Stony Brook - Graduate Courses - Topology/Geometry I - MAT 530
Topology/Geometry I
MAT 530
Fall Semester
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
Introduction to algebraic topology
Fundamental group
Fundamental group of
S
n
; 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.