Let $\tau$ be a finite closed topology on a set X . If $\tau$ is also the discrete topology, prove that the set X is finite.
Since $\tau$ is a discrete topology ,it contains all its subsets. Also $\tau$ is finite closed topology ,which implies all subsets in $\tau$ has finite complement . Combining these two we get, all the subsets of X have finite complement,i.e., X-S$_i$ = finite for all i.
How to proceed after that ?