We investigate the space ${\cal M}$ of classical solutions to Witten's
formulation of 2+1 gravity on the manifold ${\bf R} \times T^2$. ${\cal M}$ is
connected, but neither Hausdorff nor a manifold. However, removing from ${\cal
M}$ a set of measure zero yields a connected manifold which is naturally viewed
as the cotangent bundle over a non-Hausdorff base space. Avenues towards
quantizing the theory are discussed in view of the relation between spacetime
metrics and the various parts of~${\cal M}$. (Contribution to the proceedings
of the Lanczos Centenary Conference, Raleigh, NC, December 12--17, 1993.)