Classification of Compact Two-Manifolds/Lemma

Lemma for Classification of Compact Two-Manifolds
A compact, boundaryless $2$-manifold $S$ is diffeomorphic to a polyhedral disk $P$ with edges identified pairwise.

That is, for any closed, connected $2$-manifold, there exists a polyhedral disk $P$ and an equivalence relation $\sim$ such that:
 * $S \cong P \setminus \sim$