
Renzo's class 


MWF 2 – 2:50 pm 


ENGRG E 205
// in the oval with good weather 






TEXTBOOK:
Basic Topology
Armstrong
PROJECTS AND SUCH
Project 1: Dense Sets, Separation Axioms and Product
Spaces.
Project 2: OnePoint Compactifications and the
Projective Plane
Project 3: The construction of the Fundamental
Group (Functor)
The fundamental group of the circle
LaTeX CHEATSHEET:
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:
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: 

Aug 27^{th} 
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 3^{rd} 
Ex 13, 14, 15, 16, 17, 20
page 35 
Sep 10^{th} 
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 fA: A à Y is also a continuous function. Hint: you can use 1. to simplify your
life in 2. 
Sep 20^{th} 
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 27^{th} 
Ex 30, 33 page 60. 
Oct 4^{th} 
Ex 2 page 46 Ex 11 page 50 
Oct 13^{th} 
Write up one of the groups of exercises in project 2. 
Oct 25^{th} 
Prove the following statements: · The cone over an ndimensional 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 ndimenional 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 19^{th} 

Dec 1^{st} 
Writeup 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:
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.