18.901 | Fall 2004 | Undergraduate

# Introduction to Topology

## Assignments

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

Fall 2004
Lecture Notes
Problem Sets