Compact Riemann surface

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

A compact Riemann surface C that is a for more in this direction).

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