BSc Computer Science and Mathematics / Course details

A metric space is a set together with a good definition of the distance between each pair of points in the set. Metric spaces occur naturally in many parts of mathematics, including geometry, fractal geometry, topology, functional analysis and number theory. This lecture course will present the basic ideas of the theory, and illustrate them with a wealth of examples and applications.

This course unit is strongly recommended to all students who intend to study pure mathematics and is relevant to all course units involving advanced calculus or topology.

This course aims to develop basic ideas of a metric space (essentially a set with a reasonable idea of distance on it). We will study a variety of important examples, such as the space of continuous functions on [0,1] and the space of vertices of a graph. We will develop some key properties of metric spaces that allow us to extend some theorems from the real line to other settings. Finally we will prove the contraction mapping theorem, a very important theorem with applications across mathematics, physics and economics.

On completion of this unit successful students will be able to:  

• Define several standard examples of metric spaces and prove simple results related to them.
• Determine whether a given metric space has any of the following properties: openness, closedness, completeness, compactness, path connectedness.
• Prove simple results related to all of the above notions, as well as that of continuity.
• State and prove the contraction mapping theorem along with some of its applications.

1.Basic Definitions. Euclidean metric, taxicab metric, discrete metric, edge metric, word metric, sup metric, L1 metric, Hausdorff metric, l2 metric, product metrics. Examples. [4 lectures]

2.Open and Closed Sets. Interior, closure, sequences and convergence, boundary. Denseness. Equivalent metrics. Examples. [4]

3.Uniform Convergence. Sequences of continuous functions. Examples. [2]

4.Continuous maps. Extending the elementary definition. Relationship with open sets, sequences. Examples [4]

5.Compactness. Open coverings. Continuous maps on compact sets. Compactness in Euclidean space. [4]

6.Completeness. Cauchy sequences. The Contraction Mapping Theorem, Examples. [3]

• Coursework; Weighting within unit 20%

• End of semester examination; Weighting within unit 80%

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Two books are particularly relevant. The first is

Wilson A. Sutherland, Introduction to Metric and Topological Spaces, Oxford University Press (Second Edition) 2009

which contains almost all the material in the course, is beautifully written, and is highly recommended. Copies are available to purchase in Blackwells, and to borrow from the JRUL. For an alternative view, try

Micheal O'Searcoid, Metric Spaces, Springer 2006.