Continuous functions between Isbell-Mrówka spaces

García-Ferreira, Salvador. "Continuous functions between Isbell-Mrówka spaces." Commentationes Mathematicae Universitatis Carolinae 39.1 (1998): 185-195. <http://eudml.org/doc/248256>.

@article{García1998,
abstract = {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 $\{\mathbf \{S\}\}(\omega )$ 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 ) = \lbrace f \in \{\mathbf \{S\}\}(\omega ) : f[A] \in \Sigma $ for all $A \in \Sigma \rbrace $. It is shown that for every $f \in \{\mathbf \{S\}\}(\omega )$ 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 = \lbrace n+a : a \in A \rbrace $ 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 \le n < \omega $, where $n \cdot A = \lbrace n \cdot a : a \in A \rbrace $ for $1 \le n < \omega $. Several open questions are listed.},
author = {García-Ferreira, Salvador},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$MAD$-family; Isbell-Mr’owka space; MAD-family; Isbell-Mrówka space},
language = {eng},
number = {1},
pages = {185-195},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Continuous functions between Isbell-Mrówka spaces},
url = {http://eudml.org/doc/248256},
volume = {39},
year = {1998},
}

TY - JOUR
AU - García-Ferreira, Salvador
TI - Continuous functions between Isbell-Mrówka spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 1
SP - 185
EP - 195
AB - 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 ${\mathbf {S}}(\omega )$ 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 ) = \lbrace f \in {\mathbf {S}}(\omega ) : f[A] \in \Sigma $ for all $A \in \Sigma \rbrace $. It is shown that for every $f \in {\mathbf {S}}(\omega )$ 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 = \lbrace n+a : a \in A \rbrace $ 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 \le n < \omega $, where $n \cdot A = \lbrace n \cdot a : a \in A \rbrace $ for $1 \le n < \omega $. Several open questions are listed.
LA - eng
KW - $MAD$-family; Isbell-Mr’owka space; MAD-family; Isbell-Mrówka space
UR - http://eudml.org/doc/248256
ER -