Prospectus for Math 218A (Topology)
In the Fall quarter, I will be teaching the introductory graduate Topology course (218A). This is the actually the first quarter of a three-quarter course (Math 218) that is an introduction to the theory of diifferentiable manifolds, a major area of modern mathematics that every practicing mathematician needs to be familiar with, since differentiable manifolds have proved to be such an enduring source of models for the study of a great many areas of pure and applied mathematics, and in particular for physics. Topology concerns the concepts of "nearness" and "continuity" in an abstract setting. As such it treats topics that will be familiar from undergraduate analysis courses, but in a more general setting. In fact this is one of the best places for a student to become familiar with the processes of abstraction and generalization in mathematics and to get a feeling for their power.
Here are the major topics that I expect to cover:
(i) Metric Spaces: Definitions, examples, compactness, completeness, Banach Contraction Theorem, Ascoli Arzela Theorem.
(ii) Topological spaces: Definition, examples, compactness,
connectedness, separation axioms, extension theorems, and
partition of unity,
(iii) Introduction to homotopy, fundamental groups and covering spaces.
The textbook for the course will be: Introduction to Topology, by T. W. Garnelin and R. E. Greene, Saunders college publishing.
Other good references for material we will cover are:
Topology, a first course, by James R. Munkres and
Lecture notes on elementary topology and geometry, by I. M. Singer and J. A. Thorpe.
The course grade will be based on homework assignments and a take home final exam.