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Real Analysis : Connected and Disconnected Sets

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A- Find an example of a disconnected set whose closure is connected.

b- If A is connected ,is A closure necessarily connected? If A is perfect is A closure necessarily perfect?

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Solution Summary

Connected and Disconnected Sets and Closure are investigated. The solution is detailed.

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a. Q is the set of rational numbers. It is obviously a disconnected set because we can find an irrational number a, such that Q is contained in the union of the two disjoint open sets (-inf,a) and (a,inf). Because Q is dense in R, so the closure of Q is R, which is connected.

b.
(1) Yes. If A is connected, the A ...

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