Is it possible to find an example where the Minkowski sum of two open sets is not open? (If someone could think of one, could they possibly also suggest how they came up with the example? Perhaps there is a "common counterexamples" list that people usually use to approach this sort of question?)
Is there a pair of open sets whose Minkowski sum is not open?
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analysis
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0@DavideGiraudo: Thanks! What exactly does it mean to take the union of 2 open sets? How does that take care of the summing bit? Sorry this might be a stupid question :S – 2011-11-11
1 Answers
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No. In fact, $A+B$ is open even if only one of the sets (say, $A$) is open. To see this, look at any point $a+b \in A+B$. Since $A$ is open, there is an open ball $S\subset A$ with $a\in S$. Then $S + b = \{x+b\ \vert\ x\in S\}$ is again an open ball, is a subset of $A+B$, and contains $a+b$. Hence every point in $A+B$ is an interior point and $A+B$ is open.