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Let $A$ , $B$ be open subsets of $\mathbb{R}^n$ . Is $A+B$ a open subset of $\mathbb{R}^n$? Same question replacing open sets by compact sets. I know that for closed sets the result is not true.

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    thanks Moreland . each a+B is open and arbitrary union of open sets is open2012-07-08
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    The compact question can be answered by reading http://math.stackexchange.com/questions/80974/closed-sum-of-sets2012-07-08

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