Set is Closed iff Equals Topological Closure/Proof 1

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

Then $H$ is closed in $T$ iff $H = \operatorname{cl}\left({H}\right)$.

That is, a closed set equals its closure (which makes semantic sense).

Proof
Let $H'$ denote the derived set of $H$.

By Closed Set contains All its Limit Points, $H$ is closed in $T$ iff $H' \subseteq H$.

By Union with Superset is Superset, $H' \subseteq H$ iff $H = H \cup H'$.

The result follows from the definition of closure.