klien bottle

Math 430
Introduction to Topology

Spring 2009
Professor Sullivan

borromean rings
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.

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.