Topology MATH 342 Spring 2026

Topology is the study of properties of spaces and maps that are invariant under continuous deformation — the qualitative geometry captured by notions like continuity, convergence, compactness, and connectedness rather than by distances or angles. In this course we will build a rigorous foundation for these ideas and then use it as a common language for a range of modern examples, with particular attention to configuration spaces and moduli spaces, manifolds and their local-to-global structure, and profinite spaces that arise as inverse limits and in “arithmetic” topological contexts. We will also introduce homology as a first bridge from point-set foundations to algebraic topology, using it to extract computable invariants and to illuminate how spaces are built. Along the way we will emphasize categorical perspectives — functoriality, universal properties, and adjointness — as tools for organizing constructions and revealing the relationships among these different kinds of spaces.

This course does not satisfy the "primary data collection and analysis" requirement.

This course does not have an assigned textbook, but I recommend Topology (2nd ed.) by James Munkres and Topology: A Categorical Approach

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topology.mitpress.mit.edu/

as supplementary texts. I will be writing course notes during the term which will appear on this website.

You are permitted and encouraged to work with your peers on homework problems. You must cite those with whom you worked, and you must write up solutions independently. Duplicated solutions will not be accepted and constitute a violation of the Honor Principle.

You will receive timely feedback on your homework via Gradescope, either from me or the course grader (a mathematics undergraduate). Each homework problem can earn up to five points for mathematical content, and two points for the quality of writing. If your answer is incorrect, this will be reflected in the score, and there will also be a comment indicating where things went wrong with your solution. You are strongly encouraged to engage with this comment, understand your error, and try to come up with a correct solution. You are very welcome to post questions about old homework problems to the Zulip workspace (see below) and talk about them with me in office hours (see the Help section).

We will have two midterm exams and a final exam. The midterm exams will be open book, open notes, and take-home. They will have suggested time constraints and firm due dates, but you will be permitted to spend an arbitrary amount of time on the problems between when the exam is distributed and collected. Calculators, computers, phones, collaboration, the Internet, and chat bots / AI are prohibited during exams.

As members of a communal learning environment, we should all expect consideration, fairness, patience, and curiosity from each other. Our aim is to all learn more through cooperation and genuine listening and sharing, not to compete or show off. I expect diligence and academic and intellectual honesty from each of you. You should expect that I will do my best to focus the course on interesting, pertinent topics, and that I will provide feedback and guidance which will help you excel as a student.

This course will use a Zulip workspace for asynchronous communication. Use the Zulip workspace to ask questions (of me or the class), collaborate on problems, and share resources. The Zulip workspace is an extension of our classroom and the above joint expectations extend to this setting.

You will receive an email invitation to join our Zulip workspace during the first week of classes. Please use channels and threads to keep conversations organized.

Everyone is welcome and encouraged to attend my office hours 13:00-14:00 Tuesday and Thursday each week. These are an opportunity to discuss course content, get help on homework problems, or dig deeper into material. You can also schedule a one-on-one appointment with me via this link

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calendar.notion.so/meet/kyleormsby/office

.

Additionally, every Reed student is entitled to one hour of free individual tutoring per week. Use the tutoring app in IRIS to arrange to work with a student tutor.

The goal of this course is to help you think and solve problems like a mathematician. It is the instructor’s opinion that many uses of the Internet and most uses of AI chatbots (such as ChatGPT, Gemini, Claude, etc.) will impede your progress toward this goal. I encourage you to consider your priorities as a student in this course and, whenever tempted to try to look up answers or ask a chatbot, ask yourself "will this help me achieve my goals, or will it short circuit the learning process?"

It is not acceptable to turn in work copied from the Internet or generated by a chatbot; doing so constitutes a violation of the Honor Principle. Any Internet resources or chatbot sessions used to help generate a homework submission should be cited. In the case of chatbot sessions, you should include the prompt you used in your submission.

If you have a documented disability requiring academic accommodation, please have Disability & Accessibility Resources (DAR) provide a letter during the first week of classes. I will then contact you to schedule a meeting during which we can discuss your accommodations. If you believe you have an undocumented disability and that accommodations would ensure equal access to your Reed education, I would be happy to help you contact DAR.

Your grade will reflect a composite assessment of the work you produce for the class, weighted in the following fashion: 35% homework, 25% final exam, 20% exam 2, 15% exam 1, 5% class participation.

Math is hard, but we’ll get through this together!