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A set A is called clopen if and only if it is both open and closed.

a) Prove that every Euclidean space has at least two clopen sets.

b) Prove that a proper subset E of Rn is connected if and only if it contains exactly two relatively clopen sets.

c) Prove that every nonempty proper subset of Rn has a nonempty boundary.

a) Prove that every Euclidean space has at least two clopen sets.

b) Prove that a proper subset E of Rn is connected if and only if it contains exactly two relatively clopen sets.

c) Prove that every nonempty proper subset of Rn has a nonempty boundary.

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