Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics covered include: Metric spaces, convergence, completeness and the Contraction Mapping Theorem; Metric topology, open and closed subsets; Topological spaces, subspaces, product spaces; Continuous mappings and homeomorphisms; Compactness Connectedness Hausdorff spaces and normal spaces. You will learn methods and techniques of proving basic theorems in point-set topology and apply them to other areas of mathematics including basic Hilbert space theory and abstract Fourier series. By doing this unit you will develop solid foundations in the more formal aspects of topology, including knowledge of abstract concepts and how to apply them. Applications include the use of the Contraction Mapping Theorem to solve integral and differential equations.
3x1-hr lectures; 1x1-hr tutorial/wk
Quiz (10%); 2 x assignments (20%); final exam (70%)
Real analysis and vector spaces. For example (MATH2922 or MATH2961) and (MATH2923 or MATH2962)
A mark of 65 or greater in 12 credit points of 2000-level Mathematics unitsProhibitions