Boundary of Compact Set in Hausdorff Space is Compact

Theorem

Let $T = \struct {S, \tau}$ be a Hausdorff topological space.

Let $K \subset S$ be a compact subspace of $T$.

Then its boundary $\partial K$ is compact.

Proof

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

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

$\blacksquare$