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I know the proof that If A is compact and B closed then A+B is closed but would like to have an example where both are closed but not A+B.I am not able to figure out.

Myshkin
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  • By $A+B$, do you mean the union, or are you taking a sum in some sort of topological group like $\mathbb{R}^n$? – Brett Frankel Apr 26 '12 at 21:40
  • What do you mean by '+'? The union? Or... are you working in a topological abelian group, so that '+' means the set of all sums of points from A and B? –  Apr 26 '12 at 21:40
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    I am not saying about union, $A+B={x+y:x\in A, y\in B}$ – Myshkin Apr 26 '12 at 21:42
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    Any more information about the space would be helpful. Are you working in $\mathbb{R}^n$ or a more general topological group (in which case it would be nice to know what axioms you're working with)? – Brett Frankel Apr 26 '12 at 21:43
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    See also this questions: http://math.stackexchange.com/questions/124130/sum-of-two-closed-sets-in-mathbb-r-is-closed – Martin Sleziak Apr 27 '12 at 05:19

2 Answers2

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Assuming $A+B=\{a+b\mid a\in A, b\in B\}$:

$A=\{\,1,2,3,\ldots\,\}$ and $B=\{ \,-1 +{1\over2}, -2 +{1\over3} ,-3+{1\over4},\ldots\,\}$. The sum contains $\{\,{1\over2},{1\over3},{1\over4},\ldots\,\}$ but not its limit point $0$.

David Mitra
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  • when we are defining sum of two sets as above, are we defining just addition of first element of $A$ with only first element of $B$? or we can add randomly and all possible way!? I have this doubt – Myshkin Sep 02 '17 at 05:43
  • @Urgent $A+B$ is the set of all sums $x+y$ with $x\in A$ and $y\in B$. – David Mitra Sep 03 '17 at 13:56
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Let $A=\{(x,y):y\ge e^x\}$ and $B=\{(x,0)\}$. Then $A+B=\{(x,y):y>0\}$.

Julián Aguirre
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