Set is Closed iff it Contains its Boundary

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

Then $H$ is closed in $T$ iff:
 * $\partial H \subseteq H$

where $\partial H$ is the boundary of $H$.

Proof
From Boundary is Intersection of Closure with Closure of Complement:
 * $\partial H = \operatorname{cl} \left({H}\right) \cap \operatorname{cl} \left({T \setminus H}\right)$

where $\operatorname{cl} \left({H}\right)$ is the closure of $H$.

Hence from Intersection Subset we have that:
 * $\partial H \subseteq \operatorname{cl} \left({H}\right)$

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

Hence the result.