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, | . | )