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# Closed subset of Rn

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Let A and B be closed subset of Rn with A &#8745; B = Ø.
a. Prove that &#8704;u &#8712; A, &#8707;_ > 0 such that N_ (u) &#8745; B = Ø
b. Prove that there is an open set OA satisfying OA &#8835; A and OA &#8745; B = Ø
c. Prove or find a counterexample: There are open sets OA and OB satisfying OA &#8835; A,
OB &#8835; B = B and OA &#8745; OB = Ø

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#### Solution Preview

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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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