# 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.

Co-Finite Topology

2. Let X be given the co-finite topology. Suppose that X is a infinite set. Is X Hausdorff? Should compact subsets of X be closed? What are compact subsets of X?

3. Let (X,T) be a co-countable topological space. Show that X is connected if it is uncountable. In fact, show that every uncountable subspace of X is connected.

4. Let X be a set and F is separating collection of functions f: X --> Y_f, each from X into a topological space Y_f. Prove that X with the weak topology by F is Hausdorff.

5. Explain how we can think of the unit sphere (a subspace of IR^3 with its usual topology) as a subset of the Hilbert cube.

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