(i) Let $f(z)=\sum_{n=1}^{\infty} z^{2^{n}}$. Show that the unit circle is the natural boundary of the function element $(D(0,1), f)$, where $D(0,1)=\{z \in \mathbb{C}:|z|<1\}$.

(ii) Let $X$ be a connected Riemann surface and $(D, h)$ a function element on $X$ into $\mathbb{C}$. Define a germ of $(D, h)$ at a point $p \in D$. Let $\mathcal{G}$ be the set of all the germs of function elements on $X$ into $\mathbb{C}$. Describe the topology and the complex structure on $\mathcal{G}$, and show that $\mathcal{G}$ is a covering of $X$ (in the sense of complex analysis). Show that there is a oneto-one correspondence between complete holomorphic functions on $X$ into $\mathbb{C}$ and the connected components of $\mathcal{G}$. [You are not required to prove that the topology on $\mathcal{G}$ is secondcountable.]