Turn in: 1.4, 1.6, 1.14, 1.15, 1.16, 1.21
Do NOT turn in (but make sure that you know how to do them!):
1.2, 1.3, 1.9, 1.11, 1.19, 1.22
Prove that the set F(p), with addition and multiplication defined mod p, is a field if and only if p is a prime.
Due date: Monday, October 8
Turn in: 1.25, 1.26, 1.32, 1.34, 2.7, 2.12, 2.13, 2.15, 2.20
Do NOT turn in : 1.23, 1.24, 1.27, 1.29, 2.5, 2.6
Due date: Wednesday, October 17
Turn in: 3.4, 3.10, 3.11, 3.13,
3.29, 3.31, 3.32
(do 3.10-13 in general metric spaces, not just R^n)
Prove that the
distance function d(f,g) =
sup {|f(x)-g(x)|, x in X}
is a metric on the
space of bounded functions f : X -> R. Show
that the set {f, |f(x)|<1 for all x in X} is open if and only if X is finite.
Prove that the Cantor middle third set is closed and uncountable.
Do NOT turn in: 3.2, 3.3
Due date: Friday, October 26
Turn in: 3.5, 3.6,
3.24, 3.25, 3.47, 3.48,
3.52
Prove that the Cantor middle third set is perfect (see exercise 3.25).
Due date: Wednesday, November 7
Turn in: 3.16, 3.33, 3.34, 3.39, 3.40, 3.41, 3.42
Do NOT turn in: Prove that a metric space in which every infinite subset has a limit point is compact:
First prove that such a space is
separable (see exercise 3.32; to prove this, show that
for each n there are finitely many points x(j)=x(j,n)
such that every point is within distance
1/n from one of these x(j,n)). Then use 3.34.
Due date: Friday, November 16
Turn in: 4.2, 4.3, 4.10, 4.11a,b,c, 4.14, 4.15
Do NOT turn in: 4.1, 4.7, 4.8
Due date: Wednesday, November 28
Turn in: 4.16,
4.18, 4.23, 4.25, 4.30, 4.34, 4.38, 4.39
Due date: Wednesday,
December 5