M.S. Level Material
Algebraic Topology
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

EilenbergSteenrod 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 2manifolds (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
References
Lin and Lin: Set Theory
J.R. Munkres: Topology
Dugundji: Topology
Greenberg & Harper: Algebraic Topology
Hatcher: Algebraic Topology
Hu: Homology Theory
Spanier: Algebraic Topology