Boundary of Compact Closed Set is Compact

Theorem
Let $X$ be a topological space.

Let $K\subset X$ be a compact subspace of $X$.

Let $K$ be closed in $X$.

Then its boundary $\partial K$ is compact.

Proof
By Boundary of Set is Closed, $\partial K$ is closed in $X$.

By Set is Closed iff Contains Boundary, $\partial K \subset K$.

By Closed Set in Topological Subspace, $\partial K$ is closed in $K$.

By Closed Subspace of Compact Space is Compact, $\partial K$ is compact.

Also see

 * Compact Subspace of Hausdorff Space is Closed
 * Boundary of Compact Set in Hausdorff Space is Compact