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 $T$ if and only if:

$\partial H = \O$

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

Proof

From Set is Open iff Disjoint from Boundary we have that:

$H$ is open in $T$ if and only if $\partial H \cap H = \O$

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

$H$ is closed in $T$ if and only if $\partial H \subseteq H$

From Intersection with Subset is Subset‎:

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

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

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: Exercise $3.9: 30 \ \text {(ii)}$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors