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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.
