Topological Closure is Closed

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

Then $\operatorname{cl}\left({H}\right)$ is closed.

Proof
From Closure of Closure equals Closure, we have that $\operatorname{cl}\left({\operatorname{cl}\left({H}\right)}\right) = \operatorname{cl}\left({H}\right)$.

From Closed Set Equals its Closure, we have that $H$ is closed in $T$ iff $H = \operatorname{cl}\left({H}\right)$.

Thus $\operatorname{cl}\left({H}\right)$ is closed.