Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
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$\begingroup$ What do you mean by "separable" ? $\endgroup$– Guillaume AubrunCommented May 11, 2020 at 14:31
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$\begingroup$ There's a dense separable subspace, since both $A$ and $C$ are subspace of a Euclidean space $\endgroup$– ABIMCommented May 11, 2020 at 15:36
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1$\begingroup$ Then it's true, because any subset of a separable metric space is also separable. $\endgroup$– Guillaume AubrunCommented May 11, 2020 at 15:47
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$\begingroup$ Maybe @FlabbyTheKatsu had in mind that $A+C$ is simply connected? In this case the answer is negative: take for $A$ the letter $C$ and for $B$ the letter $I$ on the plane. $\endgroup$– Taras BanakhCommented May 12, 2020 at 3:07
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$\begingroup$ Yes, I meant to ask as Taras said. Thank you for the anwser. $\endgroup$– ABIMCommented May 12, 2020 at 8:32
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