General Topology
Topological spaces and continuous functions
• Basis, subbasis, subspace topology, product topology, metric topology, continuous functions and homeomorphisms, closed sets, closure and limit points, quotient spaces.
Connectedness and compactness
• Connected and disconnected spaces, path connectedness, compact spaces, continuous functions on connected and compact spaces, limit point compactness, sequential compactness, Lebesgue Lemma, uniform continuity, one-point compactification.

Algebraic Topology
The fundamental group 
• Homotopy of paths (retracts and deformation retracts, homotopy equivalences), the fundamental group , covering spaces and subgroups of, Van Kampen's Theorem.
Homology
• Simplicial and singular homology, exact sequences, long exact sequence of a pair, (X,A), excision theorem, orientation on manifolds.
Cohomology
• Duality, criteria for (H_k ) ^* = H ^k 
Applications
• Degree of a map, Euler characteristic, Stoke's Theorem, Brouwer Fixed Point Theorem, classification of closed surfaces, orientation of manifolds, Lefschetz Theorem. 

References:

1. Topology, S. Davis, McGraw-Hill.
2. Topology: The First Course, J. Munkres, Prentice Hall.
3. Algebraic Topology: A First Course, M.J. Greenberg, J.R. Harper (Math Lecture Notes)
4. Algebraic Topology, A. Hatcher www.math.cornell.edu/~hatcher