Intro to Topology/ Found to Topology Fall 2023
Type  Time  Days  Where  Date Range  Schedule Type  

FacetoFace Plus  11:00 am  12:15 pm  T  Science Math Learning Center 352  Aug 21, 2023  Dec 16, 2023  Lecture  
Remote Set Day/Time  11:00 am  12:15 pm  R  Remote Instruction MS Teams  Aug 21, 2023  Dec 16, 2023  Lecture 
Please note the following information. This page will not be updated beyond August 28, 2023. You should join the class MS Team and follow the information there. You need to be registered for the course with a @unm.edu email. Any other email will disable features of Microsoft Teams. I have sent you by email a link to join the class team. Please follow it in order to request access. You can find the link to join the team in Canvas as well. I will drop from the class all students who do not request access to Microsoft Teams by Friday, September 9 or do not come to class during this period. The remote class on Thursday will be through MS Teams. Homework, including submission, and the commucation of all course information will rely on MS Teams and OneNote.
Instructor: Dimiter Vassilev Office: SMLC 326 Email: vassilev@unm.edu Phone: 505 277 2136
Math 431 Introduction to Topology & Math 535 Foundations of Topology
Description: Metric spaces, topological spaces, continuity, algebraic topology. Basic point set topology. Separation axioms, metric spaces, topological manifolds, fundamental group and covering spaces.
Prerequisites: MATH321. Pre or co requisites: MATH322.
Text: 1. Munkres J. Topology (Pearson) 2nd ed 2. Introduction to Topology, Th. Gamelin & R.E.Greene, Dover Publ., 2nd
Final
Exam:TBD, please doublecheck with the final exam schedule.
Attendance: Attendance at UNM is mandatory, see policy.
Grades:
Accommodation Statement: In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. Accessibility Resources Center (Mesa Vista Hall 2021, 2773506) provides academic support to students who have disabilities. If you think you need alternative accessible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner.
Homework: Homework is due every Thursday at the beginning of the class in MS Teams. 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 turnin late homework! The syllabus also lists recommended 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 .
Week 
Date 
Topics/Objectives to be learned 
1 
Aug. 21 
1. Metric spaces  examples, open sets. Closed sets, limits, boundary points, limit points. Completeness, the Baire category theorem. 
2 
Aug. 28 
Separability. Total boundedness. Compactness. Continuous and uniformly continuous functions. 
3 
Sep. 4 (Labor day) 
Product Metric Spaces. Bounded/ Continuous linear operators, Principle of uniform boundedness in a Banach space. Contraction principle. 
4 
Sep. 11 
2. Topological Spaces. Continuous function. Universal properties of quotient and subspace topologies. Uniform limits of continuous functions to a metric space. 
5 
Sep. 18 
Properties of continuous functions: restrictions, locality, gluing. Base of a topology. Separation axioms. Urysohn's lemma and Tietze's extension theorem. 
6

Sep. 25 
Finish Tietze's extension theorem. Compactness. The ArzelaAscoli theorem. Locally compact spaces. One point compactification. 
7 
Oct. 2 
Products of topological spaces  subbase, base and universal property of the product metric. Separation properties and products. Alexander subbase theorem, Tychonoff's theorem. 
8 
Oct. 9

Connected, locally connected, path connected and locally path connected spaces. Fall Break Oct. 1215 
9 
Oct. 16 
Connectedness and products. Paracompactness. Metrizability. Exam 1, Thursday Oct. 19. 
10 
Oct. 23 
3. Topological manifolds. Quotient spaces  compactness, connectedness, separation properties, examples. Covering spaces. Group action with quotient map a covering. 
11 
Oct. 30 
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. 
12 
Nov. 6 
4. Homotopy Theory  the fundamental group; induced homomorphism. Homotopic maps. Deformation retracts. 
13 
Nov. 13

Homotopy equivalent spaces. The Monodromy theorem. Fundamental group vs. Aut group of a covering. Brower's theorem. BorsukUlam's theorem.

14 
Nov. 20 
The fundamental group as obstruction to existence of lifts. Classification of covers, coverings and subgroups of the fundamental group, the universal cover.
Thanksgiving 
15 
Nov. 27

Exam 2, Tuesday November 28 The Seiferet  van Kampen theorem. Free products of groups. 
16 
Dec. 4 
Amalgamated product of groups, presentation of groups, and the Seiferet  van Kampen theorem. Examples. 




Dec. 1116 Finals week 
Final Exam, Tuesday, December 12, 12:30pm2:30pm, https://schedule.unm.edu/finalexams/final_exam/fall2023.pdf 