Topology/Geometry I MAT 530 Fall Semester

1. Basic point set topology
   • Metric Spaces
   • Topological spaces and continuous maps
   • Comparison of topologies
   • Separation axioms and limits
   • Countability axioms, the Urysohn metrization theorem
   • Compactness and paracompactness, the Tychonoff theorem
   • Connectedness
   • Product spaces
   • Function spaces and their topologies, Ascoli's theorem
2. Introduction to algebraic topology
   • Fundamental group
   • Fundamental group of Sⁿ; examples of fundamental groups of surfaces
   • Seifert-van Kampen theorem
   • Classification of covering spaces, universal covering spaces; examples
   • Homotopy; essential and inessential maps

Typical references:

• James R. Munkres, Topology: a first course,
  Prentice Hall, Englewood Cliffs NJ, 1975;
• William S. Massey, Algebraic topology: an introduction,
  4^th corrected printing, Springer-Verlag, 1977.