 Boundary-bumping theorem for compact subsets

Every connected component $C$ of a nonempty compact proper subset $D$ of a connected Hausdorff space meets its boundary. If not, the Šura-Bura lemma identifies $C$ as a quasi-component in the compact Hausdorff subspace $D$. Finitely many clopen neighborhoods separate it from the compact boundary, giving a nonempty relatively clopen $H\subseteq \mathrm{int}D$. Such $H$ is open in the ambient space, and is also closed there because $D$ is closed. It is proper because $D$ is proper. This contradicts connectedness.