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On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself

Published online by Cambridge University Press:  20 November 2018

Jan J. Dijkstra  and
Jan van Mill
Jan J. Dijkstra
Affiliation:
Faculteit der Exacte Wetenschappen / Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands e-mail: dijkstra@cs.vu.nlvanmill@cs.vu.nl
Jan van Mill
Affiliation:
Faculteit der Exacte Wetenschappen / Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands e-mail: dijkstra@cs.vu.nlvanmill@cs.vu.nl
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Abstract

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In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\mathbb{R}$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\mathbb{Q}$ onto itself is homeomorphic to the infinite power of $\mathbb{Q}$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\mathbb{Q}$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ${{\mathbb{R}}^{n}}$ for $n\ge 2$, by linking the groups in question with Erdős space.

Keywords

57S05homeomorphism groupreal linecountable dense setpseudoboundaryErdős spacehyperspace
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 3 , 01 June 2006 , pp. 529 - 547
Copyright
Copyright © Canadian Mathematical Society 2006

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