MAT 578 -- Topology II 
Course Information

 
Professor: D. Bradley

Office: 322 Neville Hall  Hours: 2:00-2:50 p.m. MWF*  Phone: (207) 581-3920

Announcements

Website: http://www.umemat.maine.edu/faculty/bradley/index.html

This course is a continuation of MAT 577 (Topology I). As such, we will be reinforcing many of the concepts and topics of MAT 577 as well as forging ahead with new material in the study of point-set and algebraic topology. Grades will be assigned on the basis of in-class participation and successful completion of assignments.

Students who will profit:

Anyone contemplating the pursuit of an advanced degree in a subject with technical requirements in higher mathematics.

Text: James R. Munkres, Topology, (2th ed.) Prentice Hall, New Jersey, 2000.

Syllabus:

I plan to cover many of the following topics, and possibly others as time and class interest dictate.

• Set Theory
• Infinite Cartesian Products
• Countable and Uncountable Sets
• Transfinite Recursion/Induction
• Axiom of Choice
• Well Ordered Sets
• Maximum Principles
• Tychonoff's Theorem and its Consequences
• Nets and Moore-Smith Convergence
• Countability and Separation Axioms
• Normal Spaces
• Urysohn Lemma
• Urysohn Metrization Theorem
• Tietze Extension Theorem
• Imbeddings of Manifolds
• Complete Metric Spaces and Function Spaces
• A Space-Filling Curve
• Pointwise and Compact Convergence
• Ascoli's Theorem
• Baire Spaces
• A Continuous, Nowhere-Differentiable Function
• Algebraic Topology
• Homotopy of Paths
• Fundamental Groups
• Fundamental Theorem of Algebra
• Separation Theorems in the Plane
• Jordan Curve Theorem
• Winding Number of a Simple Closed Curve
• Cauchy's Integral Formula
• Seifert-van Kampen Theorem
• Classification of Surfaces, Homology