Sum of two closed sets need not be closed
= Sum of two closed sets need not be closed
In $\mathbb R$, the sets $X=\{n:n\ge1\}$ and $Y=\{-n+1/n:n\ge1\}$ are closed because their points have no finite accumulation point. Their <Minkowski sum> contains $1/n\to0$ 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.