Question: In any metric space X, the empty set 0 and the full space X are both open and closed. A metric space is connected if

In any metric space X, the empty set 0 and the full space X are both open and closed.
A metric space is connected if it cannot be represented as the union of two disjoint open sets. In a connected space the only sets that are both open and closed are X and ∅. This is case for R, which is connected. Also the product of connected spaces is connected. Hence Rn and ∅ are the only sets in Rn that are both open and closed.

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