
Renzo's class 


MWF 2 – 2:50 pm 


ENGRG E 204
// in the oval with good weather 







TEXTBOOK
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
notetaking, 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.
PROJECTS AND SUCH
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
LaTeX CHEATSHEET:
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:
Syllabus
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
uptodate 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!!
DATE DUE: 

August 29^{th} 
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 5^{th} 

September 19^{th} 
Each group write up one solution for the Problems in Project One 
September 26^{th} 

October 3^{rd} 

October 17^{th} 
Each group write up one solution for the Problems in Project Two 
October 24^{th} 

October 31^{st} 
Prove that T#P^2 is homeomorphic to P^2#P^2#P^2 
Novermber 7^{th} 
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 14^{th} 

Novermber 21^{st} 





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:
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.