Topology Generated by Closed Sets

Theorem
Let $X$ be a set.

Let $\FF$ be a set of subsets of $X$ such that:
 * $\O \in \FF$
 * $\forall A, B \in \FF: A \cup B \in \FF$
 * $\forall \GG \subseteq \FF: \bigcap \GG \in \FF$

Let $\tau = \set {\relcomp X A: A \in \FF}$.

Then:
 * $T = \struct {X, \tau}$ is topological space and
 * for every subset $A$ of $X$, $A$ is closed in $T$ $A \in \FF$.