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Homeomorphism groups of Sierpiński carpets and Erdős space

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Dijkstra J., Visser D.

Fundamenta Mathematicae

2010

tom 207

nr 1

1-19

Erdős space 𝔈 is the "rational" Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that 𝔈 is one-dimensional and homeomorphic to its own square 𝔈 × 𝔈, which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of 𝔈. Let $Mₙ^{n+1}$, n ∈ ℕ, be the n-dimensional Menger continuum in $ℝ^{n+1}$, also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of $Mₙ^{n+1}$. We consider the topological group $𝓗(Mₙ^{n+1},D)$ of all autohomeomorphisms of $Mₙ^{n+1}$ that map D onto itself, equipped with the compact-open topology. We show that under some conditions on D the space $𝓗(Mₙ^{n+1},D)$ is homeomorphic to 𝔈 for n ∈ ℕ ∖ {3}.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Dijkstra J.

1 Visser D.

1 2010

Wyniki wyszukiwania

Homeomorphism groups of Sierpiński carpets and Erdős space