Definition:Closed Set/Closure Operator

Definition
Let $S$ be a set.

Let $\operatorname{cl}: \mathcal P \left({S}\right) \to \mathcal P \left({S}\right)$ be a closure operator.

Then a subset $T \subseteq S$ is closed (with respect to $\operatorname{cl}$) iff:
 * $\operatorname{cl} \left({T}\right) = T$