Math 147: Topology – Francis Su

Professor Francis Su
Spring 2023
MW 2:45pm in Shan 2440

My Office: Shanahan 3416
My Email: (my last name) at math.hmc.edu
My Drop-In Hours:
Mondays 4-5pm (in person, right after class)
Also available by appointment via Zoom on Thursdays or Fridays

Graders: Clay Adams and Elizabeth Lucas-Foley (ccadams, elucasfoley)

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 I expect will 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. If you don’t yet have the book, check here for access to the first chapters.

Sakai

All course notes and other private information for the course will be posted on the course Sakai page.

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 sources that you are required to use for other courses.] The fun of the course is in struggling with ideas and enjoying challenging problems that will stretch your mind.

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 theorem proofs to be handed in weekly via Gradescope. Homeworks will be due Tuesday evenings at 10pm. The first 3 homeworks will be group assignments (I will assign the groups, you’ll turn in one assignment for the group via Gradescope). The purpose of group assignments to: get to know others and learn from one another and to discuss and agree what you will included in your solutions. I encourage you to meet with your group on Thursday or Friday to work on problems together. After the first 3 homeworks, you are encouraged to continue working in groups of your own choosing.

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!

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:

– Present in class 6 times over the course of the semester.
– Contribute 12 questions/comments in class over the course of the semester.
– Among presentations/comments: 5 should be “productive failures”: such as lessons learned from trying things that didn’t work out.
– Contribute or edit 12 proofs in the Common Solution Set [which will be sent via email].
– 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.

[The number 12 basically means once per week, and 6 means once every 2 weeks.]

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

Honor Code

The HMC Honor Code applies in all matters of conduct concerning this course. Cooperation is encouraged. On group assignments, all members should be actively engaged in producing written work (and we will discuss the best way to approach group work).

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.

Expect this course to be challenging, but also very 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!

Homeworks

Due on Gradescope on Tuesdays at 10pm. For group assignments only one person needs to submit, but you must tag the other people in your group.

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.)

Remember the rule of thumb is to keep 10 theorems ahead of where we end the prior time.

Chapter 2 ( 1 – 24, 26 -28, 30-32 ).

Section 3.1 ( 1 – 5, 7 – 12 ).

Read Section 3.2, but you don’t need to work on the problems. We will need the idea of a subbasis later so you should know the definition.

Section 3.3 ( 20 – 21 )

Section 3.4 ( 25 – 26, 28 – 31 )

Section 3.5 ( 33 – 37, 41-42 )

Section 4.1 ( 1 – 11, 14 )

Section 4.2 ( 16 – 18 )

Section 4.3 ( 20, 23 )

Section 5.1 ( 1, 4 – 6 )

Section 5.2 ( 9 , 11 )

Section 5.3 ( 14, 15, 18 )

You might read Section 5.4 and 5.5, but I won’t assign any problems in it.

Section 6.1. Many of these are review from Analysis. ( 3, 5, 8, 9, 12 )

Section 6.2. Review (but skip the exercises, which are review from Analysis).

Section 6.3. ( 18, 19, 21*, 23* ). * means an extra level of difficulty, see book discussion.

You might skim the rest of Chapter 6, but I won’t assign any problems in it.

Section 7.1 ( 1 – 4, 6, 9 – 10, 13 ). Many of these are review from Analysis.

Section 7.2 ( 15, 20, 24 ).

Section 7.3 ( 26, 28, 29, 31 ).

Section 7.4 Read 32 – 36, 41 – 42.

Section 7.5 Read all theorems but especially focus on ( 45 (part 3), 46, 48, 53 ).

Section 7.6 Read theorems 57 (Urysohn’s Lemma) and 59 (Tietze Extension). Attempt them if you dare.

Section 7.7 Read.

Skim Chapter 8, especially ( 1, 9, 35, 36 ).

Skim Chapter 9, especially ( 8, 11, 24, 48, 49, 52 ).

Read Chapter 10.

Skim Chapter 11.

Section 12.1 ( 2 – 15, 17 – 21 ).

Section 12.2 ( 23 – 26 ).

[more to come]

READ:

My handout on good mathematical writing.

In Topology Through Inquiry, Read the Introduction (pp. 1-5) and Section 2.1 (pp. 27-30). Review Section 1.1 if needed, for notation and terminology.

DO:

Theorems/Exercises 2.1 through 2.12 for Monday’s class. For Wednesday, aim to work the next 10 problems ahead of where we ended the prior class. You’ll find some problems rather straightforward applications of definitions, and others more interesting.

Your homework group should meet Friday to discuss ideas and problems, and to prepare for that you should try to work some problems in advance.

Via Gradescope, you’ll hand in the following for Tuesday:
Question A and Chapter 2( 3, 5 [finite complement only], 8, 13, 14). Question A is below.

A. Briefly summarize three things that stick with you from the reading assignments above. [Perhaps each person in the group can include 3 things that stood out to them.]

On Gradescope, only one person per group submits. Be sure to tag the other people in your group.

Don’t forget to use prior theorems (including Theorems in Chapter 1) to make your proofs simpler.

Read Chapter 1 Sections 1.1 and 1.2 and remind yourself of theorems there (they should be review from Analysis.)

Read this handout on collaboration.

Do these problems: Chapter 2 ( 15, 20, 22, 23, 26 ) + Question B (below).

B. In your group, discuss takeaways from the reading (chapter 1 + collaboration handout) and list 3 things that stood out to your group.

You will collectively submit one assignment (and receive one grade), so decide how you will collaborate to write and review the solutions you jointly submit on Gradescope.

Read Sections 3.1-3.3. Remember to look at the List of Theorems/Exercises we’ll discuss in class (see above) and to prepare for each class, keep 10 ahead of where we end class the previous time.

Do Chapter 2 ( 32 ) Chapter 3 ( 3, 8, 9, 10 [part 4], 21 [examples B and D] ).

Read Sections 3.4-3.5. Remember to look at the List of Theorems/Exercises we’ll discuss in class (see above) and to prepare for each class, keep 10 ahead of where we end class the previous time.

Do Chapter 3 ( 25, 27, 28, 31, 34, 37 ).

Read Sections 4.1 and 4.2. Remember to look at the List of Theorems/Exercises we’ll discuss in class (see above) and to prepare for each class, keep 10 ahead of where we end class the previous time.

Do Chapter 3 ( 41, 42 ) and Chapter 4 ( 1, 5, 6, 8 )

Read Sections 4.3, 4.4, 4.5. Be sure to read theorems and sections even if they aren’t ones that we’ll discuss. (You can find the List of Theorems/Exercises we’ll discuss in class above.)

Do Chapter 4 ( 10, 11, 17, 20, 23 ) and Chapter 5 ( 1 ).

Read Sections 5.1 – 5.4, and 6.1. Remember that we’ll discuss more problems than just the ones assigned for homework, and you should be keeping a notebook of all the problems that you work. See the tab below for how the notebook will be assessed. (You can find the List of Theorems/Exercises we’ll discuss in class above.)

Do Chapter 5 ( 6, 9, 11, 18 ) Chapter 6 ( 3, 12 ).

Recall that one of the course requirements is to contribute to a Common Solution Set. Over the course of the semester, you should contribute to 12 solutions either by providing a solution or editing an existing solution. In this activity, you’ll contribute to 6 of them.

The goals of this activity are:

(1) to understand that good proof writing is a constant process of revision,
(2) to work together on a common project with current and former students,
(3) to see other solutions or approaches to problems you’ve done,
(4) to develop judgment in discerning when a solution is ‘good enough’.

For this assignment, look through the Common Solution Set (an Overleaf document linked from the email I sent to the course list), and identify 6 problems whose solutions you’d like to author or edit. .

You might, for instance, provide a solution to a problem that doesn’t currently have a solution, or provide a different solution to one that already exists. Or you might edit a solution that someone else has provided, which means (a) you’ve read the solution carefully and and (b) improve it by editing it in some way (correcting, clarifying, or improving wording, etc). When you are done, you are certifying that, in your judgment, it is correct.

Read the -README file in the left sidebar for instructions on how to enter your edits. You’ll be using LaTeX. If you don’t know how to use LaTeX, there are preliminary instructions in the -README file but otherwise, you can enter your solution in text and let someone else to edit for you.

Keep track of which ones you’ve contributed/edited, and enter that into Gradescope for HW#8.

Read Sections 6.2-6.3, skim the rest of chapter 6, then read 7.1-7.3. Remember that we’ll discuss more problems than just the ones assigned for homework. (You can find the List of Theorems/Exercises we’ll discuss in class above.)

Read the Notebook Information, since your notebook is due soon (the HW after this one).

Do Chapter 6 ( 19 ) Chapter 7 ( 1[c=>b only], 10, 15, 24, 29 ).

Notebooks will be turned in physically (at my office Shan 3416) or by e-mail (not through Gradescope).

The goal: this notebook should something you look back on with pride 10 years from now. Think of it as writing your own book on topology. So it should contain enough details for you to reconstruct your thoughts. See my Guidelines for Good Mathematical Writing.

WHAT SHOULD BE IN YOUR NOTEBOOK

Summary Sheet and Evaluation that tracks your progress
The Reflection Exercise
Proofs or sketches for Theorems and Exercises you’ve done
Anything else you wish to include.

How you organize the notebook is up to you. The proofs in your notebook may be handwritten or electronic, either is fine. (For electronic files, you may find the LaTeX files useful—see the link at the top of the Assignments.) When it comes time to turn in your notebook, you may turn it in electronically if you wish.

SUMMARY SHEET and EVALUATION

Your Summary sheet should list:

theorems presented
theorems you contributed comments or questions
theorems you contributed or edited in the Common Solution Set

Both you and I will evaluate your notebook. Use the following rubric, and include on your Summary sheet an explanation of the grade you would assign yourself with this rubric.

Exceptional Pass = every theorem and exercise discussed in class has complete correct proofs or proof sketches. There are many proofs completed that were not assigned for class discussion or homework. Proofs and proof sketches are well-written. Pictures are abundant. Notation is well-chosen. Subtle points in proofs are acknowledged. Reflection Exercise included. Indicate why your Notebook should receive an Exceptional Pass.
Pass = Most theorems discussed in class have proofs or proof sketches. There are complete proofs for assigned homework. Pictures are sufficient. If you looked at it 10 years from now, you could reconstruct most of the arguments from what you have written. Reflection Exercise included. Indicate why your Notebook should receive a Pass.
Not Yet Passing = Indicate what deficiencies you have, and how you will improve your Notebook before the end of the semester in order to present it as a Pass.

REFLECTION EXERCISE for the NOTEBOOK

Reflect on the following question:
What have you learned in this class about the process of doing or creating mathematics?

Weave in potential answers to the following:
– have you experienced: joy, beauty, reward?
– what have you learned about the value of struggle?
– what have you learned about the importance of community?

Read all of Chapter 10. Skim Chapter 11 for big ideas. Read Section 12.1. As usual, we’ll discuss more problems than just the ones assigned for homework. (You can find the List of Theorems/Exercises we’ll discuss in class above.)

Do Reflections 1 and 2 (below) and 12.1 ( 2, 3, 5, 6 ).

Reflection 1. Write a paragraph, reflecting on the contents of Chapter 10 in whatever way it strikes you when you read it.

Reflection 2. One useful skill to develop as a mathematician is the ability to skim—to read mathematics for the big ideas, without worrying about the details at first. This means looking at ideas, deciding what you think is most important or interesting, and thinking about any questions it might raise for you that you could dig into on a second reading. Reflect on the contents of Chapter 11 in whatever way it strikes you as you skim it.

Read Section 12.2. (You can find the List of Theorems/Exercises we’ll discuss in class above.)

Some of these problems are ones we worked in class. Use the opportunity to write them up (more carefully than we did in class) to seek deeper understanding of those problems.

Do 12.1 ( 7, 8, 10, 13, 15, 17 ).

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.

Since I’m not able to access this website anymore, please watch your emails for the final assignment, and information about the final exam.