# Article

 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: [BV] Balcar B., Vojtáš P.: Almost disjoint refinement of families of subsets of $N$.Proc. Amer. Math. Soc. 79 (1980), 465-470. MR 0567994 Reference: [Ba] Baskirov A.I.: On maximal almost disjoint systems and Franklin bicompacta.Soviet. Math. Dokl. 19 (1978), 864-868. MR 0504217 Reference: [CN] Comfort W.W., Negrepontis S.: The Theory of Ultrafilters.Grudlehren der Mathematischen Wissenschaften, Vol. 211, Springer-Verlag, 1974. Zbl 0298.02004, MR 0396267 Reference: [GJ] Gillman L., Jerison M.: Rings of Continuous Functions.Graduate Texts in Mathematics, Vol. 43, Springer-Verlag, 1976. Zbl 0327.46040, MR 0407579 Reference: [Ka] Katětov M.: A theorem on mappings.Comment. Math. Univ. Carolinae 8 (1967), 431-433. MR 0229228 Reference: [L] Levy R.: Almost $P$-spaces.Can. J. Math. 29 (1977), 284-288. Zbl 0342.54032, MR 0464203 Reference: [Mr] Mrówka S.: On completely regular spaces.Fund. Math. 41 (1954), 105-106. MR 0063650 .

## Files

Files Size Format View
CommentatMathUnivCarolRetro_39-1998-1_19.pdf 241.7Kb application/pdf View/Open

Partner of