Boundary of Compact Set in Hausdorff Space is Compact

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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.

$\blacksquare$