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MA3F1 Introduction to Topology

Lecturer: Colin Sparrow

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 one-hour lectures

Assessment: 85% 3 hour examination, 15% assignments

Formal registration prerequisites: None

Assumed knowledge:  

MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

  • Topological spaces
  • Continuous functions
  • Homeomorphisms
  • Compactness
  • Connectedness

MA136 Introduction to Abstract Algebra:

  • Groups
  • Subgroups
  • Homomorphisms and Isomorphisms

Useful knowledge:  

Synergies: The following modules go well together with Introduction to Topology:

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content: Topology is the study of properties of spaces invariant under continuous deformation. For this reason it is often called "rubber-sheet geometry''. The module covers: topological spaces and basic examples, compactness, connectedness and path-connectedness, identification topology, Cartesian products, homotopy and the fundamental group, winding numbers and applications, an outline of the classification of surfaces.

Aims: To introduce and illustrate the main ideas and problems of topology.

Objectives:

  • To explain how to distinguish spaces by means of simple topological invariants (compactness, connectedness and the fundamental group)
  • To explain how to construct spaces by gluing and to prove that in certain cases that the result is homeomorphic to a standard space
  • To construct simple examples of spaces with given properties (e.g. compact but not connected or connected but not path connected).

Books:
Chapter 1 of Allen Hatcher's book Algebraic Topology

For more reading, see the Moodle Pages (link below). MA Armstrong, Basic Topology Springer (recommended but not essential).

Additional Resources