黎曼曲面
Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such "finite" coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves—that is, such coverings all come from finite extensions of the function field.