Catalog Description

M 411 INTRODUCTION TO TOPOLOGY (3-0-3)(S). Sets, metric spaces, topological spaces, continuous mappings, connectedness, compactness. PREREQ: M 314. Offered spring of even-numbered years, subject to sufficient demand.

Prerequisites

Jurisdiction

Learning Objectives

The objectives of Introduction to Topology coincide with three of the four departmental goals. M 411 does not stress the applications of mathematics or the impact of technology, but it does stress the ideas of abstraction, aesthetics, the development of mathematical tools and the use of the language of mathematics.

Upon completion of this course, students should:

• be familiar with basic concepts of topology,
• gain mathematical maturity,
• become competent in writing proofs,
• apply spacial imagination to theory.

Assessment of Learning Objectives

Students will be assessed by evaluating their ability to do problems based on the learning objectives. The problems will occur in several contexts:

• Periodic problem sets for homework serve both as learning and as assessment tools.

• Problems given on take-home exams are designed to evaluate a student's ability to solve more complicated and time consuming problems than can be reasonably completed on an in-class exam.

• Problems given on in-class exams are designed to see if students can use the tools that have been developed to solve straight forward problems in a limited amount of time.

• Students may also be expected to present solutions in class or to work in small groups to solve problems. Both of these situations permit the instructor to assess the student's ability to communicate effectively using the language of mathematics.

Topics and Approximate Timeline

• Metric spaces.
• Open and closed sets.
• Continuity.
• Topological spaces, separation axioms.
• Compact sets.
• Connected sets.
• Product spaces.
• Quotient spaces.

Additional topics may be covered from the following list:

• Covering spaces.
• The Jordan curve theorem.
• Classification of surfaces.
• The Euler characteristic.
• Homology.
• Homotopy.
• Differentiable manifolds and vector fields.

Text

Choice left to the instructor. Possible suggestions:

• Topology of Surfaces, C. Kinsey, Springer-Verlag, 1993.
• Topology, a First Course, J. R. Munkres, Prentice-Hall, 1975 (advanced).
• An Introduction to Topology and Analysis, G. Simmons, Mc-Graw Hill, 1963.
• A Geometric Introduction to Topology, C. T. C. Wall, Addison-Wesley, 1972.

Format, Student Activities, and Grades

To be announced by the instructor at the beginning of classes.

Letter grades are usually based on a standard scale in which 90% of the total possible points guarantees an A , 80% a B, and 70% a C, with the instructor having the discretion to lower these cut-offs if warranted.