Professor: Peter J. Nyikos

Office: LeConte 406

Phone: 7-5134

Email: nyikos @ math.sc.edu

**
The final exam in this course is on Saturday, December 10, 2:00 - 5:00 pm
**

**
Special Office hours for the first week of final exams:
Monday, Dec. 5: 10:00 - 11:00;
Tuesday, Dec. 6: 1:50 - 4:50;
Wednesday, Dec. 7: 1:30 - 4:30
Thursday, Dec. 8: 12:15 - 2:00
Friday, Dec. 9: 5:30 - 8pm
Saturday, Dec. 10: 1-2 pm
**

Usual Office Hours: Mondays, 12:45 - 1:15; Tuesdays, 2:00 -3:30; Wednesdays, 10:00 - 12:30;
Thursdays, 10:00 - 11:00 and 12:00 -12:30;
Fridays, 12:00 - 1:00; or any day by appointment (or any time I am in).

Exceptions announced in advance and posted on my office door if possible.

The first hour test was on Wednesday, October 5. It covered Chapter 1 and all but
the last section of Chapter 2.
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 two-fifths of your grade.

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

Homework for August and September can be found here.

Starred items are problems on which you get two chances.

Homework due Friday, October 8:

Section 2.4: 3(a)(b), 6(a)(b), 17

Show that a space is a Dutch door space [see Extra credit problem 2 below for
the definition] if, and only if, every subset is the intersection of
an open set and a closed set.

Homework due Wednesday, October 12, in Section 3.1:

1(a) through (d) and (h) and (i)

2(d) through (g)

3(a) through (d)

5(a) (b) (c) (e)

Homework due Monday, October 19:

Section 2.5: 1, 5, 6

Section 3.2: 3 and 5

Homework due Monday, October 26

Section 3.2, numbers 10 and 13

Section 3.3, numbers 1, 2 and 5*

(g) if and only if (h) in Theorem 3.2.9+ *

Homework due Friday, November 4:

Section 3.3, number 5 [last chance]

(g) if and only if (h) in Theorem 3.2.9+ [last chance]

Section 4.1, numbers 1, 2, 3 and 6

Homework due Wednesday, November 9:

Section 4.1 numbers 2, 4, 6, 9*

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. 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. ** Revised: **
Make a list of topologies on X = {a, b, c} such that no two
topologies on your list are homeomorphic, yet every topology on X is
homeomorphic to a topology on your list.
Make it clear why your list is complete; no need to show
that the spaces are not homeomorphic.
[15 points]

2. Call a topological space a * Dutch door space* if every subset is the union of
an open set and a closed set. Show that a space is a Dutch door space if, and only if,
every dense set is open.[16 points]

3. A space is called * irreducible * if every dense set has nonempty interior.
Show that a space is irreducible if, and only if, it cannot be the union of two disjoint
dense subsets. [8 points]