Math4201 Course Description

E-mail: adaemi@wustl.edu

Class Hours: MWF 11:00-11:50 am

Room: Mallinckrodt, Room 303

(Classes meet in person by default. Occasionally, we may hold virtual classes over Zoom. I will notify you at least a week in advance when a class will be held virtually.) Office Hours: Monday 4:00-5:00 pm and Thursday 3:00-4:00 pm, Cupples I, 212

Course website: I use Canvas as the main place to post materials related to the course.

Credit Units: 3.0 Units

Course Overview

An introduction to the fundamental ideas and results in point set topology. Course includes necessary concepts from set theory, topological spaces, subspaces, products and quotients, compactness and connectedness. Some time is also devoted to the particular case of metric spaces (including topics such as separability, completeness, completions, the Baire Caregory Theorem, equivalents of compactness in metric spaces and the metrics spaces of functions). The language of topology pervades much of modern mathematics and some of the theorems we are going to learn are required for the more advanced courses in both pure and applied mathematics you are likely to take in the near future. The prerequisite for this class is Math 4111 or permission of instructor. In particular, students should know what a proof is and how to produce one. This course is also the foundation for Topology II, where you will explore the global structure of topological spaces with the aid of algebraic methods.

Prerequisites

Math 4111 or permission of instructor. In particular, students should know what a proof is and how to produce one.

References

The primary reference for the class is “Topology” (2nd edition) by James Munkres. Each lecture of the class is based on one section of the textbook. My tentative plan is to cover Chapters 2–4 and part of Chapter 7, skipping some of the optional sections. Each section will take one to two lectures. There are several other good references on topology, some of them are listed below:

• Topology by Kläus Jänich
• Topology by James Dugundji
• Counterexamples in Topology by Lynn Steen and J. Arthur Seebach, Jr

Homework

All homework will be submitted online through Gradescope. Homework assignments will typically be due each Thursday at 10:00 pm. The first homework is due on September 14. Each problem set is posted roughly about a week before the deadline on Canvas and Gradescope. Working through homework problems is a key part of learning the material in this class. Spending enough time on homework problems helps you to have a better grasp on the materials of the class. I would start working on homework problems closer to the time that they are released, not when it is due. I understand that occasionally there might be circumstances which prevent you from completing a homework assignment by the posted deadline. I believe homework is a valuable and important part of your learning, so I’m happy to grant you a one-time extension to complete an assignment—no excuse needed. If you would like an extension, reach out to me before the homework deadline.

You are encouraged to collaborate with your classmates on homework problems, as they are one of your best learning resources. Take advantage of these discussions as much as possible, but be sure that you understand how to solve each problem afterward and write up your solutions in your own words—developing the skill of writing mathematics is an important part of the course. You are also welcome to use AI tools as a supplement to your learning, for example to brainstorm ideas or explore alternative approaches, but your submitted work must still reflect your own understanding. If you use AI tools, you should clearly acknowledge how you used them. I recommend that you first attempt each problem on your own, and if you are unable to make progress after spending sufficient time, then seek help from your classmates or, where appropriate, from AI tools. Remember that neither classmates nor AI will be available to you during exams, so homework should be treated as essential practice in developing the skills you will need to work independently.

Quizzes

There will be an in-class quiz every Friday starting in the third week of the semester. Each quiz will consist of one problem from the homework due the evening before the quiz. Occasionally, I may need to adjust the homework deadline due to midterms, Fall Break, Thanksgiving, or other scheduling conflicts. When this happens, the quiz will still take place at the beginning of class following the modified deadline.

Presentation

This course includes a presentation component based on homework problems. For this assignment, you will select one problem, write a polished solution in LaTeX, and create a video presentation explaining. In each problem set, certain problems are marked with a ∗; these problems form the pool from which you may choose your presentation topic. You will sign up for one of these problems during the semester using a signup sheet that I will share later.

Before recording your presentation, you should share your LaTeX write-up with me so I can suggest any necessary modifications. The first draft of your written solution is typically due one week after the corresponding problem set deadline. Once your written solution has been approved, you may work on your video presentation, which should be submitted by a deadline coordinated with me.

Exam and grading

There will be two in-class Midterms and a Final Exam. The dates and times are listed below; the location for the final will be announced later. Make sure that you are available at the listed times for the exams. Success on the exams will require a strong understanding of the course materials and homework problems. You should also be prepared to solve new problems during the exams that are comparable in difficulty to the homework assignments.

• Midterm I: Friday, October 10, 11:00-11:50 am, Mallinckrodt, Room 303.
• Midterm II: Monday, November 10, 11:00-11:50 am, Mallinckrodt, Room 303
• Final Exam: Tuesday, December 16, 10:30 am-12:30 pm (The location for the final will be announced later).

If you miss one of the midterms with a valid excuse, your grade for the other midterm will be used as a replacement for the missing midterm grade. The final exam will be cumulative.

Your final grade will be based on your participation and quizzes, homework, midterms and final exam. To compute your homework grade, I will drop the lowest HW grades and two lowest quiz grades. The average of the remaining homework grades/quiz grades will be your final homework/quiz grade. Then I compute the maximum of the following weighted grades:

Use the grading scale below to give your letter grade: