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    Closed subset of Rn

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    Let A and B be closed subset of Rn with A ∩ B = Ø.
    a. Prove that ∀u ∈ A, ∃_ > 0 such that N_ (u) ∩ B = Ø
    b. Prove that there is an open set OA satisfying OA ⊃ A and OA ∩ B = Ø
    c. Prove or find a counterexample: There are open sets OA and OB satisfying OA ⊃ A,
    OB ⊃ B = B and OA ∩ OB = Ø

    © BrainMass Inc. brainmass.com October 9, 2019, 3:51 pm ad1c9bdddf
    https://brainmass.com/math/synthetic-geometry/closed-subset-rn-15064

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    Let A and B be closed subset of Rn with A  B = Ø.
    a. Prove that u  A, ε > 0 such that Nε (u)  B = Ø
    Proof. We prove it by contradiction. Suppose the statement is not true. Then we know that there exists a u  A, for all such that , i.e., . So when n=1,2,..., we get a sequence such that

    which means that { } converges to u. Since B ...

    Solution Summary

    This is three proofs about a closed subset of Rn, including two about open sets.

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