||Recent advances in microstructural characterization have made it possible to measure grain boundaries and their networks in full crystallographic detail. Statistical studies of the complete boundary space using full crystallographic parameters (misorientations and boundary plane inclinations) are limited because the topology of the parameter space is not understood (especially for homophase grain boundaries). This paper addresses some of the complexities associated with the group space of grain boundaries, and resolves the topology of the complete boundary space for systems of two-dimensional crystals. Although the space of homophase boundaries is complicated by the existence of a `no-boundary' singularity, i.e., no boundary exists when the misorientation is zero, here it is shown that this singularity can be removed owing to a second special symmetry. Specifically, "grain exchange symmetry" refers to the indistinguishability of the adjoining grains at a homophase boundary, and results in symmetrically equivalent descriptions of the boundary. This symmetry affects the topology of misorientation spaces, removes the `no-boundary' singularity, and permits the identification of the space topology for two-dimensional crystals with various point symmetries. For crystals of C1and C3point symmetry, the homophase grain boundary space is shown to be D2, while for C2, C4, and C6symmetries it is \mathbb RP1.