There are two kinds of assignments: Weekly exercises that are not to be handed in for grading but are intended to prepare students for the exams, and problem sets that are to be handed in and graded.
Problem Sets
The problem sets are assigned from the textbook: Munkres, James R. Topology. 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 28 December 1999. ISBN: 0131816292.
Problem set 0 is a “diagnostic” problem set. It is designed to determine whether you are comfortable enough with the language of set theory to begin the study of topology. We will grade it in class at the second session. lf you miss more than 8 answers, you should probably take another proof-based course before trying this one. The grade on this problem set is intended for advising purposes only.
For problem sets 1-4, the solutions are to be written out carefully and legibly, in good mathematical style. “Careful” has an obvious meaning. “Legible” means to do it in LATEX or (if handwritten) in ink and double-spaced. “Good mathematical style” means the style demanded by editors of math journals. Please read these comments about what the mathematics profession means by “good mathematical style (PDF).” Follow them!
Note well: your first written solutions constitute your first draft; this is not acceptable. It will need to be rewritten, to clean up grammar and sentence structure and exposition. Treat it like a paper in a humanities class. (Unless you hand in sloppy papers there too!)
| PROBLEM SETS | DESCRIPTION |
|---|---|
| Problem Set 0 |
Sec. 1; 2. Give answers only. Your answers to (j), (k), and (l) should be “yes” or “no.” Your answers to the others should be one of the following: ⇒,  , ⇔, ⊂, ⊃, =
Sec. 1; 5. Give answers only Sec. 2; 4c and 4e. Give answers only |
| Problem Set 1 |
Sec. 3; 13
Sec. 9; 8 Sec. 13; 7 and sec. 17; 16. Give answers only Sec. 17; 18. Give answers only |
| Problem Set 2 |
Sec. 18; 13
Sec. 20; 6ab Sec. 20; 8bc, Assume 8a. Give answers only in 8c Sec. 24; 4 |
| Problem Set 3 |
Sec. 26; 9
Sec. 26; 12 Sec. 28; 4 Sec. 30; 5 |
| Problem Set 4 |
Sec. 31; 7abd
Sec. 33; 4 Sec. 34; 4 and 5. Give proofs or counterexamples Sec. 38; 9 |
| Problem Set 5 | (PDF) |
Weekly Exercises
The exercises are assigned from the textbook: Munkres, James R. Topology. 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 28 December 1999. ISBN: 0131816292. Collaboration on the weekly exercises is encouraged; you can learn a good deal from your fellow students. But the work on the problem sets is to be strictly your own. If you can’t do all of a problem, do what you can and write “here’s where I got stuck” “Faking it” is much worse than saying “I couldn’t do it.”
| WEEK # | TOPICS | EXERCISES |
|---|---|---|
| 1 | Logic and Foundations |
Sec. 1; 3
Sec. 2; 1, 2, 4, 5 |
| 2 | Relations, Cardinality, Axiom of Choice |
Sec. 3; 11, 12, 15
Sec. 5; 4, 5 Sec. 6; 3, 6 Sec. 7; 3, 4, 5 |
| 3 | Topologies, Closed Sets |
Sec. 7; 6
Sec. 13; 2, 6, 8 Sec. 16; 3, 4, 8, 10 Sec. 17; 3, 4, 5, 6, 8, 9 |
| 4 | Continuous Functions, Arbitrary Products |
Sec. 17; 10, 11, 12
Sec. 18; 2, 3, 7, 8 Sec. 19; 1, 2, 3, 6, 8 |
| 5 | Metric Topologies |
Sec. 19; 7
Sec. 20; 2, 4, 5, 8a Sec. 21; 3, 4, 6, 7 |
| 6 | Quotient Topology | Sec. 22; 2, 3, 6 |
| 7 | Connected Spaces, Compact Spaces |
Sec. 23; 2, 3, 5, 7, 8
Sec. 24; 1, 5, 8, 9 Sec. 25; 1, 2, 3 Sec. 26; 1, 4, 5, 6 |
| 8 | More about Compactness |
Sec. 26; 7, 8
Sec. 27; 1, 4 Sec. 28; 1, 2, 3 |
| 9 |
Well-ordered Sets, Maximum Principle
Midterm Exam |
|
| 10 | Countability and Separation Axioms |
Sec. 10; 2, 3, 6
Sec. 29; 5, 6, 7, 8 Sec. 30; 1, 2, 4 7, 8, 9 |
| 11 | Urysohn Lemma, Metrization |
Sec. 31; 1, 3, 6
Sec. 32; 1, 2, 3, 6, 7 |
| 12 | Tietze Theorem | Sec. 33; 2, 6, 7, 8 |
| 13 | Tychonoff Theorem, Stone-Cech Compactification |
Sec. 34; 1, 3, 7, 8
Sec. 37; 2, 3 Sec. 38; 5, 6 |
| 14 | Baire Spaces, Dimension Theory |
Sec. 38; 3, 4
Sec. 36; 1, 5 Sec. 48; 1, 2, 3, 5, 6 |
| 15 | Imbedding in Euclidean Space | |
| Final Exam |
