# Closed subset of Rn

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

https://brainmass.com/math/synthetic-geometry/closed-subset-rn-15064

#### Solution Preview

Please see the attachment.

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.