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 .
L
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
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 lea
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; th
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 metri
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(
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
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
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 ind
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
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 s