Do sets in a discrete topological space have boundaries?
Every subset of a discrete topological space has empty boundary. To see this recall the definition of the boundary of a set A as the intersection of the closure of A, and the closure of the complement of A, in a discrete space all subsets are both open and closed, so the closure of A is simply A and the closure of A^c is A^c, their intersection is empty. This should also fit our intuition of what discrete spaces are, spaces where all points are isolated, such as Z as a subspace of R (in the standard topology).
An addition to what Mandark said, it can be shown that a set has an empty boundary if and only if it is both open and closed.
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