Math task 10 hrs

Name: Math 325 GOLD Assignment # 6

Assigned: Friday, 2020.04.03 Due: Friday, 2020.04.10

Gold Homework: This will be homework you submit each week and represents YOUR OWN written work (no outside resources or collaboration allowed). Gold problems must be carefully written with excellent grammar, correct usage of mathematical symbols, and of course, with correct mathematics. Each solution you submit for gold homework should represent your best work. The following rubric indicates how problems on the gold homework will be scored.

Score Criteria 4 This is correct and well-written mathematics. 3 This is good work, yet there are some mathematical or writing errors that need addressing. 2 There is some good intuition here, but there is at least one serious flaw. 1 The grader didn’t understand this, but sees that you have worked on it; come in for help! 0 You probably haven’t worked on this problem enough or you didn’t submit any work.

Definition [Closure]. Let A ⊆ Rn be a set. The closure of A, denoted A can be defined in three different, but equivalent, ways; namely:

(i) A is the set of all limit points of A.

(ii) A is smallest closed set containing A; this means that if there is another closed set F such that A ⊆ F, then A ⊆ F .

(iii) A is the intersection of all closed sets containing A.

Definition [Interior]. Let A ⊆ Rn be a set. The set Å, called the interior of A is the set of all points x ∈ A such that there exists some � > 0 such that the neighborhood V�(x) := {y ∈ R : |x−y| < �} is contained in A. Definition [Boundary]. Let A ⊆ Rn be a set. The set ∂A, called the boundary of A. It is defined by ∂A = A \ Å; that is, the boundary is the set of all points in the closure that are not in the interior.

1. Prove that the three definitions in the definition of closure (above) are equivalent. That is, show that (i) ⇔ (ii) ⇔ (iii). (For example, you could show (i) ⇔ (ii), and then show (ii) ⇔ (iii), or you could show that (i) ⇒ (ii) ⇒ (iii) ⇒ (i), or some other order.)

2. Show that ∂A = A∩Ac. [Hint: Use the usual way to show set equality; namely, choose an element x in the left, and show it is in the right, then choose and element x in the right, and show it is in the left.]

3. Let F1,F2,F3, . . . be bounded, non-empty closed sets in Rn. Suppose that they are decreasing with respect to set inclusion, that is, suppose

F1 ⊇ F2 ⊇ F3 ⊇ ··· .

Prove that their intersection, F := ∩∞n=1Fn is closed, bounded, and non-empty. [Hint: It is essentially one line to show that F is closed and bounded. The challenge is in showing that F is non-empty. Use the Bolazno-Weierstrauss Theorem for this.]

4. Let A ⊂ Rn be a non-empty compact set, ans suppose that B is an open set and A ⊂ B. Consider the “�-dilation” A� of A given by

A� := {y ∈ Rn : ‖x−y‖ < � for some x ∈ A} .

Show that there is an � > 0 such that A� ⊆ B. [Hint: Note that for each x ∈ A, there is an r > 0 such that neighborhood Vr(x) = {y ∈ Rn : ‖x−y‖ < r} ⊆ B. Note that the set of all these neighborhoods cover A. Now, use the definition of compactness.]