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.