Math
418 continues the material from Math 417, fall 2009. In Math 417 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
418 begin with a quick review of the highlights of Sections
5.1-5.2 and then continues the discussion on connectedness for one or
two additional lectures. The course then
moves on to product and quotient spaces, more interesting "separation
axioms" (such as complete regularity and normality), and some "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 is remains
Details about
the course are given below. Homework, exams and solutions will be
posted here is the syllabus.
|
Note: Final Exam is scheduled for Tuesday, May 11,
1-3 p.m. in the usual classroom: Cupples I, 215
| Instructor | Ron
Freiwald | | My Office | Cupples I, room 203A | | My Office Hours |
M
10:30 - 12:00 and W 11:30 - 12:30 on days when classes are in
session, and by appointment. I can change these to some other
time if there's a big demand but these seem good since homeworks will
normally be due on Tuesday or Thursday.
| | Lectures |
TuTh 1-2:30 in Cupples I, room
215 (a change of room from
last semester!) We
can also schedule special meetings to talk about problems if enough
people are interested. Let me know.
For
all the lectures, you should
have read through the notes handed out
in the preceding class. Make a note to yourself
about anything you don't understand, particularly for the part of those
notes that I have already covered in a lecture.
| | Textbook |
The
textbook
I wrote continues for Math 418. As before, I will give out these
pages lecture by lecture in
class since I'm constantly revising and correcting.
The
notes will be
photocopied two-sided, on 3-hole punched paper. There will be
approximately another 125 sheets (2-sided).
So you might need to get abother (or larger) 3 ring binder.
Once
again, there will again be a charge of $10.00
for the notes to cover the cost of paper, toner and copying
time.
You can pay the receptionist in the Mathematics Department
Office (Cupples I, room 100). She will give me a list of the people who
have paid. The office will accept a check made out to “Washington
University Department of Mathematics” or cash (but the exact cash
amount is needed; the Office cannot “make change”). I will
distribute the first 20 pages or so free of charge so that there's no
rush; but please make your payment by Friday, January 29.
Some
fairly standard reference texts that are available in the library are
Kaplansky, Irving Set Theory and Metric
Spaces
Willard,
Stephen
General Topology
Munkres, James
Topology
Kahn,
Donald Topology:
An Introduction to the Point-Set and Algebraic Areas
Simmons, George Introduction
to
Topology and Modern Analysis
Eisenberg,
Murray
Topology
Each of these is quite different, and none follows the
material as I'll present it.
| | Exams |
There
will be the equivalent of four
exams in the course:
1)
Exam
1 In
class, on Thursday, February 25.
2)
Exam 2 Take-home, given
out
in class on Thursday, March 25, and due in class on
Tuesday, March 30.
3)
Exam 3 Final exam, on Tuesday, May 11, 1-3 p.m.
4)
"Exam 4" See description under "Homework" The
dates for 1) and 2) can be moved slightly if a majority of the class
wants the change. However, if there's going to be a change, I'd
like to decide that by Thiursday, January 28, so that some
students aren't upset by a sudden change later. The "in-class" exam and the final
will be "short-answer", consisting
of such things as definitions, statements of theorems, giving
examples/counterexamples,
and true/false questions.
The “take-home" exam will consist
of more substantial questions,
analogous
to homework problems.
| | Homework |
There
will be 6-8 homework sets
during the semester. Usually these will be distributed in class
and will be due in class three lectures later. Some of
the homework problems
are fairly routine, but many are quite challenging.
Most homework problems will be
read by a grader. However, on about most of the homework sets during the
semester, I will select a problem (after homework is collected) and
grade that problem myself.
Your total accumulated score on
the homework problems I grade will
count
as "Exam 4". Your accumulated score on the remaining
homework
problems will count as your homework score. Homework assignments will be posted on this
web page. Homework 1: | | 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 | During examinations "in class"
and
on take-home Exam 2, no
discussion
or consultation of any kind
with any other person (including internet or other electronic
communication) is permitted. You may consult class notes,
the text, 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
referral to the Academic Integrity Committee. If you have
questions about
what is appropriate, please ask me.
Students are encouraged to discuss
homework assignments
with each
other;
you should certainly 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 handy way to avoid
"borrowing
too much" from others is to talk together but not take
any
written notes away 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.
| | Web Pages | The following 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
Andrei
Tychonoff
Pavel
Urysohn
Heinrich
Tietze
Pavel
Alexandroff q q
The
Beginnings
of Set Theory
The
Axiom of Choice
Topology
Enters
Mathematics The
"Kuratowski 14 Problem"
|
|
