This question is motivated by this MO-problem asking if the Erdős spaces $\mathfrak E$ and $\mathfrak E_c$ admit a self-homeomorphism with dense orbits of points.

The affirmative answer would follow from the affirmative answer to the following

Problem.Are the Erdős spaces $\mathfrak E$ and $\mathfrak E_c$ homeomorphic to monothetic topological groups?

Now I recall the necessary definitions. A topological group $G$ is *monothetic* if it contains a dense cyclic subgroup. Many examples of monothetic topological groups can be found here.

The subspaces
$$\mathfrak E:=\{(x_n)_{n\in\omega}\in\ell_2:\forall n\in\omega\;\;x_n\in\mathbb Q\}$$
and
$$\mathfrak E_c:=\{(x_n)_{n\in\omega}\in\ell_2:\forall n\in\omega\;\;x_n\in\mathbb R\setminus\mathbb Q\}$$
of the real separable Hilbert space are called the *rational Erdős space* and the *complete Erdős space*, respectively. The rational Erdős space $\mathfrak E$ is a subgroup of $\ell_2$ and the complete Erdős space $\mathfrak E_c$ is homeomorphic to the closed subgroup $$G:=\{(x_n)_{n\in\omega}\in\ell_2:\forall n\in\omega\;\; x_n\in\tfrac1{2^n}\mathbb Z\}$$ by Theorem 4.11 of Dijkstra and van Mill.

However the subgroups $\mathfrak E$ and $G$ of $\ell_2$ are not monothetic.