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?