Assignment 3, Math 220
Due: Wednesday, January 30th, 2002
1 Classify each of the following sets as open, closed, neither or both.
a: n12 : n ∈ N
b: Z
c: R \ Q
d: {x : |x − π| < 1/2}
e: {sin x : x > 0}
2 Let A be a nonempty open subset of R and let Q be the set of
rationals. Prove that the intersection of A and Q is not the empty set.
3 Find the set of accumulation points of each of the following sets:
a: (−1)k (1 − 1/k) : k ∈ Z and k 6= 0
b: Z
c: R \ Q
d: {x ∈ R : 4k < x < 4k + 1 for some integer k}
e: {x ∈ R : 1/(4n + 1) < x < 1/4n for some n ∈ N}
4 Let B = [0, ∞). Give an example of an open cover of B that has
no finite subcover.
5 If A and B are nonempty sets, we define the distance between A
and B to be
d(A, B) = inf |a − b| .
a∈A,b∈B
Prove that if d(A, B) = 0, where A is compact and B is closed, then
A ∩ B 6= ∅. Give an example to show that if A and B are closed, it is
possible to have d(A, B) = 0 with A ∩ B = ∅.
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