# Closure is Closed/Power Set

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## Theorem

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\cl: \powerset S \to \powerset S$ be a closure operator.

Let $T \subseteq S$.

Then $\map \cl T$ is a closed set with respect to $\cl$.

## Proof

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

Therefore $\map \cl {\map \cl T} = \map \cl T$, so $\map \cl t$ is closed.

$\blacksquare$