Closure is Closed/Power Set

Theorem
Let $S$ be a set.

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

Let $T \subseteq S$.

Then $\operatorname{cl} (T)$ is a closed set with respect to $\operatorname{cl}$.

Proof
By the definition of closure operator, $\operatorname{cl}$ is idempotent.

Therefore $\operatorname{cl} \left({\operatorname{cl} (T)}\right) = \operatorname{cl} (T)$, so $\operatorname{cl} (T)$ is closed.