# 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

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