M.S. Topology Syllabus

Topological Spaces
• Compactifications, various characterizations of paracompactness in terms of coverings and partitions of unity
• Metrization, completion, uniform spaces
• General topological constructions such as induced, coinduced topologies, adjunction spaces, etc.

Fundamental Notions of Algebraic Topology
• Homotopy (deformation, homotopy type, fundamental group), (Universal) covering space
• Mapping cone, mapping cylinder, suspension
• Eilenberg-Steenrod axioms for (co) homology theories, uniqueness theorems
• Simplicial sets, singular, simplicial, and Cech Theories
• Derived functors, EXT, TOR, Universal Coefficient theorem, Kunneth formula

Computation of (Co) homology and Fundamental Groups
• Graphs, compact 2-manifolds (sphere, torus, projective plane, sphere with handles and cross caps), adjunction spaces, topological spaces

Applications
• Invariance of dimension, existence of extensions and retractions
• Fixed point theorems, fundamental theorems of algebra

Lin and Lin: Set Theory
J.R. Munkres: Topology
Dugundji: Topology
Greenberg & Harper: Algebraic Topology
Hatcher: Algebraic Topology
Hu: Homology Theory
Spanier: Algebraic Topology