In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the realline $R$, endowed with the compact-open topology. First, we prove that the subgroup ofhomeomorphisms that map the set of rational numbers $Q$ onto itselfis homeomorphic to the infinite power of $Q$ withthe product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundaryonto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $Q$ withthe Vietoris topology. We obtain similar results for the Cantor set but we also prove that theseresults do not extend to $R^n$ for $nge 2$, by linking the groups in question with ErdH osspace.
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机译：在本文中，我们主要考虑实线$ R $的自同胚群的两个自然子群，它们具有紧凑开放拓扑。首先，我们证明了将有理数$ Q $映射到其自身上的同胚子群与乘积拓扑的$ Q $的无限次幂同胚。其次，由伪同形映射到自身的同胚的组被证明是同维素拓扑的$ Q $非空紧子集的超空间的同胚。对于Cantor集，我们获得了相似的结果，但通过将相关组与ErdH osspace链接，我们也证明了这些结果不会扩展到$ nge 2 $的$ R ^ n $。
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