Let $G$ be a domain in $\mathbb{C}$. Define the germ of a function element $(f, D)$ at $z \in D$. Let $\mathcal{G}$ be the set of all germs of function elements in $G$. Define the topology on $\mathcal{G}$. Show it is a topology, and that it is Hausdorff. Define the complex structure on $\mathcal{G}$, and show that there is a natural projection map $\pi: \mathcal{G} \rightarrow G$ which is an analytic covering map on each connected component of $\mathcal{G}$.

Given a complete analytic function $\mathcal{F}$ on $G$, describe how it determines a connected component $\mathcal{G}_{\mathcal{F}}$ of $\mathcal{G}$. [You may assume that a function element $(g, E)$ is an analytic continuation of a function element $(f, D)$ along a path $\gamma:[0,1] \rightarrow G$ if and only if there is a lift of $\gamma$ to $\mathcal{G}$ starting at the germ of $(f, D)$ at $\gamma(0)$ and ending at the germ of $(g, E)$ at $\gamma(1)$.]

In each of the following cases, give an example of a domain $G$ in $\mathbb{C}$ and a complete analytic function $\mathcal{F}$ such that:

(i) $\pi: \mathcal{G}_{\mathcal{F}} \rightarrow G$ is regular but not bijective;

(ii) $\pi: \mathcal{G}_{\mathcal{F}} \rightarrow G$ is surjective but not regular.