Definition:Closed Set/Closure Operator

Definition
Let $S$ be a set.

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

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