Topology Examination Syllabus

Topics

Point Set Topology

• Open sets, closed sets, closures, and boundaries
• bases, dense sets, and networks
• the Baire Category Theorem for complete metric spaces and for locally compact spaces
• completion of a metric space
• connectedness
• separation axioms
• metrizable and non-metrizable spaces
• Urysohn metrization theorem
• continuous functions, quotient spaces, and quotient mappings
• Peano curves
• completely regular and normal spaces
• Urysohn's Lemma and the Tietze Extension Theorem
• compactness, characterization of compactness in various classes of spaces
• paracompactness
• A.H.Stone's theorem on paracompactness of metric spaces.

Algebraic Topology

• Fundamental group, homology of complexes, singular homology and cohomology
• polyhedra and CW-complexes
• simplicial complexes
• homology and homotopy groups of spheres
• higher homotopy groups
• Euclidean spaces (Jordan theorem, Brouwer fixed point theorem, topological invariance of open sets)
• manifolds and Poincare duality
• characteristic classes of vector bundles.

Bibliography

1. R.Engelking, General Topology (revised and completed ed.), Heldermann Verlag, 1989.
2. J.Dugundji, Topology.
3. A.Hatcher, Algebraic Topology, Cambridge University Press, 2002.
4. A.Dold, Lectures on Algebraic Topology, Springer, 1995.
5. P. May, A Concise Course in Algebraic Topology, University of Chicago Press, 1999.