Sum of a compact set and a closed set

In Euclidean space, a Minkowski sum of a compact set $X$ and a closed set $Y$ is closed. From any convergent sequence $x_j+y_j$, extract a convergent subsequence of $x_j$ using compactness. The corresponding $y_j$ then converges to the difference of the two limits, which lies in $Y$. If $Y$ is also bounded, the sum is bounded and therefore compact.