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Unit of study_

MATH4061: Metric Spaces

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.

Code MATH4061
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
An average mark of 65 or above in 12cp from the following units (MATH2X21 or MATH2X22 or MATH2X23 or MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)
Assumed knowledge:
Real analysis and vector spaces. For example (MATH2922 or MATH2961) and (MATH2923 or MATH2962)

At the completion of this unit, you should be able to:

  • LO1. Explain and apply the foundational ideas of point set topology
  • LO2. Construct metric topologies, define open/closed sets of the metric spaces, and identify and explain the relationships between these and continuity and other properties of the metric space.
  • LO3. Find and rigorously prove properties of several important classes of sequences in metric spaces
  • LO4. Synthesise knowledge from fundamental theorems of topological spaces and continuous mappings and use this to prove new results
  • LO5. Demonstrate a broad and deep understanding of important concepts in topology and exercise critical thinking to identify and use these concepts to analyse examples and create novel conclusions
  • LO6. Solve problems about differential equations using the contraction mapping theorem
  • LO7. Apply the concepts of separable spaces and separation properties in both simple and complex examples.
  • LO8. Write proofs and transfer the theory of metric spaces to other areas of mathematics.

Unit outlines

Unit outlines will be available 1 week before the first day of teaching for the relevant session.