- Department: Mathematics
- Credit value: 10 credits
- Credit level: H
- Academic year of delivery: 2022-23
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Occurrence | Teaching period |
---|---|
A | Autumn Term 2022-23 |
This module introduces students to the concept of a metric space and presents the ideas of open and closed sets, convergence, continuity, completeness and compactness in this context. It provides a foundation for more advanced courses in Mathematical Analysis and a new perspective on many of the ideas studied in Real Analysis.
Understand and appreciate the concept of a metric space and be able to recognize standard examples.
Be familiar with the fundamental notions of continuity, convergence and compactness.
Be able to utilise metric space arguments to obtain a variety of results.
Syllabus
Metric spaces; examples.
Open sets, closed sets, interior and boundary; examples.
Sequences, functions, convergence and continuity in metric spaces; examples.
Continuity in terms of preimages; examples and applications.
Pointwise and uniform convergence of sequences of functions.
Completeness and the Contraction Mapping Theorem; examples and applications in areas such as differential equations and integral equations.
Compactness in metric spaces, the Heine-Borel and Bolzano-Weierstrass theorems, existence of global extrema; examples.
Connectedness and the Intermediate Value Theorem; examples and applications.
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
W A Sutherland, Introduction to Metric and Topological Spaces, OUP.