Every nonempty proper simply connected open subset of $\CC$ is biholomorphic to the unit disc, by a unique map sending a given point to $0$ with positive derivative there.Suggest a slogan
Let \(\Omega\subsetneq\CC\) be a nonempty simply connected open set. Then for every \(z_0\in\Omega\) there exists a unique biholomorphism \(F\colon\Omega\to\DD\) such that \(F(z_0)=0\) and \(F'(z_0)>0\). In particular, any two nonempty simply connected open sets \(\Omega_1,\Omega_2\subsetneq\CC\) are biholomorphic.