5.3. Connected and Disconnected Sets

Proposition 5.3.3: Connected Sets in R are Intervals

If S is any connected subset of R then S must be some interval.


If S is not an interval, then there exists a, b S and a point t between a and b such that t is not in S. Then define the two sets Then U S # 0 (because it contains { a }) and V S # 0 (because it contains { b }), and clearly (U S) (V S) = 0. Finally, because t is not contained in S, we know that (U S) (V S) = S. Hence, we have found the required sets U and V to disconnect S. So, we have proved that if a set is not an interval it is disconnected. That is equivalent to saying that if it is connected, it must be an interval.

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