The theorem of classification of closed surfaces says that any closed surface is homeomorphic to a fundamental polygon in the plane. I was wondering if any fundamental polygon can be made into a closed surface by adjoining an appropriate atlas to it. The topological requirements of a closed surface (compactness and connectedness) are certainly met by the fund. polygon but can we give it an atlas, and if so, is the resulting 2-manifold boundaryless? Sure as hell appears so to me but I want to make sure. Side question: Say we have a polygon with sides identified, and say that some side "a" is identified with n sides. Is n a topological invariant? Thx!