(1) Show, by example, that an infinite intersection of open sets in $\mathbb{C}$ need not be an open set in $\mathbb{C}$.
Consider $\bigcap_{i}^{\infty}A_i \subset \mathbb{C}$ for $A_i$ open. Each $A_i$ has the property that for each point $a$ in the set there is an $\epsilon>0$ such that $B_\epsilon(a)\subset A_i$...
(2) Show that every open set in $\mathbb{C}$ can be written as a countable union of open balls.
Let $A\subset\mathbb{C}$ be open. Define a rational open ball with radius $r\in\mathbb{Q}$ around a point $z_0\in\mathbb{C}$ in $\mathbb{C}$ as $B_r(z_0)$=$z\in =\mathbb{C}||z-z_0|$} < r. By definition of open sets $\mathbb{C}$, $\forall a\in A$, $\exists r>0$ such that $B_r(a)\subset A$. Thus given $B_r(a_1)$,$B_r(a_2)$,...,$B_r(a_n)$, these balls cover $A$ and since each contains a rational number they are countable.
(3) Show, by example, that there are open sets in $\mathbb{C}$ for which the open balls in (2) cannot be made pairwise disjoint.
Without knowing the exact proof for (2) I'm not sure about this one.