Open mapping theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

For an open subset in the complex numbers

Suppose $U$ is an open subset of $\mathbb{C}$, and $f:U\to \mathbb{C}$ is a holomorphic function. Then, $f$ is either a constant map (i.e. maps all elements of $U$ to the same complex number) or an open map: the image of any open subset of $U$ is open.

For Riemann surfaces

Suppose $M,N$ are Riemann surfaces and $f:M\to N$ is an analytic mapping. Then $f$ is either constant, or an open map.