(a) Let $f: R \rightarrow S$ be a non-constant holomorphic map between Riemann surfaces. Prove that $f$ takes open sets of $R$ to open sets of $S$.

(b) Let $U$ be a simply connected domain strictly contained in $\mathbb{C}$. Is there a conformal equivalence between $U$ and $\mathbb{C}$ ? Justify your answer.

(c) Let $R$ be a compact Riemann surface and $A \subset R$ a discrete subset. Given a non-constant holomorphic function $f: R \backslash A \rightarrow \mathbb{C}$, show that $f(R \backslash A)$ is dense in $\mathbb{C}$.