## Fall Semester 2000

### Instructor: Prof. Joel Roberts

Description Topology can be defined as the study of those properties of figures that are preserved by homeomorphisms, that is to say, by one-to-one mappings which are continuous and also have continuous inverses. This definition is incomplete, however, because we can give the full definition of continuity only in the context of general topological spaces.

Rather than dealing immediately with the most general issues, we'll start (after doing some background material from set theory) by studying metric spaces where we have a distance function -- as in the case of Rn -- that we can use to define a concept of "closeness". In this situation, we can give a definition of continuity which is similar to the rigorous (or e - d ) definition of continuity sometimes presented in calculus courses. We also will develop some basic properties of open sets, continuous mappings, etc. that will make it easier to understand the general definition of topological space and study their basic properties.

Here are other some things about topological spaces that we'll be able to study:

• Compactness and connectedness.
• Sequences, axioms of countability, separability conditions.
• Urysohn's Theorem (giving a sufficient condition for a topological space to be a metric space).
• Quotient spaces, local compactness, complete metric spaces.

If we have time, or if we decide to skip something from the above list, then we can study some other topic, for instance classification of surfaces.

Text Topology, by D.W. Kahn.     (Dover, 1995)

Prerequisites, etc.

• The formal prerequisites are as follows:
• Sophomore level math, including multivariable calculus
• Math 2283 or Math 3283 (Sequences, Series and Foundations) [ or concurrent registration in that course ]
(If you took the honors version of sophomore calculus, then a lot of the material from Math 2283/Math3283 may have been covered there.)

• Other helpful background can include Advanced Calculus (Math 4606) or Introduction to Analysis (Math 5615/6), or some other course where a significant part was about the theory of calculus. (You won't be held responsible for knowing any specifics about what was covered in such a course, but having seen some of that material can help you to understand the significance of some of the material we will study in this course.)

For further information: Please send me an e-mail or call me at the phone number listed below.

Back to the class homepage.

Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
USA

Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017
e-mail: roberts@math.umn.edu
http://www.math.umn.edu/~roberts