620-311 Metric Spaces

Note

Students may only gain credit for one of 620-311 and 618-301 (1997 Handbook).

Credit Points

Coordinator

Prerequisites

620-221, or 620-252 with a grade of H3 or better (1997 Handbook 618-201, or 618-252 with a grade of H3 or better).

Semester

Contact

Subject Description

Students completing this subject should comprehend

• the idea of a generalised distance between elements of an abstract set, including sets of functions;
• the notion of a general topological space and the generation of such a space from a metric space, and that such spaces may be generated in other ways;

have developed the ability to

• use a number of classical results for a finite product space including products of the real numbers by using general methods for arbitrary topological spaces as far as possible, including standard results concerning compactness and connectedness;
• use the theory of completion of non-complete metric spaces;
• apply the theory of metric spaces to the approximate solution of differential equations by Picard's method;

and appreciate

• the power of more general methods free of convergence arguments where applicable, and that the more specialised the structure the richer the theorems are likely to be;
• that there will be theorems true in products of the reals but not true in every metric space, and theorems true in any metric space but not in every topological space.

Metric spaces: properties of the real line; metrics and norms, open and closed sets. Convergence: convergence, completeness, continuity, compactness, connectedness; contraction mappings; applications.

Assessment

Up to 24 pages of written assignments and a three-hour end-of-semester written examination.