# Compact Riemann surface

In mathematics, a **compact Riemann surface** is a complex manifold of dimension one that is a compact space.

## Introduction

Riemann surfaces are generally classified first into the *compact* (those that are closed manifolds) and the *open* (the rest, which from the point of view of complex analysis are very different, being for example Stein manifolds).

## As algebraic curves

Every compact Riemann surface *C* that is a connected space can be represented as an algebraic curve defined over the complex number field. More precisely, the meromorphic functions on *C* make up the function field *F* on the corresponding curve; *F* is a field extension of the complex numbers of transcendence degree equal to 1. It can in fact be generated by two functions *f* and *g*. This is a structural result on the meromorphic functions: there are *enough* in the sense of separating out the points of *C*, and any two are algebraically dependent. These facts were known in the nineteenth century (see GAGA^{[1]} for more in this direction).

A general compact Riemann surface is therefore a finite disjoint union of complex (non-singular) algebraic curves.