Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. Dijkstra¹ ; Dave Visser¹

¹Faculteit der Exacte Wetenschappen//Afdeling Wiskunde Vrije Universiteit Amsterdam De Boelelaan 1081 1081 HV Amsterdam The Netherlands

Fundamenta Mathematicae, Tome 207 (2010) no. 1, pp. 1-19

Résumé

Erdős space $\mathfrak E$ is the “rational” Hilbert space, that is, the set of vectors in $\ell^2$ with all coordinates rational. Erdős proved that $\mathfrak E$ is one-dimensional and homeomorphic to its own square $\mathfrak E \times \mathfrak E$, which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of $\mathfrak E$. Let $M^{n+1}_n$, $n \in \mathbb N$, be the $n$-dimensional Menger continuum in $\mathbb{R}^{n+1}$, also known as the $n$-dimensional Sierpiński carpet, and let $D$ be a countable dense subset of $M^{n+1}_n$. We consider the topological group $\mathcal{H}(M^{n+1}_n, D)$ of all autohomeomorphisms of $M^{n+1}_n$ that map $D$ onto itself, equipped with the compact-open topology. We show that under some conditions on $D$ the space $\mathcal{H}(M^{n+1}_n, D)$ is homeomorphic to $\mathfrak E$ for $n \in \mathbb{N} \setminus \{3\}$.

Détail

DOI : 10.4064/fm207-1-1

Keywords: erd space mathfrak rational hilbert space set vectors ell coordinates rational erd proved mathfrak one dimensional homeomorphic its own square mathfrak times mathfrak which makes important example dimension theory dijkstra van mill found topological characterizations mathfrak mathbb n dimensional menger continuum mathbb known n dimensional sierpi ski carpet countable dense subset consider topological group mathcal autohomeomorphisms map itself equipped compact open topology under conditions space mathcal homeomorphic mathfrak mathbb setminus