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

Dijkstra, Jan J. ; Mill, Jan van

Canadian journal of mathematics, Tome 58 (2006) no. 3, pp. 529-547

Résumé

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.

Détail

DOI : 10.4153/CJM-2006-022-8

Mots-clés : 57S05, homeomorphism group, real line, countable dense set, pseudoboundary, Erdős space, hyperspace

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