MATH 4450 HOMEWORK SET 1, SOLUTIONS
Problem 1 (2.8): Suppose X and Y are sets, each of which has at least two elements. Show that X Y
contains a subset that is not of the form A B for any A X, B Y .
Let X have distinct elements x1 , x2 , and let Y have di
MATH 4450 HOMEWORK SET 4, SOLUTIONS
Problem 16 (13.6): Let X, d and Y, e have the standard discrete metrics. What are the values of
( x, y , u, v ), for distinct points in X Y , for the three product metrics given above? Which of theseif
anygives you the
MATH 4450 HOMEWORK SET 2, SOLUTIONS
Problem 6 (6.2): Show no X R can have two distinct least upper bounds.
Suppose both a and b are least upper bounds for X R. Since both are upper bounds and
a is least such, we know a b. But b is least such too, so b a.
MATH 4450 HOMEWORK SET 6, SOLUTIONS
Problem 26 (20.6): Does every self-mapping on R have a xed point? What about the unit circle S1 ?
The mapping x x + 1 is an example of a xed point free map on R; the mapping z iz
(complex number notation) rotates the un
MATH 4450 HOMEWORK SET 5, SOLUTIONS
Problem 21 (17.5): Let X, d and Y, e be metric spaces, with X and Y disjoint, and both d and e bounded
by 1. Show that the function dened on X Y as above is a metric, whose open sets consist of all U X Y
such that U X T
MATH 4450, EXAM 2 (solutions), 09 MARCH, 2011
(Each of the following six problems is worth 10 points.)
(1) Let A = [0, 1) (1, 2) cfw_3, considered as a subset of the euclidean real line. Find:
(a) cl(A), the closure of A.
[0, 2] cfw_3
(b) int(A), the inte
MATH 4450, EXAM 3 (solutions), 13 APRIL, 2011
(Each of the following six problems is worth 10 points.)
(1) Let X be a T0 space. For x, y X, dene x y just in case x cl(cfw_y).
(a) Show that if x y and y x, then x = y.
Another way of saying that x cl(cfw_y)
MATH 4450, EXAM 1 (solutions), 09 FEBRUARY, 2011
(Each of the following six problems is worth 10 points.)
(1) Given a set X, a moiety of X is a subset of X that is equinumerous with its complement in X.
(a) Give an example of a subset of N = cfw_0, 1, 2,
MATH 4450 TAKE-HOME FINAL EXAM (solutions), DUE 5:00 PM, 06
MAY, 2011
(Do ve of the following eight problems; each is worth 18 points, with all parts of any one
problem having equal value. Clearly indicate the problems you want to be graded. You
may consu
MATH 4450 HOMEWORK SET 7, SOLUTIONS
Problem 31 (23.4): Prove that if X is compact, Y is Hausdor, and f : X Y is continuous, then f is a
closed mapping.
Suppose A is closed in X. We need to show that f [A] is closed in Y . Since X is compact,
we know from
MATH 4450 HOMEWORK SET 3, SOLUTIONS
Problem 11 (9.4): If x is the limit of a sequence an of distinct points in X, d , show that x is also an
accumulation point of the trace of the sequence. Does the same necessarily hold for all sequences?
Let A = cfw_an