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 Erdos 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 Erdos space.