618-301 Metric Spaces

Note:

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Coordinator:

Prerequisite/s:

Timetable:

Contact:

Objectives:

On completion of this subject, students should:

Comprehend:

• the idea of a generalised distance (metric) 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:

• 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;

• the theory of completion of non-complete metric spaces;

• applications of theory to the approximate solution of differential equations by Picard's method.

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 (even Hausdorff) space;

• the power of convergence methods in the latter case.

Content:

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

Assessment: