$University of Illinois at Urbana-Champaign$

Math 535 - General Topology
Fall 2012, Section E1

Course outline

Topology is the study of spaces. Whereas geometry studies notions like lengths, angles, and size, topology only sees the shape of a space and its qualitative features. That is why topology is sometimes called "rubber-sheet geometry".

Math 535 is a rigorous introduction to general (point-set) topology. We will review the topology of Euclidean space Rⁿ and observe that many facts from analysis are really topological in nature. This will lead to the study of metric spaces, which share most of the nice features of Rⁿ. However, many interesting spaces arising in functional analysis, algebraic topology, and algebraic geometry are not metrizable. We will therefore study general topological spaces, noting the similarities and differences with metric spaces. Throughout the course, we will discuss applications of general topology to other branches of mathematics.

Prerequisites

MATH 444 or MATH 447 or equivalent.
MATH 416 or equivalent. Depending on the specific topics, MATH 417 could be useful.
MATH 432 could be useful but will not be assumed.

List of possible topics

Topological spaces, bases, subbases
Continuous maps
Subspaces, products
Quotient topology
Metric spaces
Pseudometrics, uniform spaces
Nets, filters
Separation axioms: Hausdorff, regular, normal...
Connectedness, local connectedness, path connectedness
Compactness, Tychonoff theorem
Completeness
Urysohn lemma, Tietze extension
Countability axioms
Paracompactness, partition of unity
Metrizability
Function spaces

Time permitting:

Applications to functional analysis
k-spaces
Fundamental groupoid
Covering spaces

Lecture log and homework assignments

Course information

Instructor	Martin Frankland
Email	`franklan(at)illinois.edu`
Office	247A Illini Hall
Office hours	Monday 4-5 PM and Tuesday 3-4 PM
Lectures	MWF 1:00-1:50 PM in 443 Altgeld Hall
Course website	http://www.math.uiuc.edu/~franklan/Math535.html
Textbook	There is no required textbook. However, the following books are recommended and have been put on reserve at the Mathematics library. Stephen Willard, General Topology. James R. Munkres, Topology, 2nd edition. Glen E. Bredon, Topology and Geometry. Ronald Brown, Topology and Groupoids. See the book web page for an affordable electronic version.
Grading	The course grade is calculated as follows. Homework: 70% Final exam: 30% Bonus deal: If the final is better than the homework average (after dropping the lowest two), then the final will be worth 40% and the homework will be worth 60%.
Important dates	Mon Aug 27: First lecture. Mon Sep 3: Labor Day. No lecture. Fri Nov 16: Drop date. (Not Oct 19, which is the drop date for undergraduate courses.) Nov 19-23: Thanksgiving break. No lectures. Wed Dec 12: Last lecture. Tue Dec 18: Final exam, 1:30-4:30 PM, in 443 Altgeld (usual lecture room).
Homework	Homework assigned at each lecture will be posted on the daily log. It will be due in class, usually on Wednesday of the following week.
Grades	Registered students can view their grades here. You will need your NetID and password.

Course policies

Homework grading

In each assignment, selected problems will be graded.
The two lowest homework scores will be dropped (not one, as previously announced).

Collaborative work

You are encouraged to work on the homework in groups, although each student should submit a separate set of solutions. Please indicate which other students you collaborated with, if any.

Late homework

Late homework will not be accepted, because solutions are posted shortly after the deadline.
If you will be absent when homework is due, you must turn in your homework in advance. Note that I will check my mailbox (250 Altgeld) right after collecting homework in class.
In case of documented illness or emergency, homework may be dropped.

Make-up final exam

No make-up exam will be offered after the fact.
If the exam schedule conflicts with another exam, please email me at least one week in advance.
In case of documented illness or emergency, please contact me.

General principle

Feel free to email me or come to office hours for any administrative issues.

Back to Martin Frankland's home page.

Math 535 - General Topology Fall 2012, Section E1