MATH 338: Topology

Introduction to Topology: Pure and Applied, by Colin Adams and R. Franzosa.

We will definitely cover the first few chapters of the text but skip a few sections occasionally. We will also do our best to cover a couple of the later chapters as time permits.

Please note that we will work on developing your independent reading skills in Mathematics and your ability to learn and use definitions and theorems. I certainly won't be able to cover in class all the material you will be required to learn. As a result, you will be expected to do a lot of reading. The reading assignments will be on topics to be discussed in the following lecture to enable you to ask focused questions in the class and to better understand the material.

Topics covered: We will cover topological spaces, open and closed sets, interiors and boundaries of sets, homeomorphisms, connectedness, compactness, and manifolds. Topics are subject to change depending on the progress of the class, and various topics may be skipped due to time constraints.

By the time you take this course, you should be at the point where you are fairly comfortable with various techniques of mathematical proof. We will be using and perfecting these techniques and introducing new methods as necessary. Although this course only has "Introduction to Mathematical Proofs" as a prerequisite, BE WARNED: If this is your first proof-based 300-level math course, then this class is probably not for you. This course requires a certain amount of mathematical sophistication. You are expected to work hard and write quality, thoughtful proofs. You will be asked to write rigorous arguments in order to solve the problems. We will be doing a significant number of proofs, and everyone should ALREADY be fairly comfortable with the process. There will also be a lot of new terminology you must learn. By the end of this semester, you will be mathematical geniuses!

Topology is the study of spaces and sets and can be thought of as an extension of geometry. It is an investigation of both the local and the global structure of a space or set. The foundation of General Topology (or Point-Set Topology) is set theory. There are other areas of topology including Geometric Topology, Algebraic Topology, Differential Topology, Low-Dimensional Topology, High-Dimensional Topology, and various others. The motivation behind topology is that some geometric problems do not depend on the exact shape of an object but on the way the object is put together. For example, the square and the circle are geometrically different, but they have many properties in common: they are both one dimensional objects and both separate the plane into two parts. Similarly, a donut and a coffee cup are topologically the same even though they look completely different. Much of the study of topology comes from setting aside our preconceived notions of "shape" involving size, length, flat, straight, or curved, and realizing that a circle and a square are really the same thing.

Upon successful completion of this course, a student should be able to:

• Define and illustrate the concept of topological spaces and continuous functions,
• Define and illustrate the concept of product topology and quotient topology,
• Prove a selection of theorems concerning topological spaces, continuous functions, product topologies, and quotient topologies,
• Define and illustrate the concepts of the separation axioms,
• Define connectedness and compactness, and prove a selection of related theorems, and
• Describe different examples distinguishing general, geometric, and algebraic topology.

Your overall grade will be determined as follows:

• 25% - Homework, Quizzes and Class Participation
• 25% - Exam 1
• 25% - Exam 2
• 25% - Final Exam

Your overall grade for the course will reflect how well you are doing and will be high if you are working hard on the homework and doing well on the exams. Many of the questions on the exams will be in the same spirit as the homework questions. Therefore understanding how to do all the homework questions will enable you to do well on the exams.

In addition to learning wonderful mathematics this semester, we will also be working on three academic fundamentals that are vital to success in your education and in your career: Reading, writing, and communication.

1. Reading the textbook is extremely important. It is required, NOT optional. I certainly won't be able to cover in class all the material you will be required to learn. As a result, you will be expected to do a lot of reading. The reading assignments will be on topics to be discussed in the following lecture to enable you to ask focused questions in the class and to better understand the material. Your chances of getting a good grade in this course are infinitesimally small unless you read the textbook in addition to attending lectures.
2. Writing quality proofs is essential. You are expected to be able to effectively present mathematics with a well-organized, thoughtful, neatly written argument. What is a "quality proof"?
3. Communication with your peers will be a focal point. In addition to working in pairs on your homework assignments, there will also be regular presentations given in class. Students will be asked to present their solutions to homework problems to the rest of the class. The best way to truly understand a concept is to be able to explain it to someone else. Talking about math is just as important as thinking and writing math. Our goal is to become more comfortable with the necessary communication.

Homework: There will be regular homework assignments which must be turned in by 5:00 pm on the due date. Follow this link for a Description of Homework requirements. Pick a homework partner that you will be working with for the semester. Each student should work out every problem, but you and your partner will only turn in one combined final version of each assignment. Please note that you are welcome to work in groups of as many people as you want, but you and your partner are responsible for writing up and handing in your own pair's homework. Each pair of students should submit their own work, not a handwritten copy of someone else's. Also, you should feel free to write out your assignments by hand, but I encourage you to typeset your homework using LaTeX. Each individual student is STRONGLY ENCOURAGED to think about and attempt the problems on their own before meeting with their partner. When you are about to tackle the homework assignment, you should use this strategy: THINK, PAIR, SHARE.

Exams: There will be two Midterm Exams and one Final Exam. Exams are closed book, closed notes, closed friends, and open brain. Calculators, cell phones, iPods, and other electronic devices will NOT be permitted in exams. The dates of the exams will be decided a week or two in advance.

Class Participation: Class participation will be based on your willingness to ASK and ANSWER questions in class. There will be active discussion at certain times, and you will also be required to present some proofs to the class. It is imperative that you keep up with the reading assignments. This will help you answer my questions and help you ask more essential, thought-provoking questions during the lectures.

It is essential not to fall behind because each lecture is based on previous work. If you have trouble with some material, SEEK HELP IMMEDIATELY in the following ways: