Continuous functions between Isbell-Mrówka spaces

García-Ferreira, S.

Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 1, pp. 185-195.

Résumé

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.

Détail

Classification : 54A20, 54A35, 54C20
Keywords: $MAD$-family; Isbell-Mr'owka space