Description

Let $M$ be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let $D$ be an arbitrary countable dense subset of $M$. Consider the topological group $\mathcal{H}(M,D)$ which consists of all autohomeomorphisms of $M$ that map $D$ onto itself equipped with the compact-open topology. The authors present a complete solution to the topological classification problem for $\mathcal{H}(M,D)$ as follows. If $M$ is a one-dimensional topological manifold, then they proved in an earlier paper that $\mathcal{H}(M,D)$ is homeomorphic to $\mathbb{Q}^\omega$, the countable power of the space of rational numbers. In all other cases they find in this paper that $\mathcal{H}(M,D)$ is homeomorphic to the famed Erdős space $\mathfrak E$, which consists of the vectors in Hilbert space $\ell^2$ with rational coordinates. They obtain the second result by developing topological characterizations of Erdős space.