Boundary of Compact Set in Hausdorff Space is Compact

Theorem
Let $X$ be a Hausdorff topological space.

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

Then its boundary $\partial K$ is compact.

Proof
By Compact Subspace of Hausdorff Space is Closed, $K$ is closed in $X$.

By Boundary of Compact Closed Set is Compact, $\partial K$ is compact.