Erdős space $\mathfrak {E}$ and complete Erdős space $\mathfrak {E}_{c}$ have been previously shown to have topological characterizations. In this paper, we provide a topological characterization of the topological space $\mathbb {Q}\times \mathfrak {E}_{c}$ , where $\mathbb {Q}$ is the space of rational numbers. As a corollary, we show that the Vietoris hyperspace of finite sets $\mathcal {F}(\mathfrak {E}_{c})$ is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ . We also characterize the factors of $\mathbb {Q}\times \mathfrak {E}_{c}$ . An interesting open question that is left open is whether $\sigma \mathfrak {E}_{c}^{\omega }$ , the $\sigma $ -product of countably many copies of $\mathfrak {E}_{c}$ , is homeomorphic to $\mathbb {Q}\times \mathfrak {E}_{c}$ .

We investigate C-sets in almost zero-dimensional spaces, showing that closed $\sigma $ C-sets are C-sets. As corollaries, we prove that every rim- $\sigma $ -compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show that these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of $($ Cantor set $)\!\times \mathbb R$ is nowhere dense.

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.

In 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $\ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $\mathcal{E}$ of the Banach spaces $\ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $\mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $\mathcal{E}$ is closed in $\ell^p$, then it is homeomorphic to complete Erdős space if and only if $\dim\mathcal{E}>0$ and every $E_n$ is zero dimensional.