Exercise 9, § 1

Let $X$ be a set and let $M \to \overline{M}$ be a mapping of $\mathfrak{P}(X)$ onto itself such that: 1) $\varnothing = \overline{\varnothing}$; 2) for every $M \subset X$ we have $M \subset \overline{M}$; 3) for every $M \subset X$ we have $\overline{\overline{M}} = \overline{M}$; 4) for all $M \subset X$ and $N \subset X$ we have

Show that there is a unique topology on $X$ such that $\overline{M}$ is the closure of $M$ with respect to this topology, for all $M \subset X$ (define the topology by means of its closed sets).