Office: LeConte 406
Phone: 7-5134
Email: nyikos @ math.sc.edu
Office Hours for the remainder of this week: Wednesday May 8: 10:45 -12:00 and 1:00-6:00;
Thursday May 9: 10:00 -1:00 and 1:40-60; Friday tba
Homework corrections will be given full credit all the way to Thursday, May 9, 5pm. However, since
there may be corrections and hints on your first/next attempt, it is to your advantage to
hand problems in as soon as they are done.
This may change, but I hope to give plenty of advance notice about any changes.
Please disregard any earlier notes about deadlines that I have put into writing.
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 a very small class this semester, 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 most of 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. Much else will be covered by way of enrichment, but you are not expected to know it for the midterm or the final exam.
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 second
attempt and, for the more difficult (starred) problems, only on
the third attempt. 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.
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 (15 percent) and a final exam (30 percent). In addition, class attendance and participation can make a difference in borderline cases.
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]. I hope to be able
to offer it for the Fall Semester, 2019, but a necessary condition for that
is a bigger enrollment than this semester. That is not an unrealistic goal:
general topology, unlike most other branches of topology, is not like a tree
but like a bush with lots of stems growing out of the ground.
Math 730 is certainly an adequate preparation for Math 731, but is not
a prerequisite. Prerequisites will be a bit more demanding than for
this course, but not as demanding as for that earlier Math 731 course.
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 2020 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 Spring 2020 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.
Students are not required to come up with proofs on the midterm nor the final exam in any of these courses, 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.