Mathematics 451 (Topology) - Syllabus - Fall 2006

This is a first course in topology, designed to be taken by the student before taking any other course in topology.

Topics Covered

Included in this course are fundamental topics including:

Compactness

Connectedness

Separability

Homeomorphisms

Indecomposability

Non-metrizability

In addition, special topics such as the characterization of Jordan curves, and higher ordinal spaces may be studied.

Structure of the Course

The course begins with the axiomatic development of linearly ordered spaces, and continues to the study of metric spaces. The Lexicographic Plane and Long Line are studied at the appropriate stages. The significance of important related but non-equivalent properties such a the Heine-Borel and Bolzano-Weierstrass theorems are studied. And counter-intuitive deeper implications of the Baire Category Theorem are explored.

It is found in this course at B.Y.U., and in comparable courses at other major universities, that student’s enthusiasm for mathematics and their ability to reason soundly, and be productive researchers are stimulated by requiring students t think through to the solution of problems without looking up answers in a book. This paradigm depends upon the active involvement of the professor who prepares a sequence of theorems which the student can solve if he puts forth his best efforts. When the student needs help, the professor may suggest a lemma, or intermediate theorem. And when the student progresses quickly, the professor may suggest a more challenging problem such as research problems that the professor discovered in the course of attending professional meetings.

This involves more than the instructor lecturing, the student listening, and both of them turning their attention to other matters at the end of the class period. To the contrary, the professor continually thinks about the course material with the enrichment he gets from attending conferences where he can discourse with colleagues; the student thinks about the theorems he is trying to prove or disprove when he is out of class as much as when he is in class. As Newton said when asked how he made his great discoveries, "by thinking about it all the time."

The most famous exponent of this method in the last hundred years, R. L. Moore, who is sometimes called "the greatest math teacher ever," taught his students, for over sixty years, to think, and his students' resulting devotion to their profession was legendary, producing several Presidents of the American Mathematical Society.

Grading

Grading is based upon

1. The conventional criteria of examination results

Plus

2. The number and difficulty of theorems that the student proves throughout the course.