Summer 2006

### MTHS 664 Introduction to Topology

Mulvey,Irene

06/26/2006 - 08/10/2006
Monday & Wednesday 09:30 AM - 12:00 PM

Science Tower 139

Topology, the branch of mathematics that makes possible a rigorous study of continuity and continuous distortion, is a core mathematical discipline that provides the foundation for much of advanced mathematics and is a fascinating subject in its own right.

Our course will focus on the essential elements of the subject: topological spaces and functions between them. We will begin our study with careful examination of metric spaces using familiar examples (the real line and the plane) as well as other, more complicated examples. In this setting, we will formalize what it means for two objects to be close together. We may intuitively know what it means for objects (numbers, n-tuples, or functions) to be close together, but within the context of metric spaces we can define a general notion of distance that works in many settings. This will lead us to see that much of analysis does not depend on the specific distance at all but rather depends on the sets that are open in the metric space.

In a natural way, we will see that a topological space is the appropriate setting for describing interior, closure, and boundary of a set as well as continuous functions and the topological properties of compactness and connectedness. We will study topological subspaces, product spaces, the separation axioms and homeomorphisms concluding with surprising and surprisingly beautiful theorems.

There are no specific prerequisites for the course; all the essential topics will be introduced in the course itself, and the course should be accessible to any student with a modest background in mathematics. Students will be expected to follow carefully reasoned arguments and active participation will be encouraged. Grades will be based on regularly assigned problem sets.

Our text will be Introduction to Topology (third edition) by Bert Mendelson.

Our text will be Theodore Shifrin, Multivariable Mathematics / Linear Algebra, Multivariable Calculus, and Manifolds.

Grades will be based on regularly assigned problem sets that will be roughly an equal mixture of computational problems and theoretical problems.

There are no specific prerequisites for the course; all the essential topics will be introduced in the course itself, and the course should be accessible to any student with a modest background in mathematics.

Irene Mulvey (B.A., Stonehill College; Ph.D., Wesleyan University) is professor of mathematics at Fairfield University. Her recent publications include "Symbolic Representation for a Class of Unimodal Cycles," Topology and Its Applications (2002), and "Multi-modal Cycles with Linear Map Having Exactly One Fixed Point," International Journal of Mathematics and Mathematical Sciences (2001). Click here for more information about Irene Mulvey.

ENROLLMENT INFORMATION

Consent of Instructor Required: No

 Format: Seminar Level: GLSP Credits: 3 Enrollment Limit: 18

Texts to purchase for this course:
Bert Mendelson, INTRODUCTION TO TOPOLOGY (Dover), Paperback