Instructor
& Office Hours
Lectures
Textbook |
Ron Freiwald, Cupples
I, room 201
Tuesday 9:30-10:30, Thursday 10:30-11:30, Friday 9:30-10:30 (I will email the class in
advance, if possible, whenever I need to make a change in office hours.)
TuTh 1-2:30 in Cupples I, room 215. For all lectures, you should
be up to date on reading from the textbook, or even a bit
ahead. Make notes to yourself
about anything you don't understand so you can
raise questions.
Part II of the lecture notes for the course. It has been photocopied and is available
at Hi/Tec
Copy Center (at the intersection of Big Bend and
Forest Park Parkway). The price (about $14) is set by
Hi/Tec to cover the cost of copying and binding + whatever markup they
add for selling the notes; nothing goes to me.
Math
4181 continues the material from Math 4171, Fall 2013. In the fall course, we
covered basic set theory and cardinal arithmetic, metric spaces, an
introduction to topological spaces, complete metric spaces (including
completions, the Contraction Mapping Theorem, the Baire Category
Theorem), total boundedness, and compact spaces (with particular
emphasis on compact metric spaces). Math 417 ended with some
material about connected spaces (Notes,
Chapter 5, Sections 1-2)
Math
4181 moves on to product and quotient spaces, more interesting separation
axioms (such as complete regularity and normality), and some of the "big"
classical theorems of general topology (such as Urysohn's Metrization
Theorem, Urysohn's Lemma, and the Tietze Extension Theorem). Then we
will spend some additional time on set theory (ordered
sets and ordinal numbers) so that we can learn to
do transfinite induction and use Zorn's Lemma. A high point
of that
material comes when we give what is (by then) very simple
proof of
the Tychonoff Product Theorem. The material for the remainder of
the course will depend on how much time remains
Details about
the course are given below. Homework, exams and solutions will be
posted here in the syllabus.
|
Homework
|
There
will be 6-8 homework sets
during the semester. Homework assignments will be posted on this web page. Usually, an assignment will be due
in
class at the third
lecture after the assignment is posted online: for example, an
assignment posted on Tuesday is due in class a week from the following
Thursday.
Some of
the homework problems
are fairly routine, but others are more challenging. Usually, you can't put them off until the night before they're due.
Most
homework problems will be
read by a grader. However, on several homework sets during the
semester, I will select a problem (after homework is turned
in) that I will grade myself. Your total accumulated score on
the homework problems that I grade will
count
as "Exam 4." Your accumulated score on the
remaining
homework
problems will count as your homework score.
|
| Other Materials |
A nice little proof about the (real) transcendental numbers
A
regular space that isn't completely regular, and an infinite regular
space on which every continuous real-valued function is constant
Fundamental Groups and Covering Spaces (Elon Lages Lima)
A nice book to have. The author writes "the subjects
discussed...are well suited as an introduction to algebraic topology
for their elementary character, for exhibiting in a clear way the use
of algebraic invariants in topological problems, and because of their
immediate applications to other areas of mathematics such as real
analysis, complex variables, differential geometry...an introductory
book, with no claims of becoming a reference work."
A classic result: every infinite poset contains either an infinite chain or an infinite antichain.
Try to prove it before reading the proof posted here.
|
Exams |
There will be the equivalent of four
exams in the course:
Exam
1
In Class Tuesday, February 25 |
| Exam 1 Solutions
Exam Scores
|
Exam
2
Take Home: given out Thursday April 3, due in
class Tuesday April 8. |
| Exam 2 Solutions
Exam Scores
|
Exam
3 (Final Exam):
Tuesday, May 6, 2014, 1:00PM - 3:00PM |
|
|
| "Exam
4" |
| See
description in the Homework section |
The
dates for Exams 1 and 2) could be moved slightly if a substantial
majority of the class
wants the change. But if there's an important reason for a change,
then I'd
like to decide that within about a week so that some
students aren't upset by making a change closer to the exam
date.
The
"in-class" exam and the final
will be "short-answer" -- such things as definitions, statements of theorems, providing
examples or counterexamples,
and true/false questions.
The
“take-home" exam will consist
of more substantial questions,
analogous
to homework problems. On the take home exam, there will usually be some options for you: "answer m of the following n questions
|
Basis for Grading |
The four exam scores and the
homework
score will each count 20% of your
grade. However, homework
assignments are an essential part of the course. If
you neglect the homework, your course grade may be dramatically lowered
(regardless
of test scores) at my discretion. I will not have a
scale
for converting numeric scores into letter grades until the end of the
semester.
|
Academic Integrity |
Exams:
During all examinations, both "in class" and "take-home," no
discussion or consultation of any kind with any other person or
sources, whether in person, electronically, or via the internet, is
allowed. The only exception is for questions of clarification that you
can request from me. For
the take-home exam, you may consult class notes, the texbook, or any
other
references for ideas—but any such references must be explicitly
documented in your solutions and solutions must be
completely written up in your own words.
You
should avoid trying to "find" solutions to problems
elsewhere: that just undercuts your learning.
Any solutions taken from other sources without good documentation will
result
in a grade of 0 for the test or assignment and might
be cause for
a referral to the Academic Integrity Committee. If you have
questions about
what is appropriate, please ask me.
Homework:
Students are encouraged to discuss
homework assignments with each
other;
you should share questions and ideas. It is a powerful way to learn the
concepts. Each student, however, must write up the homework
solutions
independently
in his/her own words and notation. One good way to
avoid
"borrowing
too much" from discussions with others is to talk together but not take
away any
written notes from the conversation. Suspicious
similarities
between solution sets may be noted by the grader and may result in a
grade
of 0 for the homework.
|
History
and Biography |
These
web pages may be
give some interesting historical sidelights on
the
material.
The
MacTutor History of Mathematics Archive
George
Cantor
Bertrand
Russell
Kazimierz
Kuratowski
Kurt
Godel
Paul
Cohen
Felix
Hausdorff
Robert
Sorgenfrey
Ernst
Lindelof
Augustin-Louis
Cauchy
Rene-Louis
Baire
Pavel
Alexandroff
The
Beginnings of Set Theory
The
Axiom of Choice
Topology
Enters Mathematics
The
"Kuratowski 14 Problem"
|
Textbook & References |
In addition three fairly standard reference texts are:
1.
Munkres, James
Topology QA611
.M82 2000
2. Willard, Stephen
General Topology QA611
W55 1970
3. Kaplansky, Irving
Set Theory and Metric Spaces QA248 K36
1977
Munkres
and Willard are standard General Topology texts; Kaplansky is
a
nicely written little book; it is a "softer" introduction to set theory
and metric
spaces, with not much material about topological spaces in general.
Munkres and Willard may be of
more interest next semester. These
three books should be on
two day reserve at Olin Library.
A few other books that might
be useful. They are available in Olin Library but not on reserve:
4. Eisenberg, Murray Topology QA611
E53
5.
Kahn, Donald
Topology:
An Introduction to the Point-Set and Algebraic Areas
QA611 K32
6. Simmons,
George
Introduction
to Topology and Modern Analysis QA611
S49
Each of
these has different emphases
and perspective,
and none follows the material as I'll present it.
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