Topology — Fall 2025

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.