Introduction to Topology - 76733 - MATH 431 - 001 & Foundations of Topology - MATH 535 002, Fall 2025

(this page will not be updated after the first week of classes; use MS Teams for an up to date information)

Please note the following information. The communication of all course information, including the homework,  will rely on MS Teams and the associated OneNote. You need to be registered for the course with a @unm.edu email. Any other email will disable features of Microsoft Teams. I will send you by email a link to join the MATH431-535F25 team. Alternatively, you can use the link in the course page on Canvas to request access. 

Instructor: Dimiter Vassilev Office: SMLC 326 Email: vassilev@unm.edu Phone: 505 277 213

Class Time: 16:00-17:15 MW. Location: DSH - 229

Office Hours: MW 1:30pm - 2:30pm. Feel free to stop-by anytime if you have a quick question.

Description: Metric spaces, topological spaces, continuity, algebraic topology. Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces. 

Prerequisites: MATH431 Undergraduate level MATH 321 Minimum Grade of C and Undergraduate level MATH 322 Minimum Grade of C . 

Texts: 1. Introduction to Topology, Gamelin & R. E. Greene, Dover Publ., 2nd ed; 2. Munkres J. Topology (Pearson) 2nd ed. Course Materials Access (the following is a required UNM information only, see also https://coursematerialsaccess.unm.edu/#Complete): Your digital course materials are directly available now on the My Shelf link in Canvas. Your physical course materials, such as books and required lab/studio course kits, are available at the UNM Bookstore, and you will receive an email about how to pick them up. To simplify your course materials access, you are automatically enrolled in a Complete option at a flat rate of $279 per semester. This will show up on your bursar bill. The Complete option covers all your required course materials for all your Albuquerque campus courses, including any graduate courses you may be taking (branch campus course materials are billed and available separately). If you are interested in course materials access for only selected courses, or if you want to opt out entirely, you will need to select the option you want in the My Shelf link in Canvas. You can change your selected option in the My Shelf link in Canvas until the registrar’s “Last Day to Drop Without a ‘W’ Grade and 100% Tuition Refund.” Make sure that you review the video and information here to understand cost and the options for Complete (automatic enrollment), Select (take action), and Opt-out (take action).

 

 

 

 Please note the following guidelines for the course:

Attendance: Attendance at UNM is mandatory, see policy.

Grades: The final grade will be determined by homework (20%), two midterms (50%) and a final exam (30%). The Final Exam score will replace the midterm scores that are lower than the Final Exam score.   There will be some opportunities for bonus points which will only be announced in class. If you are not present you might miss these opportunities.  Although a small curve will be used, 90% , 80% or 70% of the possible maximum points guarantees at least an A, B or C, respectively. The students registered in MATH431 will have a more generous curve.

 Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given unless there was an emergency.

 

Accommodations: UNM is committed to providing equitable access to learning opportunities for students with documented disabilities. As your instructor, it is my objective to facilitate an inclusive classroom setting, in which students have full access and opportunity to participate. To engage in a confidential conversation about the process for requesting reasonable accommodations for this class and/or program, please contact Accessibility Resource Center at arcsrvs@unm.edu or 505-277-3506.

UAP 2720 and 2740. Our classroom and university should foster mutual respect, kindness, and support. If you have concerns about discrimination, harassment, or violence, please seek support and report incidents. Find confidential services at LoboRESPECT Advocacy Center, the Women’s Resource Center, and the LGBTQ Resource Center. UNM prohibits discrimination on the basis of sex (including gender, sex stereotyping, gender expression, and gender identity). All instructors are “responsible employees” who must  communicate reports  of sexual harassment, sexual misconduct and sexual violence to Compliance, Ethics and Equal Opportunity. For more information, please see UAP 2720 and UAP 2740.

Credit-hour statement: This is a three-credit-hour course. Class meets for two 75-minute sessions of direct instruction for fifteen weeks during the Fall 2025 semester. Please plan for a minimum of six hours of out-of-class work (or homework, study, assignment completion, and class preparation) each week.

Homework:   Homework is due every Wednesday at the beginning of the class. I encourage you to work on the homework with your classmates, but you are required to write up your own solutions in your own words. To help the grader, please write your solutions neatly using correct grammar and mathematical notation (no points will be given for work that the reader cannot follow).   The ten best homework grades will be used in computing the homework score. Please do not turn-in late homework! The syllabus also lists recommended/ extra homework problems.  These are NOT to be handed in. Work as many as it takes for you to understand the material.  You should see me as early and as often as necessary if you are having difficulties with the homework problems .

Topics/Objectives to be learned
1. Metric spaces - examples, open sets. Closed sets, limits, boundary points, limit points. Completeness, the Baire category theorem. Separability. Total boundedness. Compactness. Continuous and uniformly continuous functions. Product Metric Spaces. Bounded/ Continuous linear operators, Principle of uniform boundedness in a Banach space. Contraction principle.
2. Topological Spaces. Continuous function. Universal properties of quotient and subspace topologies. Uniform limits of continuous functions to a metric space. Properties of continuous functions: restrictions, locality, gluing. Base of a topology. Separation axioms. Urysohn's lemma and Tietze's extension theorem. 
3. Compactness. The Arzela-Ascoli theorem. Locally compact spaces. One point compactification. 
4. Products of topological spaces - sub-base, base and universal property of the product metric. Separation properties and products. Alexander subbase theorem, Tychonoff's theorem.
5. Connected, locally connected, path connected and locally path connected spaces. Connectedness and products.
6. Paracompactness. Metrizability.
7. Topological manifolds.
8. Quotient spaces - compactness, connectedness, separation properties, examples. 
9. Covering spaces. Group action with quotient map a covering. Uniqueness of lifts. The group of deck transformations; covering actions and the group of deck transformations; examples. Lifting of paths and homotopies to a cover.
10. Homotopy Theory - the fundamental group; induced homomorphism. Homotopic maps. Deformation retracts. Homotopy equivalent spaces. The Monodromy theorem. Fundamental group vs. Aut group of a covering. Brower's theorem. Borsuk-Ulam's theorem. The fundamental group as obstruction to existence of lifts. Classification of covers, coverings and sub-groups of the fundamental group, the universal cover.  The Seiferet - van Kampen theorem. Free products of groups. Amalgamated product of groups, presentation of groups, and the Seiferet - van Kampen theorem.