Office: LeConte 406
Email: nyikos @ math.sc.edu
Textbook: General Topology by S. Willard, Dover Publications.
Ordinarily we will meet in LeConte 407, the Conference Room, but will move to a room with better blackboards from time to time. On Wednesday, January 22, I have booked us for LC 312
This course includes an in-depth coverage of metrizable spaces (including complete ones, for which we will prove the Baire Category Theorem), uniform spaces, connectedness in topological spaces (including an introduction to homotopy theory), and function spaces. It also covers an assortment of other topics depending on the interests of the students.
General topology is more like a spreading bush than a tree. There are many concepts close enough to the ground, so to speak, so that a student can pick them up with a minimum of background knowledge. So, although this is the second semester of a two-semester sequence, the first semester Math 730 (General Topology I) is not a formal prerequisite for it. A semester of topology on an undergraduate level, or a semester in analysis are enough to do well in Math 731. Short reviews of basic concepts and theorems will be given where needed.
The course uses the same textbook that is used for Math 730. The book, a Dover Publications reprint, is a real bargain. The first two weeks of the course will mostly be a review of topics covered in Math 730. The rest of the course covers sections 12 and 16, and most of Chapters 6, 7, and 8, and also parts of Chapters 9 and 10 as time permits.
The material in Chapters 7, 9 and 10 are especially helpful to students studying analysis. Chapter 8 deals with connectedness and disconnectedness and includes sections on homotopy theory, the gateway to algebraic topology. In Chapter 9 uniform spaces are covered. These are topological spaces with additional structure, just as metric spaces are topological spaces with a structure where distances play a key role. In fact, each metric is associated with a unique uniformity, whereas the same topology might have many different uniformities and proximities associated with it.
Chapter 10 deals with function spaces and includes theorems about pointwise and uniform convergence of real-valued continuous functions as well as topologies on the whole function space (or the subspace of bounded functions). The chapter culminates in two classic theorems, the Arzela-Ascoli theorem and the Stone-Weierstrass Theorem, which will be discussed.
The midterm and final exam will only be counted if they pull up your grade from what you get in homework. If all three are counted, homework contributes 50%, the midterm 15%, and the final exam 35% to the overall grade. There will also be opportunities to obtain extra credit by giving presentations, either in class or in a seminar which I hope to run from time to time.
There will be one to three homework exercises each week, and for all of them you get at least one chance to get them right before the point values go down. Most of them will be treated like the starred problems in Math 730: on the third try the additional progress you make will be given 3/4 credit, on the fourth it will be half credit, and on the fifth it will be 1/4 credit.
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.