I.   Number and title of course

 MAT 471  Introduction to Topology
 

II.  Reasons for Revision

We have updated the bibliography and expanded the course pre-requisites to better address and reflect the current work in this area, and we have redefined the major objectives in terms of student outcomes.  The course continues to serve the following purposes in our program:

A. Several important branches of modern mathematics depend heavily on the fundamental concepts of point-set topology.  The trend today is to introduce these concepts at the undergraduate level.  This enables students to be better prepared for graduate study in fields which depend upon topological concepts.

B. Students who are preparing to teach mathematics at the secondary level benefit from an understanding of topology in at least two ways:  They are better able to answer the questions of high school students who are curious about the topological properties that arise in their studies.  The prospective teacher’s own understanding of the properties of the real line and the complex plane is strengthened.
 

III Major Objectives of the Course

A. Students will learn the basic definitions, examples, counterexamples, and theorems of point-set topology.

B. Students will develop skill in constructing proofs, examples and counterexamples associated with abstract mathematical concepts.
 

IV. Topical Outline

A. Sets and Functions
 1.  Indexed sets
 2.  Functions and mappings
 3.  Cardinality
 4.  Partially ordered sets

B. Topological Spaces
 1.  Definition and examples with an emphasis on metric spaces.
 2.  Open and closed sets, and neighborhoods
 3.  Bases for a topology
 4.  Closure, interior, boundary

 C. Continuous Functions
 1.  Definition and examples
 2.  Properties
 3.  Homeomorphism

D. Separation
 1.  T0, T1 spaces and Hausdortf spaces
 2.  Regular spaces and T3-spaces
 3.  Normal spaces ard T4-spaces
 4.  Completely regular spaces and Tychonoff spaces

E. Connectedness
 1.  Definition and examples
 2.  Properties
 3.  Components and loca1 connectedness

F.  Compactness
 1.  Definition and examples
 2.  Properties of compact spaces
 

V. Bibliography

Adamson, I. (1995). A General Topology Workbook.  Boston:  Birkhauser.
Alexandroff, P. (1993).  Elementary Concepts of Topology.  Boston:  Dover.
Arkhangel’skii, A.V. (1990).  General Topology I:  Basic Concepts and Constructions.  Berlin:  Springer-Verlag.
Barr, S. (1986).  Experiments in Topology.  Boston:  Dover.
Baum, J. (1987).  Elements of Point Set Topology.  Boston:  Dover.
Fomenko, A. (1994).  Visual Geometry and Topology.  Berlin:  Springer-Verlag.
Lipschutz, S. (1965). General Topology.  New York:  McGraw-Hill.
McCarty, G. (1988).  Topology, Boston:  Dover.
Munkres, J. (1974, new addition forthcoming in 1999) Topology, A First Course.  New York:  Prentice-Hall.
Sims, B. (1976). Fundamentals of Topology.  New York:  Prentice-Hall.
 

VI Presentation and Evaluation

A. Lectures and discussions.  Opportunities will be provided in class for students to present their solutions to assigned problems.

B. Evaluation by examinations, written problem assignments and class participation.
 

VII. Prerequisites

MAT 270 and (MAT 301 or MAT 417)
 

VIII. Credit

 3 Credits:  (3:0)
 

 IX. Statement of Approval

This course proposal was examined in accordance with recommended procedures and approved by the Department of Mathematics Curriculum Committee on

_____________________.  ___________________________________
     Signature of Department Chair Date
 

X. Catalog Description

An introduction to topology--sets and functions, metric spaces, topological spaces, connectedness, compactness, and separation.
 

XI. Statement of Qualifications of Faculty Who Will Teach the Course

A. A Ph.D in Mathematics or extensive background in toplogy is required to teach this course.

B. The following faculty have, at the present time, the necessary qualifications:
 Joseph Barback, Ph.D. in Mathematics
 Scott Crass, Ph.D. in Mathematics
 Daniel Cunningham, Ph.D. in Mathematics
 

XII. Support services required

 Present classroom facilities are adequate.