Boundary of compact set My question is:
Is the boundary, $\partial K=\overline{K}\setminus \operatorname{int}(K)$, of any subset $K$ of any topological space $X$ necessarily compact?

No. Take $A$ to be any infinite set and $x$ to be anything not in $A$. Now consider the space $X = A \cup \{x\}$ with the topology where a set is open iff it's empty or it contains $x$. Thus $A$ has the discrete topology as a subspace of $X$ and $\{x\}$ is compact (as it's finite). But $\overline{\{x\}} = X$ and $\{x\}^\circ = \{x\}$, so $\partial \{x\} = A$ which—being an infinite discrete space—is not compact.
