998-01-01
Classification:  54A20,  54A35,  54C20 
@article{118997,
     author = {Salvador Garc\'\i a-Ferreira},
     title = {Continuous functions between Isbell-Mr\'owka spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {39},
     year = {1998},
     pages = {185-195},
     zbl = {0938.54004},
     mrnumber = {1623018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118997}
}

García-Ferreira, Salvador. Continuous functions between Isbell-Mrówka spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 185-195. http://gdmltest.u-ga.fr/item/118997/

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