Introduction to Topology, 110.413, Spring 2020
W. Stephen Wilson

W. Stephen Wilson: or

TA: Xiyuan Wang:

Textbook: Lecture Notes on Elementary Topology and Geometry. I.M. Singer and J.A. Thorpe.
ISBN: 0-387-90202-3
ISBN: 0-540-90202-3

Little known facts about the course

The official pre-requisite for this course is just Calculus III, i.e. multi-variable calculus. This is misleading on many fronts. Where to start?

(1) This is not an introduction to proofs course. Some of the department's courses are designated as such, for example, Honors Linear Algebra, Abstract Algebra I, Real Analysis I, and Elementary Number Theory. This course assumes you have had a proof based course before this. The mathematics is very sophisticated and requires significant mathematical maturity to understand. The more math courses you've had before this, the better your chances in this one.

(2) The very course description makes it clear that you need to know about groups. Abstract Algebra I is a functional pre-requisite if not a formal pre-requisite. You will be assumed to have a knowledge of group theory if you take this course.

(3) This course generalizes many of the concepts studied in Real Analysis I. The proper order for this material is to learn the special cases in Real Analysis I and then move on to the generalization in this course. The abstraction of the generalization in this course is difficult to internalize without having seen what is being generalized. In other words, Real Analysis I really should be a pre-requisite. On the other hand, if you understand this course without that, Real Analysis is much easier when you take it.

(4) The bottom line is that to have the proper mathematical maturity and background for this course, the pre-requisites should be Real Analysis I and Introduction to Abstract Algebra I, not just multi-variable calculus. It should be considered a serious math major course for seniors who have had their math requirements filled.

(5) On the other hand, if you are mathematically mature and have had a proof based course and are willing to work hard, you can take this course and do well.

Class meets Tuesday and Thursday, 10:30 to 11:45 in Krieger 180.

Office hours are Thursday, 11:45-12:45. (Krieger 421.) This is right after class, so if you want to come to office hours, you arrange that in class, or I won't be there. You can always email me with questions or to set up a meeting if the office hours don't work for you or aren't enough.

In addition, you should all know about the math department help room that is open nearly all day and most evenings during the week. It is in Krieger 213. Your grader will be there in the help room on Monday 7-9pm with office hours 6-7pm Monfday, but most in the help room can help you with topology.

All students must attend every class and maintain a high level of consciousness during the entire class. You will have something serious to do during the entire class, every class (except the first one on Jan 28).

I am using a different textbook this year. I have been unhappy with both the textbook and the format of the course from previous years. Thus, this semester's format and syllabus is going to be a work in progress throughout the semester. Despite that, I have made up an imaginary syllabus below. It is highly unlikely that we will follow it. We will surely get through at least half of it.

These fake reading assignments are for certain days during the semester. The numbers refer to chapter and section in our textbook.

This looks nice on paper, and if I lectured and gave midterms and a final, it might even be doable. Not sure how much you'd learn that way though. So, the class won't be run like that.

There will be very little lecturing except when a consensus develops that something must be explained better. Then I'll do what I can.

Otherwise, among other things, there will be a lot of little quizzes, and they will count significantly for your grade. I will probably give at least one a week. Some will be short and simple and some might take an hour. They will not be announced in advance. I'll give a quiz whenever I get the urge. You will be prepared, of course, because you will have done the reading.

In addition, there will be homework, but perhaps on an erratic schedule. I will always give you a week to do it. You can work with other students as long as you write up your own homework. Usually, I will have 3 of the homework problems graded by the TA. I will chose the 3 and you will not be told which ones they are. The idea is that the TA will then have time to give you feedback. I will feel free to use other problems on the homework for quizzes.

To further complicate the course structure, much of our in class time that is not spent taking quizzes will consist of either individual or group problem solving. I will feel free to give a problem to the class and then call on someone to put the proof on the board, for a grade. I will give problems to groups, and when a group thinks they have a solution, I will pick someone in the group to explain it. On other days, or possibly every day, you will work problems in a group and the group will hand in one solution set at the end of class for a group grade.

Or, I might not do any of those things. I'll experiment on ways to keep you involved and learning. If something doesn't look like it is working, I'll switch to something else.

Grades will be made up of everything you do.

1.1 and 1.2 for Jan 30 (and Feb 4).

Problem set # 1.

1.3 and 1.4 for Feb 6 and 11.

Problem set # 2.

1.5 and 1.6 for Feb 13 and Feb 18.

Problem set # 3.

2.1 and 2.2 for Feb 20 and Feb 25.

Problem set # 4.

2.3 and 2.4 for Feb 27 and Mar 3.

Problem set # 5.

2.5 and 3.1 for Mar 5 and Mar 10.

Problem set # 6.

Spring break, no class Mar 17 or Mar 19.

3.2 and 3.3 for Mar 24 to Mar 26.

Problem set # 7.

4.1 for Mar 31 and Apr 2.

Problem set # 8.

4.2 for Apr 7-9.

Problem set # 9.

4.3 for Apr 14 and 16.

Problem set # 10. Due April 23.

5.1 for Apr 21-23.

5.2 for Apr 28 and 30.