If $A,B\subset \mathbb{R}$ are closed in $\mathbb{R}$, is $A+B$ also closed in $\mathbb{R}$? I think it is not, but could not find a counter example: any suggestions?
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By $A+B$, do you mean ${a+b\mid a\in A, b\in B}$? – Arthur Dec 25 '15 at 15:38
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3See this. – David Mitra Dec 25 '15 at 15:40
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Relevant threads: 1 2 3 – Viktor Vaughn Dec 25 '15 at 15:59
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What about $A= \bf Z$ and $B= \sqrt 2 \bf Z$,(of course as a subspace of $\bf R$ with usual topology) ?
Extra Exercise: If $A$ is compact and $B$ is closed then $A+B$ is closed.
Arpit Kansal
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