Set constraints are inclusions between expressions denoting set of ground terms over a finitely ranked alphabet $\Sigma$. Rational spaces are topological spaces obtained as spaces of runs of topological $\Sigma$-hypergraphs. They were introduced by Kozen in \cite{K95a}, where the topological structure of the spaces of solutions to systems of set constraints was given in terms of rational spaces. In this paper we continue the investigation of rational spaces. We give a Myhill-Nerode like characterization of rational points, which in turn is used to re-derive results about the rational points of finitary rational spaces. We define congruences on $\Sigma$-hypergraphs, investigate their interplay with the Myhill-Nerode characterization, and finally we determine the computational complexity of some decision problems related to rational spaces.