Math 445, Fall 2009
Exercise Set #1
Solutions
1. [#10, page 9] Call a subset B of a set A conite if the complement of B in A is nite. If B and
C are conite subsets of A, prove that B C is conite.
Proof. We are given that (A B ) and (A C ) are both nite se
Math 445, Fall 2009
Exercise Set #7
Solutions
1. Let X be a topological space, and suppose that A, B X are compact subsets. Show that A B
is again compact.
Proof. We must show that an open cover of A B has a nite subcover.
Let U = cfw_U | A be an open cov
Math 445, Fall 2009
Exercise Set #6
Solutions
1. [#1, page 93] Prove that the space X = B (N, R) of bounded sequences with the sup norm is
complete. [This is Example 7, page 119 of Appendix 1 in the text.]
The metric is dened by, for x = cfw_an B (N, R)
Math 445, Fall 2009
Exercise Set #5
Solutions
1. [#10, page 78] Let A, B M be subsets of a metric space M . Dene the distance between the
sets to be
D(A, B ) = inf cfw_D(a, b) | a A , b B
a) Suppose that B = cfw_x consists of a single point. Prove that D
Math 445, Fall 2009
Exercise Set #4
Solutions
1. [#1, page 78] Let M be a metric space with metric D. Prove that if cfw_xn | n = 1, 2, . . . M is
a sequence which converges to points x M and y M , then x = y .
Proof. Given > 0 we will show that D(x, y ) <
Math 445, Fall 2009
Exercise Set #3
Solutions
1. [#11, page 70] Let D1 and D2 be metrics on a single space M . Which of the following are metrics
on M : D1 + D2 , maxcfw_D1 , D2 , mincfw_D1 , D2 ?
Proof. In all three examples, the Non-degenerate and Symme
Math 445, Fall 2009
Exercise Set #2
Solutions
1. [#1, page 26] Let A be a countable set and suppose there exists a function f : A B which is
surjective. Prove that B is also countable. (Recall that a set is countable if there is a bijection with
the set o
Math 445, Fall 2009
Exercise Set #1
Solutions
1. [#10, page 9] Call a subset B of a set A conite if the complement of B in A is nite. If B and
C are conite subsets of A, prove that B C is conite.
Proof. We are given that (A B ) and (A C ) are both nite se
Math 445, Fall 2009
Midterm Exam
Solutions
You may use your textbook and class notes, but no other references.
There are 6 problems. Turn in 5 for grading.
1. Let cfw_M, D be a metric space. Let A M be a subset and suppose y A. Show that every
continuous
Math 445, Fall 2009
Midterm Exam
Turn in: October 26
You may use your textbook and classnotes, but no other references.
There are 6 problems. Turn in 5 for grading.
1. Let cfw_M, D be a metric space. Let A M be a subset and suppose y A. Show that every
co
Math 445, Fall 2009
Final Exam
December 9, 2009
1. Let X be a metric space.
a) Dene: U X is open.
b) Prove: X is separable there is a countable basis B for the metric topology on X .
2. Let f : X Y be a map between metric spaces.
a) Dene: f is continuous
Math 445, Fall 2009
Exercise Set #8
Solutions
1. Consider a Hausdor topological space T = cfw_A, T . Consider the collection of open subsets
T = cfw_U A | A U is compact in T cfw_
a) Show that T = cfw_A, T is a topological space. (i.e., that T satises t
Math 445, Fall 2009
Midterm Exam
Solutions
You may use your textbook and class notes, but no other references.
There are 6 problems. Turn in 5 for grading.
1. Let cfw_M, D be a metric space. Let A M be a subset and suppose y A. Show that every
continuous
Math 445, Fall 2009
Exercise Set #2
Solutions
1. [#1, page 26] Let A be a countable set and suppose there exists a function f : A B which is
surjective. Prove that B is also countable. (Recall that a set is countable if there is a bijection with
the set o
Math 445, Fall 2009
Exercise Set #3
Solutions
1. [#11, page 70] Let D1 and D2 be metrics on a single space M . Which of the following are metrics
on M : D1 + D2 , maxcfw_D1 , D2 , mincfw_D1 , D2 ?
Proof. In all three examples, the Non-degenerate and Symme
Math 445, Fall 2009
Exercise Set #4
Solutions
1. [#1, page 78] Let M be a metric space with metric D. Prove that if cfw_xn | n = 1, 2, . . . M is
a sequence which converges to points x M and y M , then x = y .
Proof. Given > 0 we will show that D(x, y ) <
Math 445, Fall 2009
Exercise Set #5
Solutions
1. [#10, page 78] Let A, B M be subsets of a metric space M . Dene the distance between the
sets to be
D(A, B ) = inf cfw_D(a, b) | a A , b B
a) Suppose that B = cfw_x consists of a single point. Prove that D
Math 445, Fall 2009
Exercise Set #6
Solutions
1. [#1, page 93] Prove that the space X = B (N, R) of bounded sequences with the sup norm is
complete. [This is Example 7, page 119 of Appendix 1 in the text.]
The metric is dened by, for x = cfw_an B (N, R)
Math 445, Fall 2009
Exercise Set #7
Solutions
1. Let X be a topological space, and suppose that A, B X are compact subsets. Show that A B
is again compact.
Proof. We must show that an open cover of A B has a nite subcover.
Let U = cfw_U | A be an open cov
Math 445, Fall 2009
Exercise Set #8
Solutions
1. Consider a Hausdor topological space T = cfw_A, T . Consider the collection of open subsets
T = cfw_U A | A U is compact in T cfw_
a) Show that T = cfw_A, T is a topological space. (i.e., that T satises t
Math 445, Fall 2009
Final Exam
December 9, 2009
1. Let X be a metric space.
a) Dene: U X is open.
b) Prove: X is separable there is a countable basis B for the metric topology on X .
2. Let f : X Y be a map between metric spaces.
a) Dene: f is continuous
Math 445, Fall 2009
Final Exam - Solutions
December 9, 2009
1. Let X be a metric space.
a) Dene: U X is open.
Solution: A set U is open if for every x U , there exists > 0 so that the ball B (x, ) U .
b) Prove: X is separable there is a countable basis B
Math 445, Fall 2009
Midterm Exam
Turn in: October 26
You may use your textbook and classnotes, but no other references.
There are 6 problems. Turn in 5 for grading.
1. Let cfw_M, D be a metric space. Let A M be a subset and suppose y A. Show that every
co
Math 445, Fall 2009
Final Exam - Solutions
December 9, 2009
1. Let X be a metric space.
a) Dene: U X is open.
Solution: A set U is open if for every x U , there exists > 0 so that the ball B (x, ) U .
b) Prove: X is separable there is a countable basis B