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.
need one counter example for sum of two closed set need not be closed
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general-topology
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6See also this questions: http://math.stackexchange.com/questions/124130/sum-of-two-closed-sets-in-mathbb-r-is-closed – 2012-04-27
2 Answers
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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 $0$.
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0@Urgent $A+B$ is the set of all sums $x+y$ with $x\in A$ and $y\in B$. – 2017-09-03
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Let $A=\{(x,y):y\ge e^x\}$ and $B=\{(x,0)\}$. Then $A+B=\{(x,y):y>0\}$.
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0@user47709 Yes, it is. – 2013-10-07