Download HOMEWORK 7 Due on Nov. 2 in class. Exercise 1. Consider the set

HOMEWORK 7
Due on Nov. 2 in class.
n
Exercise 1. Consider the set B = {(−1)n n+1
|n ∈ N}.
(1) Find the limit points of B.
(2) Is B a closed set?
(3) Is B an open set?
(4) Does B contain any isolated points?
(5) Find the closure B of B.
Exercise 2. For each of the following sets determine if they are open, closed, or
neither. If a set is not open find a point in the set that is not an interior point. If
a set is not closed find a limit point that is in the set.
(1) Q
(2) N
(3) {x ∈ R|x > 0}
(4) {x ∈ R|1 ≤ |x| < 2}
Exercise 3. Let A be a set and a ∈ A. Prove that a is an isolated point of A if
and only if there is some -neighborhood B (a) such that B (a) ∩ A = {a}.
Exercise 4. Let O be an open set, x ∈ O and let (xn ) be a converging sequence
with lim xm = x. Prove that all but finitely many terms of (xn ) must be in O.
Exercise 5. Prove the following topological criterion of convergence: A sequence
(xn ) converges to x if and only if any open set O with x ∈ O satisfies that all but
finitely many terms are in O.
Exercise 6. Let A, B denote two sets of real numbers and let A0 and B 0 the set of
their limit points. Similarly we denote by (A ∪ B)0 and (A ∩ B)0 the limit points of
their union and intersection respectively.
(1) Show that (A ∪ B)0 ⊆ A0 ∪ B 0 (do we also have inclusion in the other
direction?)
(2) Show that (A ∩ B)0 ⊆ A0 ∩ B 0 (do we also have inclusion in the other
direction?)
(3) Prove that A ∪ B = A ∪ B
(4) Does the result of (2) extend to an infinite unions? (Prove or find a counter
example).
(5) Is it true that A ∩ B = A ∩ B? (Prove or find a counter example).
Exercise 7. Assume that A is a bounded set and let s = sup A. Show that s ∈ A.
Exercise 8. Let A be a set and a ∈ A. Show that a is an interior point of A if
and only if A is not a limit point of the complement Ac .
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