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*show*that if [itex]X[/itex] is a non-empty set and [itex]A\subseteq X[/itex] then [itex]A[/itex] and [itex]A^c[/itex] have the same boundary?

The definition is [itex]x\in \partial A \iff [/itex] there exists [itex]r>0[/itex] such that the open ball [itex]B(x,r)[/itex] intersects both [itex]A[/itex] and [itex]A^c[/itex]

but this is precisely the statement that [itex]x\in \partial A^c[/itex]!