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A point x0 is a boundary point of S if both U ∩ S ≠ ∅...

A point x0 is a boundary point of S if both U ∩ S ≠ ∅ and U ∩ S^c ≠ ∅ for every neighborhood U of x0. The set of all boundary points of S is denoted by ∂S. Prove that a set S is open iff ∂S ∩ S = ∅.

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Replace A by S.

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