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If a set $X$ is closed then $\overline{X} = X$ and if it is open then $X^o = X$, so does this mean that for a subspace $X$ of a topological space which is both open and closed (for example in a partition) the boundary given by $\overline{X} \backslash X^o$ is just the empty set?

Conversely does this mean that all sets, in which the boundary is the empty set, are clopen sets?

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Yup! Your reasoning is correct. You can write up your solution as an answer (and accept it, if you'd like). This is explicitly encouraged by the SE network of sites; see here and here. – Zev Chonoles May 19 '12 at 18:23

This CW answer intends to remove this question from the Unanswered queue.

The reasoning is correct, as Zev Chonoles points out.

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