Metric Spaces
MATH 3260/4260 (Winter 2012)
The Tietze Extension Theorem
Lemma 1 Let (X, d) be a metric space and K a nonempty closed subset of X . Suppose
f : K R is continuous, nonconstant and bounded. Let
a = inf f (K ) and b = sup f (K )
and dene M = b
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 7
Due: Tuesday, March 6 - There are 2 pages in this problem set.
1. I Suppose that (X, d) is a compact metric space. Let Y = X 2 = X X and dene
: Y Y R by
(x, z ) = max cfw_d (x1 , z1 ) , d (x2 , z2
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 6
Due: Tuesday, February 28 - There are 3 pages in this problem set.
1. Suppose that (X, d) is a metric space and (xn ) a sequence in X . Given that
(xn ) is Cauchy; and,
a subsequence (xnk ) of (xn
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 5
Due: Thursday, February 16 - at the beginning of class. There are three pages in this
assignment.
1. Suppose that (X, d) is a metric space. Prove that the function : X X R dened
by
d (x, y )
(x, y
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 4
Midterm Exam: Thursday, Feb 2
Due: Tuesday, January 31 - at the beginning of class.
Discussion: (Problems begin on page 2.)
Recall the theorem from class.
Theorem 1 Let (X, d) be a metric space. The
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 3
Due: Thursday, January 26 - at the beginning of class.
Problems
1. Pages 98 102: 5 6 7 9 (Look at these questions. Im not asking for any of them to be
submitted because the supplied hints are essent
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 2
Due: Thursday, January 19 at the beginning of class.
All analysts spend half their time hunting through the literature for inequalities which they want to use and cannot prove.
Attributed to H. Bohr
Metric Spaces
MATH 3260/4260 (Winter 2012)
Problem Set 1
Due: Thursday, January 12 at the beginning of class.
Solve all of the problems. Submit only those problems marked with a I . One or more Fs
indicate more dicult problems. 3260 students may solve the
Metric Spaces
MATH 3260/4260 (Winter 2012)
Notes on the Baire Category Theorem
Introduction
Let (X, d) be a metric space.
Denition 1 A X is
a G set in X if A =
T
Gk where each Gk is open in X .
k=1
an F set in X if A =
S
Fk where each Fk is closed in X
Metric Spaces
MATH 3260 (Winter 2012)
Midterm Exam Preparation
Midterm Exam: Thursday, February 2
Part 1: This part is closed book. It is worth 30 points.
This part consists of two pages. The rst page is denitions from Chapters 1 and 2 of
the text. The se