Department of Mathematics Math 4181 Topology II Syllabus Spring 2014

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.

 Homework 1 Due Tu 1/28/14 Homework 1 Solutions Homework 2 Due Th 2/6/14 Homework 2 Solutions Homework 3 Due Tu 2/18/14 Homework 3 Solutions Homework 4 Due Tu 3/4/14 Homework 4 Solutions Homework 5 Due Th 3/20/14 Homework 5 Solutions Homework 6Hints Homework 6 Solutions Homework 7 Homework 7 Solutions Homework 8 Homework 8 Solutions

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 SolutionsExam Scores Exam 2   Take Home: given out Thursday April 3, due in  class Tuesday April 8. Exam 2 SolutionsExam 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

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.

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

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.