MA-547: MIDTERM Part I. (100 points) 1. (15 points) (a) Suppose and are sets. Define, i.e. and have the same size:

(b) State the Schroder – Bernstein Theorem:

- (45 points) Let be a nonempty set. (a) Define a metric d in:

(b) Suppose and Define:

(c) Suppose . Define an interior point of . Define and A is open:

(d) Given any point, define is a boundary point of. Define the closure and A is closed:

A B A ↔ B A B

X X

x0 ∈ X ε > 0 Bε(x0)

x0 ∈ A ⊆ X x0 A Ao

x0 ∈ X x0 A Ā

(e) Let. Define is compact, limit point compact and sequentially compact:

- (20 points) State the Archimedean principle of the real line. Give a necessary and sufficient condition involving a sequence for the Archimedean property to hold:
- (20 points) suppose (a) Define: M is an upper bound of:

(b) Define: a least upper bound of:

(c) What property of the real line is used to prove the Least Upper Bound property?

A ⊆ X A

A ⊆ ℝ A

A

Part II (100 points) 5. (30 points) Each of the following 3 statements is false. Give a counterexample for each: (a) A set (where is a metric space) is closed, if it is not open:

(b) A subset is compact if it is closed and bounded:

(c) If two sets and are both not finite, then

A ⊆ X (X, d )

A ⊆ X

A B A ↔ B

- (30 points) suppose is a sequence in the metric space.

Prove: If, then is compact.

(xn)n≥1 (X, d ) xn → L ∈ X {xn |n ≥ 1} ∪ {L}

- (30 points) Consider the set depicted as below:

i.e.

Find Determine if is open or closed.

A ⊆ ℝ2

A = {(x, y) | 1 2

< x2 + y2 ≤ 3 2

} ∪ {(x,0) | − ∞ < x ≤ − 3 2

}

Ao, ∂(A), Ad, Ā . A

x

y

3 2

1 2

- (10 points) State and outline the proof of the Heine-Borel theorem in. (ℝ, | . | )