Math 472  Topology



Renzo's class



MWF 2 – 2:50 pm



ENGRG  E 204  // in the oval with good weather









Office Hours





Description of the Course 

Absent. Keep reading.

There will be
no official textbook in this class. No matter which book I choose I know I will not be following it very closely. Part of the value of an advanced class is to be able to take good notes and learn from them. However it is useful to have a book to refer to for the basic notions and definitions, and anything that may escape in the process of note-taking, or just to get a different perspective from my own babbling. There is a wealth of basic textbook in topology that will all contain the relevant definitions and concepts we will be illustrating. You are encouraged to look around for one that fits your personality best. Here are some possible suggestions – I have all of these books in my office and you are welcome to come browse through them:

1.      Armstrong. Basic Topology. I find this to be the best compromise between rigor and approachability.

2.      Munkries. Topology. This one is the most complete and rigorous intro book I know of, but it ain’t always fun to read.

3.      Shick. Topology. I like the choice and ordering of topics here, but the book indulges a bit too much in pathologies.

4.      Goodman. Beginning Topology. Great pictures, very approachable, a bit wishy washy.

5.      Basener. Topology and its applications. Don’t remember much about this one, but the table of contents looks good.




Project one: on dense sets, separation axioms and the product topology.

Project two: on compactifications.

Worksheet on the classification of compact surfaces

Worksheet on the fundamental group




PDF file

LaTeX source file



Office hours : official office hours for this class are MWF at 4pm. If those don’t fit your schedule, you are very welcome to make an appointment and come ask questions, make comments, or just chat. You can also try showing up at my door anytime. But I might tell you to come back at another time if I am immersed into something else.

The tables of the law for this class are contained in the following document:


Homework:  math is not a sport for bystanders. Getting your hands dirty  is important  to make sure that things sink in and you are not just spending a semester assisting to my creative rambling. Homework will be due pretty much every Friday (but see below for the up-to-date information). I don’t guarantee I will grade all the problems I assign. If one of the problems you feel unsure about doesn’t get graded, please don’t go “Whew! Lucky one!” but rather come ask me about it. The point is you understanding, not pretending to!!



August 29th

Prove the following fact where X and Y are two metric spaces (or if you want just think euclidean spaces):

1.      a function f: X ---> Y is continuous (calculus style) if and only if the preimage of  any open set in Y is open in X.

Please: Careful with the exposition of your thoughts. Write in  either English, Spanish or Italian (but in only one  of the above languages, and make sure I can recognize which one it is!)  Please avoid the overwhelming use of math stenography.


September 5th

Homework 2

September 19th

Each group write up one solution for the Problems in Project One


Homework 3

September 26th

Homework 4

October 3rd

Homework 5

October 17th

Each group write up one solution for the Problems in Project Two

October 24th

Homework 6

October 31st

Prove that T#P^2 is homeomorphic to P^2#P^2#P^2

Novermber 7th

Write a summary of the proof of the classification of compact surface. You are not required to write complete proof of any of the steps involved in the proof. But you are requested to write a well organized strategy of proof,  including sketches of the most important arguments.

Novermber 14th

Homework 7

Novermber 21st

Homework 8







Lecture notes:

I am not going to be writing down proper notes for this course. However I do keep a skeleton of the planning of this class. These are notes I write for myself to jog my memory and not forget things that I should tell you, but are by no means intended to be complete, understandable, legible, or even in English... however if you may find them of any use, please browse them as they grow.


A few years ago I taught this class at University of Michigan and assigned as a final project the task of writing lecture notes for the course. Again, this is not a polished document. There are probably a good number of mistakes I didn’t catch and a good number of things that could be said a lot better but I just didn’t have the time/energies/ability to fix. But again, if you find them useful, you are welcome to them:

Notes from 2007

If you would like to look up some references on free groups and group presentations, you can point your attention to Chapter 2 of Milne’s Group theory Notes.