MATH 3T03
SOLUTIONS TO ASSIGNMENT 4
Each problem is worth 10 points.
Problem 10.15. Let f : X Y be continuous and surjective, and let
A X be dense. Show that f (A) is dense.
Solution. Let V Y be open; we need to show that f (A) V = . Since
A f 1 (f (A), t
Topology - Homework 7
Due Wednesday, March 9
0. Read Sections 26 and 27.
1. Determine which of the following subspaces of R2 are connected, and
which are path-connected. Carefully explain your answers.
(i) ( x, y) R2 | |y| = cos( x )
(ii) R2 \Q2
(i
Topology - Homework 2
Due Wednesday, January 20
0. Read Sections 14-17.
1. For this problem, we will say that a set S is countable if it is finite, or in
bijection with the set N of natural numbers.
(a) Suppose cfw_Si i I is a (possibly infinite) collecti
Topology - Homework 3
Due Wednesday, January 27
0. Read Sections 18 and 19.
1. Consider the half-line X := [0, ). In this problem we will be studying
the Cartesian product X 2 .
(a) Let Q be the dictionary order on X 2 induced from the standard inequality
Topology - Homework Sets 8 and 9
Due Tuesday, April 5
1. Determine which of the following subsets of R2 are compact.
(i) ( x, y) R2 | ( x/3)2 + (y/5)2 = 1
(ii) ( x, y) R2 | ( x/3)2 (y/5)2 = 1
(iii) ( x, y) R2 | x, y Q, and | x | + |y| 1
2. Define th
Topology - Homework *
Due Tuesday, April 5
Your homework is to write a final exam for this course, as well as an answer
sheet. Your exam should have 3-5 questions.
How to choose your exam problems: Choose your exam problems so that your
exam could be comp
Topology - Homework 5
Due Wednesday, February 24
0. Read Sections 21 and 23.
1. Determine which of the following functions : R R R are metrics.
Justify your answers.
(i) ( x, y) = 2| x + y|
(ii) ( x, y) = (max( x, y) min( x, y)2
(iii) ( x, y) = | x y|2
(i
Topology - Homework 1
Due Wednesday, January 13
0. Read Section 3-7 and 12-13.
1. Let X = cfw_ a, b, c, d, e (a set with five elements). Determine whether or
not each of the following collections of subsets of X is a topology on X. Justify
your answers.
(
MATH 3T03
Assignment 5
Due Tuesday, April 7, 2015
1. Show that sequential compactness (see Munkres 28, p.179) is a topological property,
that is, if X is sequentially compact and h : X Y is a homeomorphism then Y is
sequentially compact.
2. Which of the f
MATH 3T03
SOLUTIONS TO ASSIGNMENT 5
Each problem is worth 10 points.
Problem 12.11. Let A be a subset of a topological space X. Prove that A
is disconnected if and only if A has two nonempty subsets B and C such that
A = B C, B clX C = and C clX B = .
Sol
MATH 3T03
SOLUTIONS TO ASSIGNMENT 6
Each problem is worth 10 points.
Proposition 17.T. Let A Rn . Then A is compact i A is closed and
bounded.
Solution. By a Proposition from the slides, we may replace compact by
complete and totally bounded; so its enoug
MATH 3T03
Assignment 1
Due Tuesday, January 20, 2015
1. Let X = cfw_a, b, c (a set with three elements). Find all the partitions of X. For each
partition, display the corresponding equivalence relation as a subset of X X.
2. Recall that the power set of a
MATH 3T03
SOLUTIONS TO ASSIGNMENT 3
Each problem is worth 10 points.
Problem 6.12. Is int(A B) = int A int B? Is int(A B) = int A int B?
Solution. The rst equality holds: since int(A B) is an open set contained in
both A and B, and since int A is the larg
MATH 3T03
Assignment 2
Due Tuesday, February 3, 2015
1. Let X = cfw_a, b, c, d, e (a set with ve elements). Determine whether or not each of the
following collections of subsets of X is a topology on X. Justify your answers.
(i) T1 = cfw_, X, cfw_a, cfw_a
MATHEMATICS 3T03 TEST 2
Day Class Dr. A. Nicas
Duration of Examination: 50 minutes
McMaster University Midterm Examination March 17, 2015
NAME: Solg'gns STUDENT No.:
THIS TEST PAPER INCLUDES 6 PAGES AND 4 QUESTIONS. YOU ARE RESPON-
SIBLE FOR ENSURING THAT
MATH 3T03
Assignment 3
Due Thursday, February 26, 2015
1. Prove that a map f : X Y between topological spaces is continuous if and only if
f 1 (Int(A) Int (f 1 (A) for every A Y . (Recall that Int(S) denotes the interior of
a set S.)
2. Let f : X Y be a c
MATH 3T03
SOLUTIONS TO ASSIGNMENT 2
Problem 3.8. Let X be a set and be a collection of subsets of X. Prove
that is a subbase for a topology on X if and only if X = W W .
Solution. Let be the collection of all nite intersections of sets in ; we need
to sho
MATH 3T03
Assignment 4
Due Tuesday, March 24, 2015
1. Let (X, d) be a metric space, A X, and x a limit point of A (where X has the
metric topology). Show there exists a sequence, (xn ), of distinct points in A cfw_x such
that (xn ) converges to x. Note th
MATHEMATICS 3T03 TEST 1
Day Class Dr. A. Nicas
Duration of Examination: 50 minutes
McMaster University Midterm Examination February 10, 2015
NAME: Samba; STUDENT No.:
THIS TEST PAPER INCLUDES 6 PAGES AND 3 QUESTIONS. YOU ARE RESPON-
SIBLE FOR ENSURING THA
Topology - Homework 0
Due Wednesday, January 6
Read Sections 1 and 2, and hand in solutions to the following problems.
Section 1, Problem 2: Choose any 3 problems from (a)-(q) listed in the
statement of this problem.
Section 1, Problem 8
Section 1, Pro