Let $ W \subset X$ is an open subset of a Riemann surface $X$ , we will denote the set of holomorphic functions on $W$ by $O_{X}(W)=\{ f : W \rightarrow \mathbb{C }| f $ is holomorphic $\}$. How to show that $O_{X}(W)$ is a $ \mathbb{C}-$ algebra. My aim is to show there exists a ring homomorphism $h :\mathbb{C}\rightarrow O_{X}(W)$.
Basically for each $z\in C $ we have to get a holomorphic map. How can I get that map ?