Open mapping theorem

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.