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).