Spring 2003 Syllabus

Instructor:	Dr. Jan Pearce		C.P.O.:	1815
Office:	304-B Draper Hall (859) 985-3569		Office Hours:	MTWRF 1:00 to 1:50 pm
Email:	jan_pearce@berea.edu		Feel free to send email for appointments at other times.

The Course Goals To gain experience with the fundamental concepts of general topology. To further develop skills in proving mathematical theorems. To practice communicating mathematical ideas to others in a precise way. To become a more independent learner, logical thinker, and stronger mathematician.

Mobius Strip II by M.C. Escher

The Class Atmosphere The members of this class constitute a learning community. Learning in such a community best takes place in an atmosphere in which instructor and the students treat everyone with mutual respect. Students need not always raise their hands in order to ask questions or to make comments, but they should not interrupt the instructor or fellow students in doing so. Students typically find the atmosphere set by the instructor to be a sometimes playful and nearly always relaxed one, but students will still need to work hard and consistently both in and out of class in order to do well. If at anytime you have thoughts, comments, or suggestions about how the class atmosphere could be improved or made into one which is more supportive of your learning, please come by or drop me a note about it. I welcome such suggestions.

The Texts and Other Resources
The primary required text is the third edition of Introduction to Topology by Bert Mendelson, an undergraduate introduction to the fundamentals of topology. This text cover employs an axiomatic approach to topology. We will cover chapters 1-5 and some additional topics.
Chapter 1, Theory of Sets This chapter serves as a review of some of the material encountered in MAT 315. We will review the formal properties of functions, relations, and sets, establish notations, and remind ourselves about some of the important techniques needed for proving theorems.
Chapter 2, Metric Spaces In this chapter we will learn that a metric is roughly a function which measures distance, such as on the real line or in the plane. The concept of the metric space grew out of the study of the real number line and the Cartesian plane.  The concept of the metric will also allow us to look at concepts such as the limit with more formality.
Chapter 3, Topological Spaces The word "topology" is derived from the Greek word "," which means position or location. This name is appropriate for this branch of mathematics which deals with the geometric properties which are not destroyed by shrinking, stretching, or twisting, but not cutting or puncturing.  In this chapter we will learn more formally what is meant by "a topology," and we will encounter many examples of topological spaces as well as concepts such as closure and continuity.
Chapter 4, Connectedness The property of connectedness introduced in this chapter arises from generalizing an important idea of Calculus. The property of connectedness, which is roughly speaking the property of being "unbroken" or "all one piece," is one of the most important topological properties we will encounter. It is the property which allows us to prove theorems such as the intermediate value theorem from Calculus. The property of compactness is more difficult to explain in rough terms, but it is the topological property which generalizes the crucial properties of the closed and bounded interval of the real line, [a, b]. Compactness of intervals allows us to prove important theorems such as the maximum value theorem from Calculus.
Chapter 5, Compactness The property of compactness introduced in this chapter also arises from generalizing an important idea of Calculus. The property of compactness is difficult to explain in rough terms, but it is the topological property which generalizes the crucial properties of the closed and bounded interval of the real line, [a, b]. Compactness of intervals allows us to prove important theorems such as the maximum value theorem from Calculus.
Additional Topics, Countability and Separations Axioms The two ideas introduced in this chapter do not arise from the study of Calculus, but rather from deep studies of topological ideas. The main goal of the chapter is to prove the Urysohn Metrization Theorem which says that if a topological space satisfies certain countability and separation axioms, then the space is able to be thought of as a metric space.
The second required text is Intuitive Concepts in Elementary Topology by B.H. Arnold. This text uses an intuitive approach to topology, serving to give us intuition about the various topics we cover in the primary text.

The World Wide Web Our course home page is located at http://www.berea.edu/Math/Faculty/Jan/MAT436/. Use this page as a resource to find this syllabus and other course-related information.

The System of Evaluation
Evaluated Items Points Grading Percentages Test 1 Test 2 Test 3 Quizzes/Presentations Homework Final Exam 100 100 100 100 200 200 12.5 % 12.5 % 12.5 % 12.5 % 25.0 % 25.0 %		Maximum 90-100 % 80-89 % 70-79 % 60-69 % 0-59 % Scale A's B's C's D's F

Please refer to the GRADING section of the current Berea College Catalog for the College-wide interpretations of these letter grades.

The Grading Policies For the benefit of the students in the class, all course grade computations are continually updated by the instructor, so students may check frequently on their in-progress course grade during the term.
	The lowest score earned on one 100 point exam score, quiz total will be dropped before computing the grade. If the lowest percentage score is earned on a 200 point item, then one half of the score will be dropped.
	A student's final grade may be raised above her or his earned percentage grade if in the instructor's opinion the student shows significantly improved work in the course or on the comprehensive final exam.

The Tests and Quizzes Tests and short quizzes will be given in this course. Approximately one announced quiz will be given each week in which there is no test. In general, the announced quizzes will consist of questions on the assigned text readings or homework-like problems. The three test dates are not pre-scheduled as the instructor believes it is very important that students have input into when the tests are held. However, the tests will fall approximately in the following weeks:
Week of March 3 Week of April 7 Week of May 12
Problems that appear on the tests will be similar to the types of proof on the homework, but may also include definitions.

The Final Exam The comprehensive final exam may be a take home final, but if it is an in-class comprehensive final then it will be at the regularly scheduled time of 3:00-4:40 PM Thursday, May 22, the very last day of final exams. By Berea College policy, no instructor can reschedule a final exam on his or her own, so please plan now to take it then.

The Attendance Policy Class lectures and discussions are considered to be vital to success in this course. It is the hope of the instructor that class sessions are both informative and useful, therefore attendance is expected at each class session unless a specific exception is made. This policy will be enforced in several ways. Quizzes may be announced or occasionally "popped," and because the lowest quiz grade will be dropped, under nearly all circumstances, make-up quizzes will not be given. Likewise, make-up tests will under almost no circumstances be given, so missed tests will therefore count as the student's dropped 100 points. Absences from class are noted, and repeated absences will adversely affect the student's grade. The final grade may be lowered by one third of a letter grade for each absence after the fourth. Thus, it is the responsibility of the student to speak to the instructor about each absence from class. This should be done as soon as possible, and if at all possible before the absence occurs. Students who miss class are held responsible for all of the material covered, assigned, and collected during their absence.

On Homework

Through homework, students get the needed practice of proving theorems. Homework will be assigned regularly, since doing homework thoughtfully and conscientiously is one of the keys to success in this course. One half of the 200 point homework grade will be calculated by grading the quality of all of the submitted problems and the other half of the homework grade will be calculated from the quantity of homework submitted.

On Homework Collection All written work should be neat, organized, and should show sufficiently many steps to demonstrate a clear understanding of the techniques used. Homework is due at the beginning of class on the announced date due. If a student must miss class due to either a sickness or a planned absence, homework is still expected to be submitted on time. Assignments may be requested in advance. Homework assignments may be turned in before class or at the instructor's office, but should NOT be sent through the CPO. Assignments not meeting the above standards may receive reduced credit.

On Teamwork Learning to collaborate effectively is not only welcomed but is strongly encouraged. Some homework assignments will be specifically designed for teamwork, others for individual work, but on most homework you can choose to work alone or in a team. However, every student must hand in their own individual handwritten completed assignment. On any assignment where some work was completed collaboratively, the assignments should be handed in with each author acknowledging all of the help received for each problem. This includes significant help received from the instructor or by the Math Lab Consultants. Remember that thoughtful practice, not (even mindful) copying, is ultimately the best way to learn. Warning: Please be careful to conform to these standards for teamwork, since they are designed to encourage good learning practices. (Furthermore, copying another's work or otherwise failing to adhere to these standards may even result in a charge of academic dishonesty.)

For Additional Help There will not be a teaching assistant for this course. Students are strongly encouraged to work together and to make use of the help available in the instructor's office hours. Best results are obtained trying to solve problems alone or in a group before asking for help, so students should be prepared to show what they have already tried. Topics in this course build throughout the course, so students should be sure to do their best to keep up with the class, so as to not get behind and possibly forever lost. Remember, no question to which one does not know the answer is ever "dumb" unless it goes unanswered because it remained unasked.