Let $A\subset \mathbb R^n$ be compact and connected, where $n\ge2$. Is the boundary $\partial A$ connected?
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3If $A$ is not connected, the answer is clearly no. For example, take $A={ a,b }$ a set with two points. – Crostul Jul 17 '15 at 13:20
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No, but see here for the question, whether the boundary of every compact convex set in $ℝ^n$ , $n>1$ is connected. – Dietrich Burde Jul 17 '15 at 13:25
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No. Counter-example: $A = [0, 1] \subset \Bbb R $.
No. Counter-example: $A = \{(x,y) \in \Bbb R^2 \mid 1 \leq x^2 + y^2 \leq 2\} \subset \Bbb R^2$.
Ivo Terek
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