• On the other hand, some subsets of $\mathbb{R}^n$ (e.g. a sphere, a torus) are accepted as manifolds w/o specifying an atlas. I think the situation is that if there is a "well-known" submersion such with this subset is one of the fibers, then this subset is accepted by community as a manifold. (A neccesary condition then is for this subset to have a unique tangent space at each point, so tetrahedron fails to satisfy this condition and you don't need to try to find a relevant submersion.) – Paul Yuryev Mar 12 '10 at 13:47