Math 4431

MATH 4431

Introduction to Topology

Fall 2015

MWF 12:05-12:55 Skiles 257

INSTRUCTOR

Stavros Garoufalidis
Room 105, Skiles
Telephone: 404-894-6614
email
Office hours: Wed 3:30-4:30 or by appointment.

GRADER

Shane Scott.

TEXTBOOK

James Munkres, Topology (2nd Edition), Prentice Hall.

COURSE DESCRIPTION

We plan to give a self-contained course in point set topology, after introducing the relevant concepts of set theory and logic. This is a course with proofs, and abstract mathematical concepts (such as topological spaces, compactness, continuity). Despite its abstraction, topological spaces and continuous maps are very useful in applications, and form the background of applied mathematics. We promise to give proofs, examples and counter-examples of many theorems. A goal of the course, aside from topology, is to learn what is a mathematical proof, and how to present it. This requires basic mathematical maturity, and familiarity with math 4317.

WHAT IS A PROOF?

See a note by Chris Heil.

PREREQUISITES

MATH 4317.

HOMEWORK

Homework will be due on Fridays, collected in the beginning of the class, and returned back to you the following Monday. I will not be assuming that you know how to think, write and present a mathematical proof, but achieving this by the end of the semester, will be an important gain for the class. You are welcomed to work together, but please write down your homework separately . Please staple your homework, and write clearly, each problem on a new page. The two lowest homework grades will be dropped. Any missed homework will be counted as zero.

MIDTERM-EXAMS

There is one in-class midterm exams on Wednesday October 14. I do not accept make-up exams unless there is a serious documented reason.

FINAL EXAM

Conflict in final exams means having 3 final exams in the same day. In that case, you can reschedule the one in the middle. In case of a final exam conflict, I need a two week advance notice to accomodate your request.

ATTENDANCE

Although attendance is not mandatory, you are strongly encouraged to attend. Students who do not attend the lectures statistically do below average--and usually end up with a C or lower grade. Attendance is an important part of learning. During the lectures, you should turn all powered devices (including phones, laptops, music players) off.

GRADES

The final course grade will be calculated according to the following scheme:

Homework : 25 %
Midterm : 30 %
Final Exam : 45 %

If the average course grade (out of 100) is

at least 80, then you will get an A,
at least 70, then you will get a B or A,
at least 60, then you will get a C, B or A,
at least 50, then you will get a D, C, B or A,
if your final exam grade is less than 40, then you will get an F.

ACADEMIC HONOR CODE

All students are required to review and accept the Honor Code of Georgia Tech, which may be found here.

You are not allowed to use calculators, formula sheets, notes or texts during the midterms and final exams.
Identification is required in taking tests.

ACADEMIC CALENDAR

August 17, First day of class.
September 7, School Holiday: Labor Day.
October 25, Last day to withdraw with W.
October 10-13 Fall recess.
October 14, Midterm I.
November 26-27 School Holiday: Thanksgiving.
December 5, Last Day of classes.
December 11, Final Exam. Friday 11:30am - 2:20pm
December 15, Grades available.

HOMEWORK ASSIGNEMENTS

Homework 1 Due Friday August 21, in the beginning of the class. Sec.1.1: 2. Sec.1.2: 5. Sec.1.3: 6, 15.
Homework 2 Due Friday August 28. Sec.5: 5. Sec.6: 3,6. Sec.1.7: 6. Sec. 1.9: 2,3.
Homework 3 Due Friday September 4. Sec.13: 7,8. Sec.16: 3,8,9,10.
Homework 4 Due Friday September 11. Sec.16: 3,9,10. Sec.1.17: 8,9,20. Sec.2.18:2,6,8,13. Problem: (a) If A is a subset of a topological space X, show that X is the disjoint union of the interior of A, the boundary of A and the interior of X-A. (b) Show A is closed if and only if A contains its boundary. (c) Show that A is open if and only if A and its boundary are disjoint.
Homework 5 Due Friday September 18. Sec.19:8. Sec.20:2,4,5. Sec.2.21:4,7,8,9,11.
Homework 6 Due Friday September 25. Sec.22:2,4,5.Sec.23:2,3,5,6,7.
Homework 7 Due Friday Oct 2. Sec.24: 2,3,9. Sec.25: 2.
Homework 8 Due Friday Oct 9. Sec.26: 1,7,12. Sec.27: 3,4,5. Sec.28: 6,7. Problem: Suppose (X,d) is a metric space, S a compact subset of X and x in X. Show that there exists s in S such that d(x,S)=d(x,s). Here, d(x,S)=inf{d(x,y)| y in S}. Problem: (a) Suppose (X,d) is a metric space and A,B two disjoint nonempty compact subsets. Show that there exists delta >0 such that d(a,b) > delta for all a in A, b in B. (b) Find a counterexample to the above statement if A,B are disjoint nonempty closed subsets of a metric space.
Homework 9 Due Friday Oct 16. Sec.28: 6,7. Sec.29: 1,4,8.
Homework 10 Due Monday Oct 26. Sec.30: 6,11,12. Sec.31:1,2.
Homework 11 Due Monday Nov 2. Sec.32.1. Sec.34.7. Sec.35.7.
Homework 12 Due Monday Nov 9. Sec.36:2,3,4. Problem: Let S be the subset of real numbers in [0,1] with decimal expansion x=0.a1b1c10a2b2c20a3b3c3... where ai,bi,ci in 0,1,...,9. (a) Show S is closed. (b) Let phi_1: S->[0,1] be defined by phi_1(x)=0.a1a2a3... Likewise phi_2 and phi_3. Show phi_i i=1,2,3 are well-defined and continuous. (c) Since [0,1]-S is a countable union of open intervals extend phi_i linearly on these intervals to define f_i:[0,1]->[0,1] continous and onto for i=1,2,3. (d) Let F:[0,1]->[0,1]^3 be F(x)=(f_1(x),f_2(x),f_3(x)). Show F is continous and onto. It is a space-filling curve. (e) Compute the values of F for all rational numbers in [0,1] with denominator at most 5.
Homework 13 Due Monday Nov 16. Sec.37:2,3. Sec.43:2,5.Sec.44:3.
Homework 14 Due Monday Nov 23. Sec.37:2,3. Sec.45:1,2,3,4. Problem: Show that the Cantor set is homeomorphic to {0,1}^N, with the product topology.