What is Topology? In Abstract Algebra courses you study
addition, subtraction and multiplication in abstract settings: groups,
rings and fields. The basis of Topology is the abstract study of continuity.
What are the basic structures one needs to talk rigorously about a function
from one set to another (or itself) being continuous? For example, the
derivative maps the set of "smooth" functions to itself. Is it continuous?
The operation of taking an infinite sequence to its limit is a function.
Is it continuous? Does a small change in parameter of a differential equation
cause only a small change in the solution space? These are all questions
where our abstract discussion of continuity can be applied.
Math 430Spring 2009
Introduction to Topology
In Algebra, once one understands what groups, rings, etc., are, one
then asks: given two groups, are they the same, i.e., are they
isomorphic? In Topology, once continuity is understood, we classify
sets (really topological spaces) by asking if there is a continuous
bijection with continuous inverse between them. If so, we say they are
homeomorphic. We will only touch on this. Much more will be said
in Math 530.
Almost all phenomena studied in modern Mathematics is either discrete
(algebraic) or continuous (topological). So, Algebra and Topology might be
said to span most of Mathematics.
The branch of topology that has been most widly used in other areas of
mathematics and in applications is the theory of manifolds. Most beginning
level topology textbooks just touch on manifolds. Lee's text gives the
basics of point set topology then applies it to manifolds throughout.
You may wish to read the reviews on Amazon.
- Textbook: Introduction to Topological Manifolds by John Lee
- Syllabus: We will cover what we can.
- Grades will be based on a midterm (30%), a final (50%), and
homework (20%). You can expect to be assigned about 5 problems a week.
Students may be asked to give short presentations of homework in class.