2025 March 19,

Math 354: Topology

Carleton College, Winter 2026, Dr. Joshua R. Davis, , CMC 324, x4095

Introduction

It's hard to describe what topology is in a simple paragraph, but here's an attempt. Topology is the basic mathematics of space. By "space" I don't mean outer space, although topology is used in physics and astronomy. In topology, we define a notion of topological space. There are infinitely many topological spaces, although we tend to focus on a few kinds that appear frequently throughout math and its applications. By "basic" I don't mean easy; rather, I mean fundamental. Topology is the math of space, upon which other maths of space are built. A topological space has just enough structure to define what a continuous function is. Topology is the study of continuous functions.

Topology comes in three flavors: point-set (which is the core of the subject), algebraic (which connects point-set topology to abstract algebra), and differential (which connects to both calculus/analysis and abstract algebra). The first third of this course surveys the most important topics in point-set topology, including compactness, connectedness, and Hausdorffness. The bulk of the course surveys a classic first topic of algebraic topology, which is the fundamental group and the classification of closed surfaces. At the end of the course, we might glimpse some differential topology too.

The prerequisite for this course is Math 236: Mathematical Structures. We prove a lot of theorems about sets and functions. I am happy to admit students who have taken CS 202: Math of Computer Science instead. Talk to me if you are concerned about your background.

Responsibilities

The College's accreditation says that a 6-credit course is 150 hours of work. That's about 15 hours per week or 5 hours per class meeting. Those 5 hours break down into about 1 hour for class itself and 4 hours for homework, reading, studying, etc. If you find yourself spending much more time, then talk to me.

Participation

Our course meets in CMC 319 during period 3A. That's Monday 11:10-12:20, Wednesday 11:10-12:20, Friday 12:00-1:00. You are expected to attend every class meeting promptly, take notes on paper or a tablet, participate in discussion and group work, and ask and answer questions. You can make up for a deficiency in class participation by talking with me in office hours.

Laptops, phones, photos, and recordings are prohibited (except by special arrangement). Why? This course's material can't be typed easily on a keyboard. Using a laptop or phone measurably distracts the students around you. Photographing a chalkboard is not as educational as taking notes. I want our class to be a safe space, where students don't feel that they're on stage.

It is important that our course be welcoming to all students, regardless of their identities, backgrounds, and experiences. We all sometimes say and do things that make life worse for others, and we should all strive not to. Please let me know if the class feels hostile to you, because of something that I or someone else has done.

Homework

Although class meetings may seem like the core of the course, homework assignments are actually where you learn the material. It is essential that you attempt each homework promptly, before the next class meeting. For then you better understand that next class meeting!

On homework, you are encouraged to figure out the problems with other students. However, you should always write/type your solutions individually, in your own words. You may not copy someone else's work or allow them to copy yours. You may not use AIs such as ChatGPT. Presenting someone else's work as your own is an act of academic dishonesty. Presenting someone else's work as your own is a violation of Carleton's Academic Integrity standards.

Writing is not just for literature and history majors. Written and oral communication skills are essential to every academic discipline and are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that the grader can ascertain whether you yourself understand the material.

Homework is usually due two class meetings after it was assigned. This schedule tries to keep you on track, so that work doesn't pile up. But we all have bad weeks, where we can't get everything done, right? If you need to submit an assignment late, then write "LATE" at the top of the front page and submit it as soon as you can. If the grader hasn't graded the assignment yet, then they can grade your paper with the others for full credit. If the grader has graded the assignment already, then you might not get credit. There are limitations to how much delay and complication a grader can handle.

When handing in an assignment, please mark it with the day that it was assigned (e.g., "Day 11") and staple your pages into a single packet, in the correct order. Is there a stapler in the classroom? Often not, so staple ahead of time. Is a paper clip just as good? Sorry, no.

Depending on time constraints in any given week, perhaps not all of your homework will be graded. In order to ensure full credit, do all of the assigned problems by their due date. As far as your grade is concerned, homework serves primarily as the first step in studying for exams.

Exams

We have three exams. They are given roughly 1/3, 2/3, and 3/3 of the way through the term. The last exam is given during our official final exam period, which is Saturday March 14 8:30-11:00 AM in our usual room. I haven't decided whether any exams are take-home.

Are the exams cumulative? No and yes. No, in that each exam is focused on the material that has not been tested yet. Yes, in that much of the material is inherently cumulative. Also, the last exam might have some questions that explicitly address material from earlier in the course.

Grading

For better or worse, we are required to measure your learning using grades. Your numerical grade is based on the responsibilities above: participation 5%, homework 10%, Exam A 25%, Exam B 25%, Exam C 35%.

Numerical grades are converted to letter grades only at the end of the term. Grades are not curved, so students are not in competition with each other. Grades are also not based on predetermined percentages (90%, 80%, 70%, etc.), because Math 354 problems are difficult to tune so precisely. (I could accidentally write a difficult exam and wreck everyone's percentages.) Rather, I assign grades by comparing the students to the course goals. For example:

• A student who earns an A is not perfect, but they can speak fluently about nearly all of the course content. They exceed expectations.
• A student who earns a B learns most of the big concepts pretty well, but they are missing some concepts. They lack the depth of understanding needed to appreciate subtleties in proofs, etc.
• A student who earns a C gets some of the material but has significant gaps in their understanding of major concepts or their ability to prove important theorems.

Talk to me, if you are concerned about your grade.

Resources

I want all of my students to work hard and learn a lot. I try to give them all of the resources that they need. Here are the basics:

• Our textbook is Topology by Munkres, 2nd edition. This is a deservedly popular introductory topology textbook. It is often, but not always, the textbook of Math 354. It gives a thorough overview of point-set topology and the fundamental group. It covers more material than we have time for. It will serve you well in the future, if you continue in mathematics.
• Here are the exams from this term, annotated with quartiles of student scores.
  • Exam A, Solutions (41/48, 37/48, 29/48)
  • Exam B, Solutions, In-Class Portion, Solutions (48/54, 46/54, 40/54)
  • Exam C, Solutions (49/56, 46/56, 34/56)
• Here are the exams from my winter 2024 edition of this course.
  • Exam A, Solutions (46/48, 42/48, 38/48)
  • Exam B, Solutions (36/39, 31/39, 28/39)
  • Exam C, Solutions (45/48, 42/48, 37/48)
• Groups tutorial

Remember that I encourage you to solve problems with classmates (even if the work that you submit must be your own). If you want help in finding a study partner, then e-mail me, perhaps describing some of your habits: working in the middle of the night, not waiting until deadlines, etc.

I hold several office hours per week. Our grader, Luca, holds one office hour too. No appointment is needed; just drop in! Here's our schedule:

If you want to meet with me but can't make office hours, then consult my weekly schedule and e-mail me, listing several possible meeting times.

If a health condition or other personal matter affects your participation in class, homework, exams, etc., then please let me know as soon as possible. Depending on the situation, we might want to confer with Accessibility Resources, Assistive Technology, Student Health and Counseling, or Title IX. When you ask me to help, I do my best to help. :)

Schedule

This schedule is tentative. It is tweaked and filled in as the course progresses.

To help you decode the schedule below, here is an example. Monday January 5 is Day 01 of the 28-day course. During class, I give an intuitive overview of topology. You have a homework assignment called "Day 01", which you should attempt immediately, but which is due at the start of class on Day 03. You are expected to skim Sections 1-7 of the textbook after (or before) class.