Set is Clopen iff Boundary is Empty

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

Then $H$ is both closed and open in $T$ :
 * $\partial H = \O$

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

Proof
From Set is Open iff Disjoint from Boundary we have that:
 * $H$ is open in $T$ $\partial H \cap H = \O$

From Set is Closed iff it Contains its Boundary we have that:


 * $H$ is closed in $T$ $\partial H \subseteq H$

From Intersection with Subset is Subset‎:


 * $\partial H \subseteq H \iff \partial H \cap H = \partial H $

From the above it follows that $\partial H = \O$ $H$ is closed and $H$ is open.