Aims

To introduce the language of metrics, a formalisation of the intuitive notion of distance. To illustrate how the framework of metric and topological spaces makes central concepts from real analysis, such as convergence of sequences and continuity of functions, applicable to a vast number of different settings.


Module Summary

The notion of distance between two objects is ubiquitous throughout mathematics and the sciences. From the familiar distance between two points in Euclidean space to the length of a shortest path between two computers in a network, there are various ways of making sense of the concept of distance depending on the context. This module presents the basic theory of metric spaces, that is, sets equipped with an abstract notion of distance. This approach leads to a widely applicable theory, which is why metric spaces form a foundational part of the modern mathematics curriculum.

The module will discuss basic analytic notions (convergence, Cauchy sequences, continuity) that will put the familiar theory from real analysis into a broader context, but also introduce some new topological and geometric ideas (connectedness, compactness, isometries, contractions).

Outline Of Syllabus

Metric spaces, subspaces, product spaces. Convergence of sequences. Cauchy sequences, complete metric spaces. Functions between metric spaces (continuous functions, isometries, contractions), the Banach Fixed-Point Theorem. Open sets and closed sets, convergence and continuity in terms of open sets. Interior, closure, boundary. Compact spaces, the Heine-Borel Theorem. Connected metric spaces. Topological spaces.

Teaching Rationale And Relationship

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lecture materials (video recordings, lecture notes) are used for the dissemination of theory and explanation of methods, illustrated with examples. Brief worksheets or online quizzes guide students’ independent study of the material and provide instant feedback on their understanding. Problem Classes provide the students with the opportunity to train the intended skill outcomes, receive feedback on their work, and ask questions. Drop-Ins provide students with the opportunity to fill the remaining gaps in their understanding and to receive guidance and hints to tackle the most difficult parts of the material.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.