Math 147: Topology Spring 2025

Professor Francis Su
Spring 2025
MW 1:15pm in Shanahan 3485

My Office: Shanahan 3416
My Email: (my last name) at math.hmc.edu
My Drop-In Hours:
Wednesdays 2:30-4pm or by appointment.

Grader: Shreya Balaji (sbalaji) at g.hmc.edu

Prerequisites: Math 131 (analysis). Math 171 (algebra) is recommended as a co-requisite.

Topology is the study of properties of objects preserved by continuous deformations (much like geometry is the study of properties preserved by rigid motions). Hence, topology is sometimes called “rubber-sheet” geometry. This course is an introduction to point-set topology with additional topics chosen from geometric and algebraic topology, including: topological spaces, separation properties, compactness, connectedness, and the fundamental group.

Inquiry Based Learning

This course will be taught in an IBL (inquiry-based learning) format. This is a guided discovery process, which can be one of the most rewarding mathematical experiences you’ll have. We learn best when we discover things for ourselves. The book contains theorems, definitions, examples structured so that you’ll naturally discover things. Your goal is to prove a collection of theorems in collaboration with classmates and with guidance from me.

You will take turns presenting proofs of theorems in class, while other students will determine if it is correct. I will provide perspective on the material, and motivating examples if you get stuck.

Text

The text for the course is Topology through Inquiry by Michael Starbird and Francis Su. The publisher sells a hardcopy and e-book version here. Or compare here.

Course Structure

NO OUTSIDE SOURCES: As is customary in any IBL course, you are forbidden from consulting any outside sources, including textbooks, the internet, or artificial intelligence to solve these problems. [Exceptions are solutions that you encounter while doing work for other courses.] The fun of the course is learning how to wrestle with challenging ideas. Consulting outside sources stifles the development of your own creativity.

NOTEBOOK: You should create a notebook (a loose-leaf binder works well for this). As you prove theorems in this course (or see them proved in class), you will write up these proofs and add them to this notebook. In a sense, you are writing your own book on the subject, filled with your own proofs. As part of your grade for the course, these notebooks will be evaluated. More importantly, a nice notebook will be something you can be very proud of years from now, as you look back on your experience.

HOMEWORK: Though you should be attempting the proof of many theorems, you will be asked to write up 5 selected problems to be handed in weekly via Gradescope. Therefore it will be important to pay attention to proofs presented in class, as you will write them up for credit. Homeworks will be due Thursdays at 5pm.

EXAMS: There will be no midterm, but there will be a final examination.

GRADES: I’d rather you not worry about grades. I hope you will concentrate on learning. But since we must, grades will be determined in the following ways. 50% of your grade will be specifications-based. This means that you must meet each of the following specifications to pass the course:

– Make class presentations at least 12 times (roughly once/week).
– Participate in class discussions at least 12 times (roughly once/week).
– Share a story of a “productive failure” in class at least 3 times (e.g., lessons you learned from trying things that didn’t work out).
– Keep a well-organized notebook, which will be turned in 2/3 of way through the semester. These will be graded exceptional pass/pass/no pass.

Homeworks (30%) and Final Exam (20%) will count the remaining portion of your grade.

The learning you are doing in this class takes place in a larger framework of school and life. Even if I am excited about teaching and you are excited about learning, work is not the most important thing, and sometimes life can take precedence. I can be somewhat flexible in accommodating requests for homework extensions and absences for other important events. Please make these requests 24 hours in advance, if possible.

Similarly, ‘success’ by whatever measure is not the most important thing in this course either. Every assessment of your work in this class is a measure of progress, not a measure of promise. Joy, wonder, productive struggle, having your mind expanded—these are more important!

Honor Code

The HMC Honor Code applies in all matters of conduct concerning this course. Cooperation is encouraged. You may not consult outside mathematical sources without my permission, unless required for some other course. Your solutions should acknowledge the assistance of other people or resources of any kind.

You can expect this course to be challenging and rewarding. The value of the IBL format is that when you prove theorems by yourself, you will never forget the proofs you came up with, and you will gain confidence in your abilities as mathematicians!

Homework Assignments

Due on Gradescope by 5pm on the assigned day, generally Thursdays.

You may find these LaTeX files of all the theorems in the book handy.
(The main tex file in this folder is topology-thms.tex.)

There’s also a PDF of all theorem statements: one per page, or all together.

Remember the rule of thumb is to keep 10 problems ahead of where we end the prior class. In general:
– All theorems (in assigned sections) should be attempted.
– Exercises are encouraged, as they lead to greater understanding.

Week 1: (just Wednesday) we discussed goals of course and upcoming ideas.
Week 2: We discussed 2.1-2.7, 2.8-2.14.
Week 3: through 2.31.
Week 4: through 3.10.
Week 5: through 3.31 ( skip all theorems/exercises involving ordinal numbers).

Upcoming:

Here are a list of theorems/exercises to focus on, and try to stay 10 ahead on this list for each class session.
Be sure to read all theorem/exercises, even ones you skip, as you may see interesting ones to work. Exercises are good for building intuition and can often be useful in proving later Theorems.
Bolded problems are assigned in one of the HW’s below.
Starred problems* will not be discussed in class, but one of the final exam problems will be to solve a starred problem of your own choosing.

Chapter 3 ( 34, 36, 39, 41 )

Chapter 4 ( 1, 4, 5, 6, 7, 8, 14, 16, 17, 18, 23, 29)

+ more added later

READING:

Read my handout on good mathematical writing.
In the text Topology Through Inquiry,
– read the Introduction (pp. 1-5) and Sections 2.1 (pp. 27-30),
– review Section 1.1 if needed, for notation and useful theorems you can cite.

ASSIGNED PROBLEMS:

Via Gradescope, turn in the following:
Question A (below)
and Chapter 2( 3, 5, 8, 13 ).

Question A. Briefly summarize three things that stood out to you from readings above.

Your solutions should include the statements of the assigned problems. For LaTeX’ed problems, see the LaTeX files linked above. For handwritten problems, you might print out pages with the desired theorem statements: one per page.

READING:

Review Section 1.2 and remind yourself of theorems there (they should be review from Analysis.)
In Chapter 2, try to keep 10 problems ahead of where we ended the last meeting.
Especially do all Theorems. Exercises are encouraged, as they lead to greater understanding.

ASSIGNED PROBLEMS:

Via Gradescope, turn in the following:
Chapter 2 ( 14, 20, 22, 23, 26 )

Remember: use prior theorems (including Theorems in Chapter 1) to make your proofs simpler!

READING:

Read Sections 3.1-3.3. Rule of Thumb: Keep 10 problems ahead of where we ended the last time. Especially do all Theorems. Exercises are encouraged, as they lead to greater understanding.

ASSIGNED PROBLEMS:

Via Gradescope, turn in the following:

Chapter 2 ( 28 ) Chapter 3 ( 3, 8, 9, 10 [part 4 only] ).

Note that the LaTeX file for Chapter 3 may differ (in wording and in theorem numbers) from what’s in book, so please refer to the book as you do these problems (and don’t necessarily trust the LaTeX file).

READING:

Read Sections 3.4 and 3.5. Rule of Thumb: Keep 10 problems ahead of where we ended the last time. Especially do all Theorems. Exercises are encouraged, as they lead to greater understanding. See guidance above HW#1.

ASSIGNED PROBLEMS:

Via Gradescope, turn in the following:

Chapter 3 ( 16, 21 [D only], 25, 28, 31 ).

I have updated the LaTeX files above to fix the earlier inconsistencies, but it’s always good to check them against the book.

READING:

Read Sections 3.6, 4.1, 4.3. Work on the specific problems outlined in the tab above HW#1.

ASSIGNED PROBLEMS:

Via Gradescope, turn in the following:

Chapter 3 ( 36, 39, 41 ). Chapter 4 ( 6, 8).

I have updated the LaTeX files above to fix the earlier inconsistencies, but it’s always good to check them against the book.

Below this line, all homeworks are TENTATIVE. This means they are likely to be assigned, but there is no guarantee that they will until you see them moved above this bar.