- #1

- 4,802

- 29

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.

Thx!

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!

Last edited: