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Closed subset of Rn

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Please see the attached file for full problem description.

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

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Solution Summary

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

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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 provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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