# 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:

$S \cong P \setminus \sim$