Let $X$ be a topological space and $B$ is a Borel set of this space, i.e. $B\in\mathcal{B}(X)$ where $\mathcal{B}(X)$ is the smallest $\sigma$-algebra which contains all open subsets of $X$.
Let $B\subset A$ where $A$ is compact. Is it true that A' = (A\setminus B) \cup\partial B is a compact set?
If it is not true in general, does it hold if $X$ is also metrizable (separable)?
Finally, is a condition that $B$ is a Borel set is crucial?