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# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Homeomorphism groups of Sierpiński carpets and Erdős space

### Tom 207 / 2010

Fundamenta Mathematicae 207 (2010), 1-19 MSC: Primary 57S05. DOI: 10.4064/fm207-1-1

#### Streszczenie

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 $n \in \mathbb{N} \setminus \{3\}$.

#### Autorzy

• Jan J. DijkstraFaculteit der Exacte Wetenschappen//Afdeling Wiskunde
Vrije Universiteit Amsterdam
De Boelelaan 1081
1081 HV Amsterdam
The Netherlands
e-mail
• Dave VisserFaculteit der Exacte Wetenschappen//Afdeling Wiskunde
Vrije Universiteit Amsterdam
De Boelelaan 1081
1081 HV Amsterdam
The Netherlands
e-mail

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