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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](http://meta.stackexchange.com/questions/12513/should-i-not-answer-my-own-questions) and [here](http://blog.stackoverflow.com/2011/07/its-ok-to-ask-and-answer-your-own-questions/).2012-05-19

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


The reasoning is correct, as Zev Chonoles points out.