Fall 2023
Math 4330: General Topology



Course format

This course will be primarily instructed face-to-face. On rare occasions we may switch to an online format.

Tue/Thu 2:00 pm - 3:20 pm. Bruce and Gloria Ingram Hall 03105.

Course description

This class will be an introduction to the basic language necessary to study topology. Some buzz-words you can Google include: topological spaces, continuity, compactness, metric spaces, connectedness, and Hausdorffness.

Beyond a fluency with the above topics, another goal of this class is for you to become familiar with mathematical thinking---questioning and understanding why definitions exist, identifying when you or another communicator is being precise or imprecise (and for what purpose), developing tastes that are rooted in practice and informed experience, exploring the mathematical landscape on your own.

Prerequisite: MATH 3330 with a grade of C or higher.


Textbook and resources.

You do not need to buy a textbook for this class. The following are freely available resources:

  1. Here is a link to the Fall 2019 version of this course and the Fall 2020 version of this course. There you will find past course notes, past homework assignments, et cetera.
  2. Allen Hatcher's notes on point-set topology.
  3. Stefan Waner's notes on Elementary Topology.
  4. Sidney A. Morris's Topology without Tears.

Resources for proof (all freely available):

  1. Book of Proof by Richard Hammack. You will need to be comfortable with all the material in this book.
  2. Introduction to Proof in Analysis by Steve Halperin. You will need to be comfortable with Chapters 1-3 of this book.
  3. Introduction to mathematical arguments by Michael Hutchings. You will need to be comfortable with all this material.



The syllabus

The syllabus can be found here.

Important dates

Midterm Exam: Thursday, October 5.
Final Exam: Thursday, December 7.

Collaboration policy

I strongly encourage all of you to collaborate. Please do so. If you do, you must indicate clearly on every assignment that you have collaborated, and indicate with whom. However, write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing your homework, you must do so without assistance from another. This is because the act of solving something and writing a mathematical proof are two different skills, and I want you to also hone the latter. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated. To reiterate, you may not write solutions together.



Readings (read before class)

  1. Tue, Aug 22. Sets and power sets.
    Notes from Lecture One, taken by a classmate.
    Notes from Lecture One, taken by another classmate.
  2. Thu, Aug 24. Some important examples in topology.
    Notes from Lecture Two, taken by a classmate.
    Notes from Lecture One, taken by another classmate (coming soon).
  3. Tue, Aug 29. Partially ordered sets.
    Notes from Lecture Three, taken by a classmate.
  4. Thu, Aug 31. Unions and intersections.
    Notes from Lecture Four, taken by a classmate.
  5. Tue, Sep 5. Open subsets of Euclidean space.
    Notes from Lecture Five, taken by a classmate.
    Notes from Lecture Five, taken by another classmate.
    Notes from Lecture Five, taken by yet another classmate.
  6. Thu, Sep 7. Closed subsets.
    Notes from Lecture Six, taken by another classmate.
    Notes from Lecture Six, taken by another classmate.
  7. Tue, Sep 12. Topology, continuity, and homeomorphism.
    Notes from Lecture Seven, taken by another classmate.
    Notes from Lecture Seven, taken by another classmate.
  8. Thu, Sep 14. Subspaces.
    Notes from Lecture Eight, taken by a classmate.
    Notes from Lecture Eight, taken by another classmate.
  9. Tue, Sep 19. Compactness, I. Open covers and finite subcovers.
    Notes from Lecture Nine, taken by a classmate.
    Notes from Lecture Nine, taken by another classmate.
  10. Thu, Sep 21. Compactness, II. Heine-Borel Theorem
    Notes from Lecture Ten, taken by a classmate.
    Notes from Lecture Ten, taken by another classmate.
  11. Tue, Sep 26. Compactness, III. Extreme Value Theorem.
    Notes from Lecture Eleven, taken by a classmate.
    Notes from Lecture Eleven, taken by another classmate.
  12. Thu, Sep 28. Equivalence relations and quotient sets. (Hiro not in class; substitute.)
    Tue, Oct 3. Question/answer/exercise session before exam.
    Thu, Oct 5. Midterm exam
  13. Tue, Oct 10. Quotient spaces. (In-class STEP UP exercise.)
  14. Thu, Oct 12. Product Spaces.
    Notes from Lecture Fourteen, taken by a classmate.
    Notes from Lecture Fourteen, taken by another classmate.
  15. Tue, Oct 17. Hausdorfness.
    Notes from Lecture Fifteen, taken by a classmate.
  16. Thu, Oct 19. Hausdorffness group activity.
    Notes from Lecture Sixteen, taken by a classmate.
  17. Tue, Oct 24. Metric spaces (notes updated).
    Notes from Lecture Seventeen, taken by a classmate.
    Notes from Lecture Seventeen , taken by another classmate.
  18. Thu, Oct 26. Isometries.
    Notes from Lecture Eighteen, taken by a classmate.
  19. Tue, Oct 31. Path-connectedness.
    Notes from Lecture Nineteen, taken by a classmate.
  20. Thu, Nov 2. Invariance of domain and pi-nought.
    Notes from Lecture Twenty, taken by a classmate.
  21. Tue, Nov 7. Connectedness.
    Notes from Lecture Twenty-One, taken by a classmate.
    Notes from Lecture Twenty-One, taken by another classmate.
  22. Thu, Nov 9. STEP UP exercise on connectedness.
  23. Tue, Nov 14. Stereographic projection and one-point compactification. (Exercises have been updated.)
    Notes from Lecture Twenty-Three, taken by a classmate.
  24. Thu, Nov 16. Interiors, neighborhoods.
  25. Tue, Nov 21. Density, closures.
    Notes from Lecture Twenty-Five, taken by a classmate.
    Thu, Nov 23. No class. Thanksgiving Break.
  26. Tue, Nov 28.

    27. Thu, Nov 30. Last day of class.
    Thu, Dec 7. Final Exam, 2 PM - 4:30 PM.


Homeworks

All assignments are due on Canvas, and must be uploaded in PDF format.
All homework assignments may be shared with your classmates, so I recommend that you remove your names from your scanned/typed/written assignments. (When you upload your homework, I will know which assignments belong to whom, thanks to Canvas.)