Intro to Topology/ Found to Topology
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: Introduction to Topology, T. Gamelin & R.E.Greene, Dover Publ., 2^{nd} edition
Final
Exam: Wednesday, December 11, 3:00pm  5:00pm, please doublecheck with the final exam schedule.
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 (25%), two midterms (50%) and a final exam (25%). The Final Exam score will replace the midterm scores that are lower than the Final Exam score. 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. All grades will
be posted on UNM Learn.
Missed
Exams: Makeup 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.
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 related material including solutions of some problems will be posted on UNMLearn. The general rule is that homework assigned in one week is due at the beginning of the Wednesday class of the following week. The "Extra Problems" listed in the homework assignments each week should not be turned in for grading. No late homework will be accepted. The best ten homework grades only will be counted. You can work together on the homework, but you do need to write up your own solutions in your own words. Feel free to ask me for help or hints if you are stuck on a particular problem for some time. The homework is an opportunity for you to practice solving problems, so talking to me should be your first place for help. Please write your solutions neatly and clearly (no points will be given for work that the reader cannot follow) and staple the sheets.
Homework – Fall 2019 (please check after class as advanced postings of homework could change)
Week 
Date 
Topics/Objectives to be learned 
Homework Problems are due Wednesday the following week (the "Extra Problems' are not due) 

1 
Aug. 19 
1. Metric spaces  examples, open sets. 

2 
Aug. 26 
Closed sets, limits, boundary points, limit points. Completeness, the Baire category theorem. 

3 
Sep. 2 (Labor day) 
Separability. Total boundedness. Compactness. Continuous and uniformly continuous functions. 

4 
Sep. 9 
Contraction principle. Principle of uniform boundedness in a Banach space. 2. Topological Spaces. Continuous function. 

5 
Sep. 16 
Universal properties of quotient and subspace topologies. Properties of continuous functions: restrictions, locality, gluing. Base of a topology. Separation axioms. 

6

Sep. 23 
Urysohn's lemma and Tietze's extension theorem. Uniform limits of continuous functions to a metric space. Compactness. The ArzelaAscoli theorem. 


7 
Sep. 30 
Locally compact spaces. One point compactification. Products of topological spaces  subbase, base and universal property of the product metric. 

8 
Oct. 7
Oct. 1013 
Seperation properties and products. Alexander subbase theorem, Tychonoff's theorem. Exam 1, Wednesday Oct. 9. Solutions Fall Break 
Exam scores are posted on https://learn.unm.edu/ 

9 
Oct. 14 
Connected, locally connected, path connected and locally path connected spaces. 


10 
Oct. 21 
3. Topological manifolds. Quotient spaces  compactness, connectedness, seperation properties, examples. Disjoint unions and gluing along a subspace. Covering spaces. Group action with quotient map a covering; uniqueness of lifts. 

11 
Oct. 28 
The group of deck transformations; covering actions and the group of deck transformations; examples. Lifting of paths and homotopies to a cover. 

12 
Nov. 4 
4. Homotopy Theory  the fundamental group. Homotpic maps and homotopy equivalent spaces. Deformation retracts. 

13 
Nov. 11 
The Monodromy theorem. Fundamental group vs. Aut group of a covering. Brower's theorem. BorsukUlam's theorem. 

14 
Nov. 18 
The fundamental group as obstruction to existence of lifts. Classification of covers, coverings and subgroups of the fundamental group, the universal cover. 

15 
Nov. 25
November 28 
Exam 2, Monday November 25 Solutions The Seiferet  van Kampen theorem. Free products of groups. Thanksgiving 

16 
Dec. 2 
Amalgamated product of groups, presentation of groups, and the Seiferet  van Kampen theorem. Examples. 




The Jordan curve theorem (reference Article) 



Dec. 914 Finals week 
Wednesday, December 11, 3:00pm  5:00pm 