(Topics)Point-set topology: bases, metric spaces, continuous maps, homeomorphisms, connectedness, compactness, seperation and countability axioms, product and quotient spaces, projective spaces, topological groups. Algebraic topology: homotopy, fundamental group, covering spaces, retractions and fixed points, one-dimensional homology and cohomology. Manifolds: definitions, low-dimensional manifolds, smooth manifolds, immersions, embeddings, tangent vectors, vector bundles, orientation, transversality and cobordisms.

(Topics)Fundamental group, path homotopy, Van Kampen's theorem, covering spaces, simplicial and singular homology, cohomology groups, cup product, Poincare duality, homotopy groups and connections with cohomology, and additional advanced topics.

(Topics) Topological spaces, closed sets and closure, continuity and homeomorphism, compactness, connectedness, separability, nets, first countable spaces, product spaces, seperation axioms.