There is no bijective holomorphic map between $\mathbb{D}-\{0\}$ and $\{z|0

Question

This problem, stated in the title, comes from Schwarz Lemma part in complex analysis. It usually makes sense to ask of the existence of a holomorphic bijection(constructing one with some familiar holomorphic maps). However, how to figure out that there isn't any such map? Which proposition/theorem/lemma should I consider? Thanks!

Answer 1

Let $f: D \backslash \{0\} \rightarrow C=\{z|0<r <|z| < R\}$ be a holomorphic bijection. Then $f$ is bounded in the neighborhood of $0$ so that the singularity is removable. As the extension of $f$ is open with discrete fibers, $f(0)$ is a point that has a neighborhood $V$ such that $V \backslash \{f(0)\} \subset C$. So $f(0) \in C$.

So there is some $z \in D$ with $z \neq 0$ such that $f(z)=f(0)$. Then $0$ and $z$ have some disjoint open neighborhoods $U,W$, and $f(U) \cap f(W)=\{f(0)\}$ since $f$ is injective on $D \backslash \{0\}$. But $f$ is open so $f(U) \cap f(W)$ is a neighborhood of $f(0)$. We reach a contradiction.