Topology — Fall 2025

Here is the homework 6, in LaTeX hw6.tex and pdf: hw6.pdf (Corrected typos Sunday Nov 16 and Nov 20). Aim to have these done in about a week (they don't involve anything we haven't covered yet).

Also, for this weekend, read sections 6.1-6.5 in Chapter 6. There are some interesting adjunctions (eg: cylinder / free path space).

Guest lecture: I'm very happy to say that on Thursday, Nov 6 there will be a guest lecture by applied and computational topologist Mikael Vejdemo-Johansson about Adjunctions in functional programming. Will someone please volunteer to take notes (email me to let me know)!

Topologies on sets of functions: Please be aware of this video on exponential topologies (13 min at 1x). It tracks the beginning of section 5.6 in Chapter 5 in the book and what I discussed in class.

Reading due Thursday, Nov 13: The next assignment will be reading section 5.5 in Chapter 5 and watching this video on compactifications (12 min at 1x).

Presentations for hw5 (updated Sunday Nov 9): All of the problems are taken --- I'm looking forward to the presentations on Tuesday!

This weekend, read through Chapter 4 of the book. This chapter is about limits and colimits, which covers a huge number of constructions in topology and throughout mathematics. For a bonus, watch this video (~7 min on 1x) which proves that limits are unique up to unique isomorphism. I'll give a lecture on this material on Tuesday. To test your understanding, I encourage you to prove that colimits are unique up to unique isomorphism by giving the argument that is parallel to the one for limits in the video. For some more exercises, try these:

• Solve Problem 6 on Spring 2021
• Solve Problem 4 on Fall 2022
• Express the space in Problem 5 of Fall 2023 as the (co)limit of a diagram.
• Express the space in Problem 5 on Spring 2023 as the (co)limit of a diagram.
• Express the situation in Problem 3 of Fall 2025 as the (co)limit of a diagram.

Here is the homework 5, in LaTeX hw5.tex and pdf: hw5.pdf. If you grab the LaTeX source, grab the picture genus2_octagon_two_arcs.pdf as a separate file. Correction: Wed Oct 22: fixed a typo in problem 3, there was a 'g' that was supposed to be an 'f'.

If you like, you can treat this as a midterm exam: that means study for it, when you feel ready, give yourself about 90 minutes to take it without reference material.

Plan on student presentations of solutions on Thursday, Nov 6.

So you have a clear idea of what material is covered: all the material from the class so far, plus the material I plan to cover this week. That includes filters and ultrafilters, and there's one question about a pushout. For references:

• You'll want to be familiar with the material in the second video on Filters and Ultrafilters (~ 20 min on 1x). Which gives all the definitions and states and proves everything you'd need to know: A space is compact iff every ultrafilter converges to at least one point, a space is Hausdorff iff every ultrafilter converges to at most one point, a function is continuous iff it maps convergent filters to convergent filters, Tychnoff's theorem...
• Pushouts are just a convenient way to describe certain quotients of unions. They're described in section 4.3.3 of Chapter 4 of the book. I'll talk about these in class (probably on Thursday Oct 23). Pushouts are one of the main constructions in topology and essential for the Seifert-Van Kampen theorem, which is one of the most important theorems used to compute the fundamental group of spaces.

Here are some problems from past qualifying exams that involve surfaces --- have fun with them this weekend:

• Problem 5 in Part II of Spring 2013
• Problem 7 in Part II of Spring 2014
• Problem 10 on Fall 2015
• Problem 10 on Spring 2016
• Problem 7 on Spring 2023
• Problem 9 on Fall 2022

Also, remember that Tuesday, Oct 14 is a CUNY Monday, so our next class is Thursday, Oct 16. I'll start talking about convergence and topological properties, first using sequences and then with filters and ultrafilters. To prepare:

• Read through section 3.2 (pg 55-60) in Chapter 3 of the book.
• Watch the following video (<13 min on 1x) for an introduction to Filters and Ultrafilters (Part I - no topology).

Just an fyi: I'm going to present the material on filters and ultrafilters a little differently than in the book, so no need to read beyond section 3.2 (the video is good though!)

The next topic is the classification of compact, connected surfaces. I’ll follow Massey’s proof in Algebraic Topology: An Introduction (in Springer's Yellow GTM series). For reading, please use the following bachelor’s thesis by Ana da Silva Rodrigues (supervised by Ana Cannas da Silva, ETH Zürich):

• Classification of Surfaces (pdf)

These notes give a crisp, Massey-style treatment and are a pleasure to read. I'm looking forward to presenting this material in class!

I forgot to include a due date on Homework 4 (pdf). Please submit it by Friday, October 10, 2025. If you collaborated, you may submit a single write-up with all contributors’ names. Please don’t just divide the problems up --- make sure everyone understands every solution.

Reminder: there is no class on Thursday, Oct 2. The next class is Tuesday, Oct 7. Over the next week try to catch up

• any unfinished homework assignments
• complete your peer grading of the second problem set
• be sure you've read and studied the material through Chapter 2 of the book
• be sure you understand the details of today's lecture: the definitions of fundamental groupoid and group, the fixed point theorem we proved, the topological properties Hausdorff and compact and how they work together.

Also, I made a few corrections to problems in homework 3 (definition of mixing vs forward mixing, replaced simeq with cong in a couple places). Get the corrected version here: hw3.tex and pdf: hw3.pdf.

For homework 4, I won't collect solutions, but let me know if you have questions. The next assignment will be a "take-home" midterm.

Here's the fourth homework assignment. In LaTeX: hw4.tex and PDF: hw4.pdf. As usual, collaboration is encouraged. This problem set is a bit shorter— something I'd be comfortable using for an in-class exam (after studying!). If you're tackling it before you've reviewed the material as for a midterm, expect it to take a bit longer since you'll be studying and working at the same time.

Also, next class is Tuesday, September 30. There will be no class on Thursday, October 4.

Please read through Chapter 2 of the book. This includes material on connectedness which I mostly covered in class on Thursday Sep 18. It also includes material on hausdorff/compactness which I'll begin in the next classe. Before I start on hausdorff and compactness though, I'll talk about homotopies and the fundamental groupoid.

Here are a couple of short videos to watch (~10-15 minutes at 1x)

• Paths
• Compact & Hausdorff

Remember, there's no class on Tuesday, Sep 23. Next class is Thursday, Sep 25.

Peer Grading — HW 2

We’ll be peer grading the second problem set.

• Read your classmate’s solutions carefully.
• Give some written feedback (what’s correct/clear, what’s unclear, and any gaps).
• Have a short meeting to discuss the problems and feedback. Make sure any questions are resolved and if you solved the problems differently, talk about the different approaches.

If you’re unsure who you’re grading—or who is grading yours—please let me know.

Be aware of the CUNY Academic Calendar. In particular, there's no Topology class on

• Tuesday, Sep 23
• Thursday, Oct 2
• Tuesday, Oct 14 (CUNY follows Monday schedule)
• Thursday, Nov 27

I put together some nice problems about the product topology and dynamics. In LaTeX: hw3.tex and pdf: hw3.pdf.

• I will not collect full written solutions to everything. We’ll discuss a few in class; others may reappear on future sets or exams.
• By Sept 25, choose one of the following
  1. a quick oral check during office hours
  2. email me a one-page summary of which problems you solved and any questions you still have
  3. if you do want to submit written solutions and get some written feedback, turn in solutions to problems 2b, 3, 7b, 8b, and 12a.

Here is some homework for the weekend: read and study chapter 1 of the book. I've covered some but not all of this material in class. What hasn't been covered should be fun to read and it's always a good idea to read a bit ahead to prepare for the next class.

I wrote some notes to clarify the idea of a basis for a topology and the idea of a topology generated by a basis. The definition in the book is correct, but there are some more details checked in these notes: basis_notes.pdf.

At the end of class, I began talking about the subspace topology. Check out this video (~8 min at 1x) on the subspace topology.

Next class, I'm going to continue to discuss the subspace topology and other constructions like the product topology, the quotient topology, and the coproduct topology. This material is in chapter 1 of the book.

Here's a new homework assignment (the second written assignment). In LaTeX: hw2.tex and pdf: hw2.pdf. Collaboration is encouraged! (problem 3,7 corrected Sep 3).

Here is some homework for the weekend: read and study chapter 0 of the book. Almost all of this material was covered in class, so you should be in a good position to read it carefully. Reading carefully means when you read something like the following

It is straightforward to check that for any topological space X, the identity id : X → X is continuous, and for any topological spaces X, Y, Z and any continuous functions f : X → Y and g : Y → Z, the composition g f := g ◦ f : X → Z is continuous, and moreover that this composition is associative.

I will tag section 0.2.3 Natural transformations and the Yoneda Lemma as optional for now --- read it only if interested.

Exercises 1-5 are strongly recommended, but no need to turn them in --- I'm going to create a problem set of problems to work on over the next couple of weeks and turn it. I'll post that when it's ready.

Also, you may want to check out this video (~8 min at 1x).

Here's the homework assignment I passed out in class yesterday. In LaTeX: hw1.tex and pdf: hw1.pdf. Solutions in LaTeX preferred, handwritten okay. If you wrote your solutions in LaTeX, you can email a pdf. Turn in handwritten solutions in class. Collaboration is encouraged! Please write your own solutions and acknowledge collaborators.

MATH 70700: Topology
When/Where: Tuesday & Thursday, 9:30 AM – 11:00 AM, Room 6417
Instructor: Prof. J. Terilla • Credits: 4.5

Course Description. This is the first semester of a two-part course in topology. We begin with a brisk treatment of point-set topology, covering concepts such as connectedness, separation, and compactness. These ideas form the topological foundations essential in fields ranging from analysis to number theory.

Beyond fluency with these core ideas, this course is designed to prepare students for success in algebraic topology. To that end, we emphasize the categorical perspective early on, using point-set topology as a setting in which to develop the structural thinking that algebraic topology and many areas of contemporary mathematics require. Through theory and examples, students will gain technical tools and conceptual insight, building a foundation that supports deeper study in topology and related areas.

Text. Topology: A Categorical Approach (MIT Press). Publisher page →

In addition to the material in the book, particular attention will be spent on the theory of covering spaces. Open-access chapter PDFs and additional materials are available here: Book page →

Pre‑requisites. A good working knowledge of linear algebra, abstract algebra, and undergraduate real analysis.

Office hours. Tue 1:15–2:15 and Thu 11:00–12:00 in 4208.01 (or the math lounge 4214 if we need more board space); and by appointment.