* https://en.wikipedia.org/wiki/Surface_(topology)\#Classification_of_closed_surfaces
* http://www.proofwiki.org/wiki/Classification_of_Compact_Two-Manifolds

So simple!! You can either:
* cut two holes and glue a handle. This is easy to visualize as it can be embedded in <\R^3>: you just get a <Torus>, then a double torus, and so on
* cut a single hole and glue  a<Möbius strip> in it. Keep in mind that this is possible because the <Möbius strip> has a single boundary just like the hole you just cut. This leads to another infinite family that starts with:
  * 1: <real projective plane>
  * 2: <Klein bottle>

A handle cancels out a <Möbius strip>, so adding one of each does not lead to a new object.

You can glue a Mobius strip into a single hole in dimension larger than 3! And it gives you a Klein bottle!

Intuitively speaking, they can be sees as the smooth surfaces in N-dimensional space (called an embedding), such that deforming them is allowed. 4-dimensions is enough to embed cover all the cases: 3 is not enough because of the Klein bottle and family.


Discussion on "Classification of closed surfaces" by Ciro Santilli 32