# Closed subset of Rn

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Let A and B be closed subset of Rn with A ∩ B = Ã˜.

a. Prove that ∀u ∈ A, ∃_ > 0 such that N_ (u) ∩ B = Ã˜

b. Prove that there is an open set OA satisfying OA ⊃ A and OA ∩ B = Ã˜

c. Prove or find a counterexample: There are open sets OA and OB satisfying OA ⊃ A,

OB ⊃ B = B and OA ∩ OB = Ã˜

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#### Solution Preview

Please see the attachment.

Let A and B be closed subset of Rn with A ïƒ‡ B = Ã˜.

a. Prove that ï€¢u ïƒŽ A, ï€¤Îµ > 0 such that NÎµ (u) ïƒ‡ B = Ã˜

Proof. We prove it by contradiction. Suppose the statement is not true. Then we know that there exists a u ïƒŽ 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.