Content: Roughly speaking, a metric space is any set provided with a sensible notion of the ``distance'' between points. The ways in which distance is measured and the sets involved may be very diverse. For example, the set could be the sphere, and we could measure distance either along great circles or along straight lines through the globe; or the set could be New York and we could measure distance ``as the crow flies'' or by counting blocks. Or the set might be the set of real valued continuous functions on the unit interval, in which case we could take as a measure of the distance between two functions either the maximum of their difference, or alternatively its ``root mean square''.
This module examines how the important concepts introduced in first year analysis, such as convergence of sequences, continuity of functions, completeness, etc, can be extended to general metric spaces. Applying these ideas we will be able to prove some powerful and important results, used in many parts of mathematics. For example, a continuous real-valued function on a compact metric space must be bounded. And such a function on a connected metric space cannot take both positive and negative values without also taking the value zero. Continuity is readily described in terms of open subsets, which leads us naturally to study the above concepts also in the more general context of a topological space, where, instead of a distance, it is declared which subsets are open.
Aims: To introduce the theory of metric and topological spaces; to show how the theory and concepts grow naturally from problems and examples.
Objectives: To be able to give examples which show that metric spaces are more general than Euclidean spaces, and that topological spaces are yet more general than metric spaces. To be able to work with continuous functions, and to recognize whether spaces are connected, compact or complete.
W A Sutherland, Introduction to Metric and Topological Spaces, OUP.
Other books worth consulting:
E T Copson, Metric Spaces, CUP.
W Rudin, Principles of Mathematical Analysis, McGraw Hill.
G W Simmons, Introduction to Topology and Modern Analysis, McGraw Hill. (More advanced, although it starts at the beginning; helpful for several third year and MMath modules in analysis).
A M Gleason, Fundamentals of Abstract Analysis, Jones and Bartlett.
D Epstein, Metric Spaces Lecture Notes, 1999-2000, Mathstuff.