# Closure is Closed/Power Set

## Theorem

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

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

Let $T \subseteq S$.

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

## Proof

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

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

$\blacksquare$