Set is Subset of its Topological Closure

Theorem
Let $T$ be a topological space, and let $H \subseteq T$.

Let $\operatorname{cl}\left({H}\right)$ be the closure of $H$ in $T$.

Then:
 * $H \subseteq \operatorname{cl}\left({H}\right)$

Proof
From the definition of closure, we have:
 * $\operatorname{cl} \left({H}\right)$ is the union of $H$ and its limit points.

From Subset of Union it follows directly that:


 * $H \subseteq \operatorname{cl}\left({H}\right)$