Math 751 - General Topology, Spring 2017 Semester

The final exam is available here and its solutions are here. It is a take-home exam: The exam is due by 7:30pm on Monday, May 22 (either by e-mail or in person; I'll be in my office at 7:30pm on May 22).

The final exam is cumulative and covers the following topics:
Ch 1.1-1.6, 2.1-2.6, 2.8-2.13, 3.1-3.2. 
The exam consists of ten problems. Two of these problems are multiple choice questions where you only need to provide the answers, without explaining why those answers are correct. There are also eight technical/proof type problems, where you need to provide complete and detailed solutions.

For those of you doing the optional course projects, we'll do the project presentations during the last week of classes, May 15-May 19.
Here is the schedule for the project presentations (based on the Doodle poll)

Monday, May 15, HE 920, 7pm-9:30pm
1. David Meretzky
2. Yoseph Feit
3. Weiyan Lin

Wednesday, May 17, HE 921, 1pm-3pm
1. George Soloveychik
2. Sardar Singh

Thursday, May 18, HE 1042, 5pm-9pm
1. Jarret Petrillo 
2. Harrison Tietze
3. Kevin Fellner 


The midterm exam has been graded and the scores have been posted to Blackboard.

Starting from Monday, Feb 27 our class will meet in HE 920.

Instructor: Prof. Ilya Kapovich  

I am an Ada Peluso Visiting Professor at Hunter for the Spring 2017 semester. Here is a link to my main page at the University of Illinois at Urbana-Champaign

Class time and place: Mondays, 7:35pm - 9:25 pm, room HE 922 HE 920
My e-mail:  (preferred way of reaching me)

Office phone: 212-772-5303

Main TextIntroduction to Topology (2nd Edition),  by Theodore Gamelin and Robert Greene; Dover 1999

Office hours:  Mondays, 5pm-7pm, HE 917

Website of the course:

Course info:

1. There will be weekly homework collected at the start of each class. The h/work will be both collected and graded (several problems from each h/wk will be graded). At the end of the course one low h/work score will be dropped for each student.

2. There will be one midterm in-class exam, which is tentatively scheduled for Monday, March 20.

3. The final exam for this course is scheduled for Monday, May 22, 2017, 7:35pm - 9:25 pm in HE 922

4. Before every class you will receive a reading assignment for that class; they will be posted about a day before every class online.  While these reading assignments are not mandatory, I highly recommend that you do them as it will make it easier for you to follow the course.

5. In addition to the required course components (midterm, final, h/wks, worksheets), you can earn extra credit by doing a course-related project. These projects are optional.  A project involves writing a 5-8 page project paper and doing a blackboard presentation (approximately 30 minutes long) on that paper. 

6. How this course is graded: The final exam counts as 45% of the grade, the midterm exam is 28% of the grade, the h/wk is 27% of the grade. An extra credit course project counts as an additional possible 12% of the grade. 


  1.  H/wk 1, due Monday, February 6 is available here. and its partial solutions are here.
  2.  H/wk 2, due Wednesday, February 15 is available here and its partial solutions are here. 
  3.  H/wk 3, due Monday, February 27 is available here  and its partial solutions are here.
  4.  H/wk 4, due Monday, March 6 is available here and its partial solutions are here.
  5.  H/wk 5, due Monday, March 13: Ch 2.1, problems 1, 3, 4, 5, 6, 8, 9, 12 (on pp. 63-64 in the book); see partial solutions here.
  6.  No h/wk is due on Monday March 20 because of the midterm exam. The midterm exam is available here.
  7. H/wk 6, due Monday, March 27: Ch 2.4, problems 5, 6; Ch 2.5 problems 4, 5, 8, 9; Ch 2.6 problems 6, 7. Partial solutions are available here.
  8. H/wk 7, due Monday, April 3, is available here and solutions are available here.
  9. H/wk 8, due Thursday, April 20, is available here and its solutions are available here.
  10. H/wk 9, due Monday, April 24, is available here and its solutions are available here.
  11. H/wk 10, due Monday, May 1, is available here and its solutions are here.
  12. H/wk 11, due Monday, May 8, is available here and its solutions are here.

Pre-class reading assignments:
  1. For Monday, February 6: Ch 1.1-1.2 in the book, and the handout on complex numbers.
  2. For Monday, February 13: Ch 1.2 and 1.4 in the book.
  3. For Monday, Feb 27: Ch 1.5 in the book (compactness), and the handout on cardinality and countability
  4. For Monday, March 6: Ch 2.1 in the book (topological spaces).
  5. For Monday, March 13: Ch 2.3 (continuous functions) and Ch 2.4 (base for a topology) in the book
  6. For Monday, March 20: Ch 2.5 (separation axioms), Ch 2.6 (compactness)
  7. For Monday, March 27: Ch 2.8 (connectedness) and Ch 2.9 (path connectedness)
  8. For Monday, April 3: Ch 2.9 (path connectedness) and Ch 2.10 (product topology)
  9. For Thursday, April 20: Ch 2.11 (axiom of choice and Zorn's lemma)
  10. For Monday, April 24: Ch 2.13 (quotient topology), and the handout in equivalence relations
  11. For Monday, May 1: Ch 3.1 (groups) and 3.2 (homotopic paths).
  12. For Monday, May 8: Ch 3.3 (the fundamental group) and Ch 3.4 (induced homomorphism)

Approximate Syllabus:

  1. Definition of a metric space; open and closed sets; limits of sequences.
  2. Completeness for metric spaces.
  3. The real line; least bound axiom and completeness of the real line.
  4. Products of metric spaces.
  5. Compactness; Heine-Borel Theorem; separability.
  6. Continuous functions.
  7. Normed vector spaces (time permitting)
  8. Topological spaces: definition, examples, basic properties.
  9. Subspaces and subspace topology.
  10. Continuous functions between topological spaces.
  11. Base for a topology.
  12. Separation axioms; Hausdorff spaces.
  13. Compactness for topological spaces.
  14. Connectivity and path connectivity.
  15. Finite product spaces.
  16. Set theory: axiom of choice and Zorn's lemma.
  17. Infinite product spaces (time permitting)
  18. Quotient spaces; gluing constructions.
  19. Groups
  20. Homotopy for continuous maps and for paths.
  21. The fundamental group.
  22. Covering spaces; index of a loop.
  23. Applications of index.
  24. Time permitting: Maps into the punctured plane and the Fundamental Theorem of Algebra; Vector fields and the Jordan Curve Theorem.


My March 15, 2015 talk at the Hunter College GRECS Seminar, Groups as Geometric objects