Sum of two closed sets need not be closed

In $\mathbb{R}$, the sets $X=\{n:n\ge 1\}$ and $Y=\{-n+1/n:n\ge 1\}$ are closed because their points have no finite accumulation point. Their Minkowski sum contains $1/n\to 0$ but does not contain zero, since $m-n+1/n=0$ has no positive-integer solution. Compactness of one summand is a useful sufficient condition missing from this counterexample.