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

Abstract

AbstractIn this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line ℝ, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers ℚ onto itself is homeomorphic to the infinite power of 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 ℚ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ℝn for n ≥ 2, by linking the groups in question with Erdős space.