Topological Closure is Closed

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

Then the closure $\operatorname{cl}\left({H}\right)$ of $H$ is closed in $T$.

Proof
From Closure of Topological Closure equals Closure:
 * $\operatorname{cl}\left({\operatorname{cl}\left({H}\right)}\right) = \operatorname{cl}\left({H}\right)$

From Set is Closed iff Equals Topological Closure, it follows that $\operatorname{cl}\left({H}\right)$ is closed.

Also see

 * Topological Closure is Closure Operator