Course information: MA 571, Elementary Topology, Purdue University, Fall 2025
Course description: This is a first graduate course on topology. We will emphasize those aspects of the subject that are fundamental to modern mathematics. The topics will include the basics of point set topology and a brief introduction to some of the ideas of algebraic topology.
Instructor: Prof. Manuel Rivera (manuelr at purdue dot edu), Office: Mathematics Building 708
Textbook: Topology by J. Munkres and the instructor’s course notes
Prerequisites: We will assume familiarity with rigorous proofs and set theory. In particular, we will assume Chapter 1 of the textbook.
Course schedule: Monday, Wednesday, Friday 9:30-10:20pm at Mathematical Sciences Building 215
Topics: We will discuss the following topics
1: Basic notions: topology, open sets, basis; order, product, and subspace topologies
2: Closed sets and limit points, continuous functions, more on product topology
3: Metric topology and quotient topology
4: Connectedness
5: Compactness
6: Function spaces and the compact-open topology
6.5: Normal spaces and Urysohn’s Lemma
7: Manifolds
8: Classification of surfaces [self study]
9: Homotopy, homotopy equivalence
10: Fundamental group, covering spaces
11: Calculating the fundamental group
12: Van Kampen’s theorem
Office hours: W 1:30PM
Homework: Working out exercises by yourself is essential for learning and understanding mathematics. There will be about six problem sets they will be posted on the course website. Each problem set will contain exercises from the book as well as exercises not in the book. Solutions to the exercises can probably be found online. Do not look them up; try each problem as hard as you can and if you get stuck discuss with me or your colleagues. Then write clear and complete solutions to all of them. Collaboration is encouraged but you must write up your own solutions and indicate who you worked with. Homework should be sent directly through email to the grader and the instructor. These can be either hand written or tex’ed.
Here are some “Comments on style” written by J. Munkres and here are some “Guidelines for good mathematical writing” written by Francis Su. Please read these during the first week of the course and keep them in mind while writing your homework and exam solutions.
Problem Set 1 – due September 12
Problem Set 2 – due September 26
Problem set 3 – due October 10
Problem set 4 – due October 24
Problem set 5 – Read the proof of the classification of surfaces
Problem set 6 – due November 30
Problem set 7 – do not turn in, take it as practice for the final.
Ethical Use of Generative AI in this Course
One of the fundamental goals of higher education is to develop one’s understanding of a subject and ability to express original thoughts. While Generative AI (GAI) tools, such as ChatGPT, Google Gemini, DALL-E 2, and others, can aid in providing information and understanding, students are reminded that the value of their education comes from developing their own voice and original ideas. Using AI tools should not overshadow the importance of personal intellectual growth.
In this course, you may utilize GAI as an editor, translator, data visualization tool, or to improve grammar and spelling. GAI can be a powerful learning tool, but like all tools, it has limitations and weaknesses that you should be aware of. GAI can fabricate seemingly credible data and generate wholly inaccurate content that is nonetheless highly persuasive. This is especially true when asking it for references, quotations, citations, and calculations. Therefore, it is imperative that you carefully read and verify GAI-generated content when incorporating it into your learning experience.
Purdue Libraries offers a guide and resources on the use of GAI for research and other uses.
While GAI can be used to enhance your work, inappropriate and/or unethical use of GAI can negatively impact your education and professional development.
Grader: Liam Ashton Morgan (lashton at purdue.edu)
Exams: This course will have a midterm and a final exam. The midterm will be a take home exam. You will have three days to work on the midterm. Consulting sources other than Munkres’ book will not be permitted.
The midterm will be sent through email on October 29 and you must turn in your solutions on or before October 31 at 11:59PM.
The final exam will be an in-person individual oral exam in the instructors’ office during finals week.
Grading scale: Homework 20%, midterm 40%, final exam 40%.
Students who get at least 97% of the total points in this course are guaranteed an A+, 93% guarantees an A, 90% an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C-, 67% a D+, 63% a D, and 60% a D-; for each of these grades, it’s possible that at the end of the semester a somewhat lower percentage will be enough to get that grade.
Special accommodations:
Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247, as soon as possible.
If the Disability Resource Center (DRC) has determined reasonable accommodations that you would like to utilize in this class, you must send your Course Accommodation Letter to the instructor. Instructions on sharing your Course Accommodation Letter can be found by visiting:
https://www.purdue.edu/drc/students/course-accommodation-letter.php
Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.
riveramanuel.com 