If $U$ is open in the complex plane $\mathbb{C}$, and $A$ is any subset of the complex plane $\mathbb{C}$, then $U \cap A$ is open in $A$.
I have re-expressed $U$ as an arbitrary union of open neighborhoods of the elements of $U$ and can reason how this intersection is open in $U$, but not how it is open in $A$.
Any help would be appreciated.
Note that an open ball in A is by definition the intersection of an open ball in C and A.
Since U is a union of open balls in C, the intersection of U and A is a union of open balls in A, hence it is open in A.