Math 472  Topology



Renzo's class



MWF 2 – 2:50 pm



ENGRG  E 205  // in the oval with good weather








Office Hours





Description of the Course 


Basic Topology





Project 1: Dense Sets, Separation Axioms and Product Spaces.

Project 2: One-Point Compactifications and the Projective Plane

Project 3: The construction of the Fundamental Group (Functor)

The fundamental group of the circle




PDF file

LaTeX source file



Office hours : there are no official office hours for this class. However, 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!!



Aug 27th

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

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

2.      a function f: X à Y is continuous (calculus style) if and only if the preimage of  any closed set in Y is closed in X.

Hint: You can use 1. to simplify your life in 2. !!

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.


Sep 3rd

Ex 13, 14, 15, 16, 17, 20 page 35

Sep 10th

Prove the following theorems:

1.      Let X be a topological space, and A a subset of X. Prove that the induced topology on A is the coarsest topology on A that makes the inclusion function i:AàX continuous.

2.      Suppose f: X à Y is continuous and A is a subset of  X that we make into a topological space by giving it the induced topology from X. Prove that the restriction of f to A  f|A: A à Y is also a continuous function.

Hint: you can use 1. to simplify your life in 2.

Sep 20th

Write up one of the block of exercises (by that I am meaning all exercises in one of the three sections) of the projects we did in class.

Sep 27th

Ex 30, 33 page 60.

Oct 4th

Ex  2 page 46

Ex  11 page 50

Oct 13th

Write up one of the groups of exercises in project 2.

Oct 25th

Prove the following statements:

·        The cone over an n-dimensional sphere is homeomorphic to an (n+1) dimensional closed disk

·        The identification space corresponding to making all points in the boundary of  an n dimensional disk equivalent is homeomorphic to the n-dimenional sphere. Show that this is also the one point compactification of the open disk.

·        The one point compactification of an open Mobius strip is homeomorphic to the projective plane.

·        Let X=the real line (with Euclidean topology). Give two different group actions on X (possibly with two different group) and describe the corresponding identification spaces.

Nov 19th

Homework on surfaces

Dec 1st

Write-up the problems in Project 3.









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 feel the whirlwind presentation on free groups and group presentations was too…whirlwindy, and would like to look up some references, you can point your attention to Chapter 2 of Milne’s Group theory Notes.