# Math 534 Section 001 Elements of General Topology

Fall, 2017

### Lectures in LeConte 115 MWF 9:40 -- 10:30 AM

Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

Office hours until further notice : 10:40 - 12:35 and 2:10 -3:30 MWF; also you may see me any time I am in, or by appointment. Exceptions will be announced in advance if possible and posted on my office door.

There will be a quiz on Monday, Oct. 30, on definitions from 2.3, 2.5, and 3.1, and problems like the practice problems: 4 and 6 in Section 2.5 and 2, 4, and 6ab in Section 3.1.

The final exam is on Wednesday, December 13, at 9am. Information on all final exam times can be found outside of "Self Service Carolina" here. Textbook: Introduction to Topology , by Crump W. Baker, supplemented by numerous handouts that will be used in class discussion and some extra credit problems. The syllabus includes the first three chapters; Sections 4.1 and 4.4; Sections 5.1 and 5.2; Chapter 6; Sections 8.1 and 8.3; and excerpts from sections 4.2, 5.3, 8.2, 8.4, and from all four sections of Chapter 7, as time permits.

Learning Outcomes: Students will master concepts and solve problems based upon the topics covered in the course, including the following concepts: topological space; metric space; continuous, open, and closed functions; homeomorphism; product topologies; compactness; connectedness; Hausdorff, regular, and normal space. Students will need to understand the generalizations of theorems such as the Heine-Borel Theorem, Bolzano-Weierstrass Theorem, and the Intermediate Value Theorem to arbitrary topological spaces.

For some of you, this will be the first course besides MATH 300 in which you are asked to come up with proofs on your own. And the proofs will mostly be on concepts that are very different from those in MATH 300. On the other hand, they are very similar to many in MATH 554, which is one of the required courses for all math majors.

This will be done in the homework, very sparingly at first, beginning with very simple proofs and only gradually working up to more challenging ones. Most of the homework problems in any one assignment are routine and most do not require the working of proofs. You will not be required to come up with proofs on tests, but only on homework; some memorization of proofs is expected, and you will be tested only for (some of the) proofs that have been gone over in class with plenty of opportunity for asking of questions.

This gradual acclimatization to proofs makes this course a good preparation not only for Math 554 (for which the course content is also very relevant) but also for any higher- level course like Math 546 which emphasizes proofs.

Because of the emphasis on proofs, homework counts for one third of your grade. Another third will be based on a midterm and quizzes, and the final exam counts for the remaining third.

Practice problems, not to be handed in:

Section 1.2: 1bdfh, 2bdg, 4bc, 5, 9bc, 10, 11, 13

Homework due Friday, September 8:
6, p. 11
Rewrite the proof in the book of the first direction in 1.2.16, in complement notation.
9d, p. 11
12*, p. 11

On starred problems, you have two chances to get it right, with partial credit and hints if you don't get it the first time around.

Homework due Wednesday, Nov. 1: A* Prove Theorem 2.4.16 with `iff'
Section 3.1: 5, 7*, 8, 13 (if a set is in the base, find an open interval B such that the intersection of B and A is the desired set.

Homework due Friday, November 10:
Section 3.2: Exercises 3, 6, 7*, 9, 11*
Section 3.3: Exercises 5 and 8.

#### Extra Credit Problems

From time to time, extra credit problems will be assigned in class. Extra credit problems are to be done strictly on your own, except that I am willing to give you advice. You are not to discuss them with anyone else.