Topology Prelim topics

Topics

  • Topological spaces: basis for a topology, order topology, product topology, subspace topology.

  • Closed sets and limit points. Interior, exterior, closure and boundary.

  • Continuous functions.

  • Quotient topology. Quotient constructions. Cut and Paste analysis of Torus, projective space, Klein bottle, Mobius band and surfaces via quotient constructions.

  • Connectedness, local connectedness, path connectedness, components.

  • Compactness, limit point compactness, sequential compactness and local compactness. One point compactification construction.

  • Countability axioms: 1st and 2nd countability, Lindelof spaces, separable spaces.

  • Separation axioms: Hausdorff, T1, regular, normal and completely regular spaces. Urysohn Lemma, Urysohn Metrization Theorem, Tietze Extension Theorem.

  • Manifolds, tangent spaces and derivatives, smooth maps, inverse and implicit function theorems, local immersion and submersion theorems, regular value theorem.

  • Transversality and intersection. Transversality homotopy theorem, Sard's theorem, Mod-2 intersection and degree theory.

Main References

  1. J. R. Munkres, Topology, (2nd edition), Prentice Hall Inc, 2000. (Chapters 1 - 9)
  2. J. Dugundji, Topology Allyn and Bacon, Boston, 1966.
  3. S. Willard, General Topology. Addison-Wesley Publishing Company Reading Mass. 1970.
  4. V. Guillemin and A. Pollack, Differential Topology, American Mathematical Society, 2010