Topology/Geometry I (MAT 530)

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.