MA 651-Spring 2010

Ma 651 - Topology I - Spring 2010

Announcements

Office hours updated 2/8/2010.
You may find Topology without Tears a helpful reference.
On Monday, Feb 1 class will start 15 minutes late at 6:30.
Classes will meet in K-360 on the third floor of the Kidde Building.

Textbook

Elementary Topology Problem Textbook, O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, and and V. M. Kharlamov, American Mathematical Society, 2008.

We will cover material from the first four chapters of the textbook with additional material as time permits.

Course Organization

Homework assignments will include new material from the textbook and problems to be prepared for class. Problems discussed in class may be assigned to be written up and handed in for grading.

There will be one in class exam and a final exam. Each exam will count 30% of the course grade, and homework will count 40%. In other words the sum of the homework grades is scaled to 0-40 and the two exam grades are scaled 0-30. The sum of the scaled grades is converted to a letter grade as follows.

[100-90%] = A, <90-85%] = A-, <85-80%] = B+, <80-75%] = B, <75-70%] = B-, <70-65%] = C+, <65-60] = C, <60-55%] = C-, <55-0%] = F

except that letter grades my be raised at the discretion of the instructor, typically in the case of students who start slowly but show mastery by the end of the course.

There is no grade of D in graduate courses. Graduate students must have at least a B average to graduate. For additional information consult the Student Handbook at the Office of Graduate Academics web site.

Contact Information

The best way to reach me is by email at rgilman@stevens.edu.
Office hours: Monday 3-5:30 in Peirce 304. Phone: (201) 216-5440.

Homework and Syllabus

(Homework is listed under the class in which it is assigned. Unless otherwise noted, homework is due the the next class.)

January 25. Chapter I.2: Review of set theory, definition of a topology, historical digression, examples of topological spaces. Homework: Chapter I.2 Problems 2.1 - 2.7.

February 1. The rest of Chapter I.2: Open and closed sets, open subsets of the real numbers. Chapter I.3: Bases. Homework: Chapter I.2: Problem 2.I; Chapter I.3: Problems 3.5 - 3.7.

February 8. Read Chapter I.4 Metric Spaces, and work actively on the problems. Homework: 2.Ix, 3.6, and the following problem.

Problem 1. Let A = a₁,a₂,... and B = b₁,b₂,... be infinite sequences of 0's and 1's. Define p(A,B) to be 0 if the two sequences are identical and otherwise 2^-n where n is the least integer for which a_n is not equal to b_n. Show that p is a metric on the set of all infinite sequences of 0's and 1's.

February 16. Read Chapter I.5 Subspaces and I.6 Position of a Point with Respect to a Set. Homework: 4.1, 4.2, 4.7, 5.2, 5.5, 5.6, 6.A, 6.D, 6.F

February 22. Definition of continuity. Homework: Write up 5.6, 6.A, 6.D, 6.F to hand in. Read Chapter II.9 and II.10. Prepare 9.A, 9.M, 10.2, 10.G for class.

March 1. Properties of continuous maps. Homework: Write up 9.A, 9.M, 10.2, 10.G to hand in. Read II.11. Prepare 11.A, 11.D, 11.3, 11.4.

March 8. Midterm exam.

March 22. Connectedness. Read Section 12. Prepare 12.1, 12.9, 12.B

March 29. Write up 12.18, 12.19, 12.20, 12.Q to hand in. Read Sections 13 and 14. Prepare 13.2 and 14.B.

April 5. Separation axioms; countability. Write up 13.4, 14.8, 14P, 14Q to hand in. Read Sections 15 and 16.

April 12. Compactness. Write up 15.4, 15.11, 15.17, 15.21, 16.6, 16.L to hand in. Read Sections 17 and 18.

April 19. Write up 17.7, 17.I, 17.14, 18.K, 18.L to hand in.

April 26. Product and quotient spaces. Read Sections 20 and 21. Write up 20.8, 20.9, 20.10, 20.M, 21.C, 21.3, 21.4. May 10. Final Exam.