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Why is it necessary for one of the sets in the Minkowski sum to be bounded (given that both sets are closed) in order that the Minkowski sum be closed?

[Edit: In view of @robjohn 's comment (thanks!), I should perhaps change the question to: "Why is it necessary for one of the sets in the Minkowski sum to be bounded (given that both sets are closed) in order to guarantee that the Minkowski sum be closed?]

Also, what are the conditions for the Minkowski sum to be open?

Thanks.

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Let $A=\{(x,0):x\ge0\}$ and $B=\{(0,x):x\ge0\}$. $A+B=\{(x,y):x,y\ge0\}$ is closed, yet neither $A$ nor $B$ is bounded. –  robjohn Nov 10 '11 at 9:27
    
Thanks, @robjohn. I have edited the question. –  bart Nov 10 '11 at 9:47
    
So you are looking for closed $A$ and $B$ so that $A+B$ is not closed. Of course, both $A$ and $B$ must be unbounded. –  robjohn Nov 10 '11 at 10:54
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1 Answer

up vote 1 down vote accepted

Let $A:=\{1,2,3,...\}$ (that is, $\mathbb{N}$, not including 0), and $B:=\{(-k+1/k\mid k\in \mathbb{N}, k\ge 2\}=\{-2+1/2, -3+1/3,-4+1/4,... \}$. Then both $A$ and $B$ are closed (in fact discrete) subsets of $\mathbb{R}$, but the sum $A+B$ includes all numbers of the form $1/k$ for $k\in\mathbb{N}, k\ge 2$, but it does not include 0. Hence $A+B$ is not closed.

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