= Solution
\b[False.] In $\mathbb R$, let
$$
X=\{n:n\ge1\},\qquad Y=\{-n+1/n:n\ge1\}.
$$
Each is closed: its distinct points escape to infinity, so it has no finite accumulation point. Their <Minkowski sum> contains $1/n$, tending to zero. However zero is not in the sum. If $m-n+1/n=0$ with positive integers $m,n$, then $n-m=1/n$. For $n>1$ the right side is strictly between zero and one, while the left side is an integer; for $n=1$ the equation would require $m=0$. Thus \b[the sum of two closed sets need not be closed]. The same counterexample embeds along a coordinate axis in $\mathbb R^n$.
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