Can someone give an example of a domain $G$ which has a boundary different from the boundary of its complement $G^c$?
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This is impossible: every set has the same boundary of its complement. – Crostul Dec 17 '16 at 12:04
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The boundary of a subset is always equal to the boundary of the complement. – Leon Sot Dec 17 '16 at 12:04
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This question may be wrong because of a typo in the e-book. This was taken from Silverman, "Introductory complex analysis", Dover, Amazon e-book version, Chapter 2, problem #5. As written there, the question is "Given an example of a domain G such that G and G have different boundaries". For me, this does not make sense, so I guessed that it could be speaking of $G^c$. As per the reference given in Silverman, this question comes originally from Markushevich, "Theory of Functions of a Complex Variable", p. 68, however I don't have this source to verify. – IRO Dec 17 '16 at 14:11
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Actually the print edition of Silverman shows $\overline{G}$, so I suppose it's the complement. – IRO Dec 24 '16 at 00:15