Boundary of Set is Closed

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

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

Then $\partial H$ is closed in $T$.

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

From Closure is Closed, both $\operatorname{cl} \left({X}\right)$ and $\operatorname{cl} \left({T \setminus X}\right)$ are closed in $T$.

From Topology Defined by Closed Sets, the intersection of arbitrarily many (in particular $2$) closed sets of $T$ is a closed set of $T$.

As $\partial X$ is the intersection of $\operatorname{cl} \left({X}\right)$ and $\operatorname{cl} \left({T \setminus X}\right)$ the result follows.