# Topological Closure is Closed

## Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

Then the closure $\map \cl H$ of $H$ is closed in $T$.

## Proof

$\map \cl {\map \cl H} = \map \cl H$

From Set is Closed iff Equals Topological Closure, it follows that $\map \cl H$ is closed.

$\blacksquare$