Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. Dijkstra, and Dave Visser. "Homeomorphism groups of Sierpiński carpets and Erdős space." Fundamenta Mathematicae 207.1 (2010): 1-19. <http://eudml.org/doc/282718>.

@article{JanJ2010,
abstract = {Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . 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 $(Mₙ^\{n+1\},D)$ 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 $(Mₙ^\{n+1\},D)$ is homeomorphic to for n ∈ ℕ ∖ 3.},
author = {Jan J. Dijkstra, Dave Visser},
journal = {Fundamenta Mathematicae},
language = {eng},
number = {1},
pages = {1-19},
title = {Homeomorphism groups of Sierpiński carpets and Erdős space},
url = {http://eudml.org/doc/282718},
volume = {207},
year = {2010},
}

TY - JOUR
AU - Jan J. Dijkstra
AU - Dave Visser
TI - Homeomorphism groups of Sierpiński carpets and Erdős space
JO - Fundamenta Mathematicae
PY - 2010
VL - 207
IS - 1
SP - 1
EP - 19
AB - Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . 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 $(Mₙ^{n+1},D)$ 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 $(Mₙ^{n+1},D)$ is homeomorphic to for n ∈ ℕ ∖ 3.
LA - eng
UR - http://eudml.org/doc/282718
ER -