Question 1
(a) Give an example of a space where the discrete topology is the same as the finite complement topology.
Let $\displaystyle X$ be a finite set. Then the discrete topology is simply $\displaystyle \mathcal{P}(X)$. But since $\displaystyle X$ is finite it means the complement of any $\displaystyle Y\subseteq X$ is finite. Thus, the finite complement topology consists of all subsets of $\displaystyle X$ i.e. $\displaystyle \mathcal{P}(X)$.