Columbia University GU4051 section 001
Topology, fall 2022

Basic information

Call number: 12826
Room/Time: TuTh 2:40pm--3:55pm, 417 Math
Instructor: Mikhail Khovanov
Office: 620 Math
Office hours: Tentative: Zoom Wednesdays 3-4pm and Office (620 Math) Thursdays 1-2pm
Graduate Teaching Assistant:    Song Yu,   email:
Graduate TA office hours: Wed 12-3pm. Sept 14, 21 via Zoom, then in Help Room, 406 Math
Undergraduate Teaching Assistant:    TBA
Undergraduate TA office hours: TBA in Help Room, 406 Math
Midterm 1: Thursday, October 6
Midterm 2: (Tentative) Thursday, November 17

Prerequisites: The first semester of "Introduction to Modern Algebra" (Math W4041) or equivalent is strongly advised, since we'll be using basic group theory in the second half of the course. You also need to be familiar with how to write proofs. Familiarity with analysis (Honors Math A,B or Introduction to Modern Analysis) is recommended as well.

Textbook: Topology, by James R. Munkres, second edition. Available on Amazon. We are using the more expensive Pearson edition (listed around $100 plus tax). Cheaper (and probably identical except for the paper quality) Prentice-Hall edition is available on Amazon as well. To be on the safe side, my recommendation is to get Pearson edition (first link).

Syllabus: Topological spaces and continuous maps. Examples of topologies.
Closed sets, interior, limit points. Hausdorff spaces.
Product topology.
Connectedness and path-connectedness.
Metric spaces and their topology.
Countability and separation axioms
Normal spaces, the Uryson lemma.
Homotopic maps.
The fundamental group. Examples.
Retraction and homotopy equivalence.
Groups via generators and relations.
Free groups and amalgamated products.
The van Kampen theorem and its applications.
Surfaces and their classification.

Lecture 1 (Tu, Sept 6): Topology, basis for a topology (Sections 12, 13). Notes
Lecture 2 (Th, Sept 8): Order topology, product topology, subspace topology (Sections 14, 15, 16) Notes
Lecture 3 (Tu, Sept 13): Closed sets, closure and interior of a set, limit points, Hausdorff spaces (Section 17) Notes
Lecture 4 (Th, Sept 15): Continuous functions, homeomorphisms (Section 18) Notes
Lecture 5 (Tu, Sept 20): Topology on the image of a continuous map, metric spaces
Lecture 6 (Th, Sept 22): More metric spaces, product topology for infinite products (Sections 19,20), Cantor set as an infinite product. Notes
Lecture 7 (Tu, Sept 27): Continue with Sections 19-21. More about box and product topologies. Uniform topology. Convergence of sequences. First-countability of metric spaces. Notes
Lecture 8 (Th, Sept 29): Clarification about uniform topology. Continuity of sums and products of functions. Start on connected spaces (Sections 23, 24). Notes

Homework: Homework will be assigned on Thursdays, due Thursday evening the next week, via an online upload on Courseworks. The first problem set is due September 15. The lowest homework score will be dropped.
The numerical grade for the course will be the following linear combination:
22% homework, 20% each midterm, 35% final, 3% quizzes.

Homework 1 Due September 15.      

Homework 2 Due September 22.      

Homework 3 Due September 24.