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# Metric Spaces - MAT00037H

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• Department: Mathematics
• Module co-ordinator: Dr. Jason Levesley
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2022-23
• See module specification for other years: 2021-22

• None

• None

## 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.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Metric Spaces
2 hours 100

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Metric Spaces
2 hours 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.