Metric Spaces - MAT00037H

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Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

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.

Module learning outcomes

• 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.

Module content

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.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

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.

Indicative reading

W A Sutherland, Introduction to Metric and Topological Spaces, OUP.