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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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    https://brainmass.com/math/real-analysis/real-analysis-connected-and-disconnected-sets-27562

    Solution Preview

    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 ...

    Solution Summary

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

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