Math 447 - Elementary Topology

Instructor

Volker Runde

Office hours

TBA

Syllabus

Set theory

• sets and functions
• cardinals
• Cartesian products

Metric spaces

• definitions and examples
• open and closed sets
• convergence and continuity
• completeness, and Baire's theorem
• compactness for metric spaces

Point set topology

• topological spaces
• continuity and convergence of nets
• compactness (Tychonoff's theorem)
• connectedness
• separation properties

Function algebras

• the Tietze-Urysohn theorem
• the Stone-Czech compactification
• the Stone-Weierstraß theorem

Basic algebraic topology

• homotopy
• the fundamental group
• covering spaces

Textbooks

The required text is
• G. F. Simmons, Introduction to Topology and Modern Analysis, Krieger Publishing, 1983 (original edition: 1963).

Part One of this book covers the material on set theory, metric spaces, point set topology, and function algebras. I plan to supplement the text with some notes of mine (mainly on nets) that will be made available through this webpage. The material on basic algebraic topology corresponds to paragraphs one to five of Chapter II and one to four of Chapter V of

• W. S. Massey, A Basic Course in Algebraic Topology, Springer Verlag, 1991.

This text is not not required, and I will make notes available on those topics.

Further recommanded books are:

1. G. L. Cain, Introduction to General Topology. Addison Wesley, 1994.
2. J. Dixmier, General Topology, Springer Verlag, 1984.
3. K. Jänich, Topology, Springer Verlag, 1984.
4. J. L. Kelley, General Topology. Springer Verlag, 1975.
5. J. R. Munkres, Topology (Second Edition). Prentice Hall, 2000.

Grading

The grade will be based on weekly homework assignments (30%), an in-class-midterm (20%) on February 13, and a final (50%). For the final, you will be a allowed a crib sheet: you may bring one letter sized sheet, on which you may write from both sides whatever you like. Of course, only handwritten crib sheets are acceptable; xeroxes or computer printouts will be considered cheating. The crib sheet has to be turned in with your exam.