Problem: Let $X$ be a topological space with co-finite topology (that is, $U$ is open in $X$ if it is either $\emptyset$ or $X$ or $U^c$ is finite). Suppose that $X$ has at least three distinct subsets which is closed as well as open. Prove that $X$ is a finite set.
Solution: We know that for any topology the sets $\emptyset$ and $X$ are open and closed simultaneously. It is given that $X$ has at least three distinct subsets which are open and closed. Therefore, there exists $\emptyset \neq U\subsetneq X$ such that $U$ is open as well as closed. Since, the topology is cofinite, and $U$ is open implies that $U^c$ is finite. Since $U$ is also closed, so its complement must be open therefore, $\left( U^c \right) ^c = U$ must be finite. Since \[ X = U \sqcup U^c \] and $U,U^c$ are finite, so $X$ is also a finite set.