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.
To introduce and illustrate the main ideas and problems of topology.
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 (eg compact but not connected or connected but not path connected).
Chapter 1 of Allen Hatcher's book Algebraic Topology
MA Armstrong Basic Topology Springer (recommended but not essential).