Math 418, Introduction to Topology and Modern Analysis
Spring 2004
Instructor Ron Freiwald
Meetings Tu-Th 1-2:30 in Cupples I, rm 199 (except for March 16 and March 18: location those days TBA)
We can also schedule special meetings to talk about problems if that would be helpful.My Office 203A Cupples I
My Office Hours M,W 1:30-2:30, Th 2:45-3:30 (but I'm also happy to try to make appointments at some other times)
Office Phone 935-6737No text is assigned. I will provide outlines for all the lectures, and detailed written notes at certain points in the course.
In 417, we covered most of the material in Kaplansky's book, Set Theory and Metric Spaces; the omitted material consisted mostly of Chapter 3.
Those topics will be covered in 418. In addition, we covered some topics in much more depth and added some material about general topological spaces.
In Math 418, we will cover material about products, quotients, embedding, metrizability. This includes some of the classic theorems of general topology
such as Urysohn's Lemma, Tietze's Theorem, Urysohn's Metrization Theorem and the Tychonoff Product Theorem. There will be some additional material
from set theory (ordinal numbers and "transfinite methods"--transfinite induction and Zorn's Lemma). I haven't yet decided on the material to include in the
last few weeks of the course.
__________________________________________________________________________Exams There will be the equivalent of four exams in the course:
1) Exam 1 In class, Tuesday, March 2
2) Exam 2 Take-home, given out in class on Thursday, April 1
due on Tuesday, April 6 by 4 pm.
3) "Exam 3" See under "Homework"
4) Exam 4 Final exam, on Tuesday, May 11, 1-3 pm.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 approximately 8 homework sets during the semester. Some of problems are fairly routine, but many are quite challenging. During the course
of the semester, I will choose one problem from each homework set after the homeworks are submitted and grade those problems myself. The remaining
homework problems will be handled by a grader.Your total accumulated score on the homework problems I grade will count as "Exam 3". Your accumulated score on the remainder of the homework problems
will count as your homework score.
Oral Presentation About midway through the course, I will give you a short list of topics and possible references for a one-hour oral presentation.
The oral presentation is in some sense optional: doing a presentation is a necessary but not sufficient condition to receive a grade >= A- . Presentations will be made outside of class; any students not presenting the same topic are welcome to also attend.
Basis for Grading
The four exams and the homework will each count about 20% of your grade. Homework assignments are an essential part of the course. If a student neglects
these, the course grade may be dramatically lowered (regardless of test scores) at my discretion.
Academic Integrity
During examinations "in class" and on take-home Exam 2, no discussion or consultation of any kind with any other person is permitted.
It is understood that on any take-home work (tests or homework) a student may consult class notes, the text, or any other references—provided the other
references are explicitly documented. Generally speaking, you should avoid trying to "find" solutions to problems elsewhere. Any solutions taken from other
sources without documentation will result in a grade of 0 for the test or assignment. If you have questions about what is appropriate, please ask me.Students are encouraged to discuss homework assignments with each other; 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 sessions
with 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 sidelights on the material. The are in roughly the order in which the characters appear in the course.
The MacTutor History of Mathematics ArchiveGeorge Cantor
Bertrand Russell
Kazimierz Kuratowski The Kuratowski "14 Problem"
Kurt Godel
Paul Cohen
Robert Sorgenfrey
Felix Hausdorff
Ernst Lindelof
Augustin-Louis Cauchy
Rene-Louis Baire
Pavel Alexandroff
August Ferdinand Moebius
Andrei Tychonoff
Paul Urysohn
Heinrich Tietze
Ernst Zermelo
Max Zorn
Julius Konig
Richard Dedekind
The Beginnings of Set Theory
Home Page for the Axiom of Choice
Axiom of Choice (the music)
Topology Enters Mathematics
BibliographyThe following is a brief bibliography you may find useful. I have requested that all of the books be placed on 2 day reserve at Olin Library.
Some of these books are "classics" and some (such as Munkres) are more recent.
Eisenberg Topology
Kamke Theory of Sets
Kelley General Topology
Dugundji Topology
Munkres Topology
Willard General Topology
Sierpinski Cardinal and Ordinal Numbers
Gillman & Jerison Rings of Continuous Functions