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Core Module in Topology

Dr Francesco Russo
1st semester 2018
20 credits / 30 lectures

The course will introduce basic concepts and ideas from the field of topology. It will try to take a balanced view between analytic topology (that is, the generalization of the theory of metric spaces, which is mainly based on methods from set-theory and analysis) and geometric topology (which often investigates the connectivity properties of topological spaces by making use of methods from algebra).

  • Axiomatic set-theory/ordinals
  • Abstract topological spaces, continuous functions, homeomorphisms
  • New spaces from old: products and quotients
  • Convergence
  • Compactness, connectedness
  • Metrizability
  • Homotopies, retracts, deformation retracts
  • Covering maps and the fundamental group (with emphasis on the example of the real line covering the circle)
  • Topological manifolds

Prerequisites:

Students will be assumed to have taken 3MS (Metric Spaces) and a 3rd year module in Analysis (such as 3CA). Students who have not done this may be admitted to the module at the discretion of the Honours Program Convenor, but must assume responsibility for covering the missing material on their own.