Definition:Isolated Singularity/Riemann Surface

Definition
Let $U$ be an open set of a Riemann surface.

Let $z_0 \in U$.

Let $f: U \setminus \set {z_0} \to \C$ be a holomorphic function.

Then $f$ has an isolated singularity at $z_0$.

In most applications, the Riemann surface in question is the complex plane or the Riemann sphere.