Is there any compact surface having $\mathbb{R}^2$ as its (universal, of course)covering space other than torus and the Klein bottle?
While I was thinking of the tiling the plane with a regular polygon, I came to suspect there is only two surfaces mentioned which have $\mathbb{R}^2$ as its covering space.
Every closed orientable surface of genus at least 1, and every non-orientable surface of non-orientable genus at least 2 has the plane as its universal cover(though if one considers the geometry too, then really you're working with the hyperbolic plane, rather than Euclidean space, but they are homeomorphic).
The only other closed surfaces are the sphere, which is its own universal cover, and the real projective plane, which has the sphere as its universal cover. This gives a total classification for closed surfaces.
