Definition:Topology Defined by Closed Sets

Definition
Let $S$ be a set.

Let $F \subseteq \mathcal P(S)$ be a subset of its power set satisfying the closed set axioms.

The topology defined by $F$ is the topology whose open sets are the complements of elements of $F$:
 * $\tau = \{U \subseteq S : S \setminus U \in F\}$

Also see

 * Topology Defined by Closed Sets is Topology