Problem: Suppose that $X$ is an infinite set equipped with the cofinite topology. Then for any finite set $A \subseteq X$ and infinite set $B \subseteq X$, find $\bar{A} $ and $\bar{B} $.
Solution: Note that the complement of A in $X$, that is $X\setminus A$ is open as its complement contains only finitely many points. Therefore, $A$ must be closed and hence, $\bar{A} = A$.
On the other hand, we claim that the closure of $B$ will be $X$. To show this, take any point $x \in X\setminus B$. Note that any open set $U$ containing $x$ contains all but finitely many points of $X$. Thus, $U$ must contains some points of $B$. Therefore, $\bar{B} =X$.