Math 730 (TTh 5:00 -6:15 PM, in LeConte 303B)
General Topology I

Fall 2009

Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @

Home page: Peter J. Nyikos.

Office hours: TTh 1 - 3, and any time when I am in, and by appointment; exceptions posted on my door and announced in advance whenever possible.

Extended office hours beginning December 3:
Thursday, December 3: 1-4 pm
Friday, December 4: 10-11:50 am
Monday, December 7: 2:30 - 5:30
Tuesday, December 8: 1:30 - 4:30
Wednesday, December 9: 12:00 - 3:00
Thursday, December 10: 9:00 - 11:00am and 4:20 - 5:20pm
Office hours for the rest of finals week posted later.

For homework assignments, click here.

Math 730 makes a natural follow-up to a basic real analysis course and is a highly desirable preparation for any course in functional analysis. It is also the foundation for additional courses in topology as well as a good introduction to the independence results of modern set theory. Since it is not a very large class, it will be tailored to a considerable extent to the needs of those taking it. So, besides the official textbook for this course, there will also be short excerpts from other books and other notes distributed in class from time to time.

The official texbook, Willard's General Topology, is not only at a bargain price but is, in my estimation, the only really good textbook of general topology for a beginning year of general topology. There are other excellent books for lower and higher levels, and some that include a great deal of extra material, but this one is unique in being tailored to a course at this level.

The course covers Chapters 1 through 5 in depth and some parts of the first three sections of Chapter 6 (Sections 17-19) and of Section 26 in Chapter 8.

Objectives of the course include a good basic understanding of metric spaces and topological spaces, especially compact, countably compact, and first countable spaces; of techniques of construciting new spaces from old, including subspaces, quotients, and product spaces; of continuous functions between topological spaces, including quotient maps and homeomorphisms; and of convergence of sequences, nets, and filters. The student is expected to be able to come up with some proofs in homework and to understand numerous proofs of important results in the text and lectures, as well as to understand individual examples of spaces and continuous functions.

There are no formal prerequisites, although a course in real analysis at either the undergraduate or graduate level would be very helpful, as would an undergraduate topology course. Anyone who has taken even a semester of functional analysis should be familiar with a good fraction of the concepts and will be able to pick up most other concepts easily. But students can still do well if they just have the sort of mathematical sophistication that comes from taking abstract algebra (Math 546 or Math 701) or graduate linear algebra (Math 700).

Students get several opportunities to get homework problems right: they hand in what they can do and I give hints for how to finish the problems they could not finish. Points start getting deducted only on the third and subsequent attempts. In the past, most students have gotten a big majority of the problems right the second time around. New homework is collected about once a week, usually on Tuesday.

Extra credit problems will be given from time to time; these only expire when some student has been handed back a completely correct solution, and there is no deduction of points for later attempts.

The grading is based on homework (55 percent) a midterm test (15 percent) and a final exam (30 percent). In addition, class attendance and participation can make a difference in borderline cases. Students are not required to come up with proofs on the midterm nor the final exam, but only on homework; some memorization of proofs is expected, and students will be tested only for (some of the) proofs that have been talked about in class or gotten right by everyone on the homework, with plenty of opportunity for asking of questions.

Students who do especially well on the homework and the midterm can exempt the final. This is designed to be the first course in a 2-semester sequence, so it is necessary to have high standards for the "especially" part.

The most natural follow-up to this course is Math 731 [see (General Topology II) for a course description for the last time it was offered]; it will be offered for the Spring Semester, 2010.
Math 730 is certainly an adequate preparation but is not a prerequisite.

Another natural follow-up is a course which has been a topics course up to now but I am hoping to make into a regular numbered course: Set-Theoretic Topology. There is no adequate textbook for that course but by Spring 2010 I hope to be far enough along in the writing of a textbook so that I can put in the paperwork to make it a regular course, which will be offered for Fall 2010 if there is enough interest in it. Until the textbook is finished, students can use what I have along with a booklet by Mary Ellen Rudin, Lectures in Set Theoretic Topology. It is dated but is still a fine introduction to the material, and is still quite inexpensive.