Let $X$ be an infinite set with a topology $T$, such that every infinite subset of $X$ is closed. Prove that $T$ is the discrete topology.
I have somewhat of an answer but I don't think it's enough to prove it, especially with respect to the subsets being infinite.
Let $S$ be contained in $X$, then $X \setminus S$ is also contained in $X$. Therefore we can say that $X \setminus S$ is closed, therefore $S$ is open for any $S$ contained in $X$. Hence $T$ is the discrete topology.