Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

The final exam is on Saturday, December 14, at 9am. Information on all final exam times can be found outside of "Self Service Carolina" at this good old fashioned Registrar's website.

Office hours for week of December 8-14:

Monday 10:30 - 11:30 and 1:20 - 5:30;

Tuesday 9:00 -12:30 and 1:00 - 4:00

Wednesday 9:00 -12:30 and 1:00 - 4:15

Thursday 9:30 - 1:00 and 1:30 - 5:00;

Friday 12:30 - 1:30 and 2:00- 4:30pm

The midterm was on Wednesday, October 30, on Chapters 1 and 2 and
a bit of Chapter 3; a detailed list of definitions
and proofs from Chapters 1 and 2 for which you are
responsible was given Friday, October 25; for Chapter 3 these are the
definitions and proofs that were given Monday Oct. 21 through Friday, Oct. 25.
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 in which you are asked to come up with proofs on your own. 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{but 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.

For more information on how the course is graded, click here.

There was a quiz on Monday, September 30, on the definitions in Sections 2.2 and 2.3, and on proofs of 2.2.9 and 2.3.7.

There was a quiz on Wednesday, October 9, on the definitions in Sections 2.2 and 2.4, and on proofs of 2.1.5, 2.4.14, and 2.4.16.

Practice problems from Section 1.1,, not to be handed in:

numbers 1(a)(c)(i), 2(b)(h), 3(a), 4(a)(c)

Homework from Section 1.1 handed in on Friday, August 30:

numbers 1(d), 2(d)(g), 3(b)(d), 4(b)

Homework handed in Wednesday, September 11:

p.11 numbers 12, 13

p.14 numbers 2, 4, 6

p.20 number 2

Homework handed in Friday, September 27:

Section 2.1 numbers 1 (c) (g) and 4

Section 2.2 numbers, 2 (e) and 4

Homework handed in Friday, October 25:

Section 3.1, numbers 1 (d) (e) (f); 2 (d) (e) (f);
3 (d) (e) (f) 4

Homework handed in Wednesday, November 6:

Section 3.2, numbers 3, 5, 8, 13, 14, 16

Homework to handed in Monday, November 11:

Section 3.1, numbers 5 (g) (h) (i), 6 (a) (b)

Homework to be handed in Monday, December 2:

Section 6.1, numbers 1, 2, 5, 11, 13.

There is no due date for extra credit, but once a fully correct solution
is handed back, the problem is no longer eligible for extra credit.
** This is true of any problem crossed out below.
**

If you can't quite get the solution but have some ideas, hand them in for partial credit. I will keep adding to your score as you improve your work on it.

1. [8 points each part] Let X and Y be sets and let f:X -> Y be a function.

~~(a) If there is a subset V of Y such that f(f~~ ^{ -1}(V) does not equal V,
must there be a subset A of X such that f^{ -1}(f(A)) does not equal A?

(b) If there is a subset A of X such that f^{ -1}(f(A))
does not equal A,
must there be a subset V of Y such that f(f^{ -1}(V)
does not equal V?

2. On the top half of page 26, do:

~~(a) [8 points] number 8 ~~

~~(b) [10 points] number 11~~

3.~~ Also on the top half of page 26, do number 7 [6 points]~~

4. Do problem 1, page 56. [1 point apiece for (b), (c) (f) and (g); 4 points apiece for (a), (d), (e) and (h).]

5.~~ Do 5 on p. 88 [see last sentence in the first paragraph of the
proof of Theorem 5.1.11.] [10 points]~~

6. See handout "Preimages and inverse images." [10 points]

7.problem 8, p. 70 [4 points]

8. problem 4, p. 74 [6 points]

9. problem 3, p. 96 [6 points]

10. problem 5, p. 137 [12 points]