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# Metric Spaces - MAT00051I

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• Department: Mathematics
• Module co-ordinator: Dr. Jason Levesley
• Credit value: 20 credits
• Credit level: I
• Academic year of delivery: 2023-24

## Module summary

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 from Analysis studied in the first year.

## Related modules

• None

### Prohibited combinations

• None

This module is the first part of the Pure Mathematics stream, and as such must be taken with the second part (Groups, Rings & Fields)

Pre-requisite modules:

• Foundations & Calculus,
• Multivariable Calculus & Matrices
• Introduction to Pure Mathematics

## Module will run

Occurrence Teaching period
A Semester 1 2023-24
B Semester 1 2023-24

## 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 from Analysis studied in the first year

## Module learning outcomes

By the end of the module, students will be able to:

1. determine if a given space is a metric space and do calculations that involve the metric structure.

2. determine the continuity or otherwise of functions in abstract metric spaces and utilise continuity in a metric space setting.

3. determine the convergence or otherwise of sequences in abstract metric spaces.

4. determine when a metric space is complete and use completeness in calculations that involve theorems such as the contraction mapping theorem.

5. show a space is compact and prove results about compactness in abstract metric spaces.

## Module content

• Properties of a metric.

• Open sets, closed sets, interior and boundary.

• Sequences, functions, convergence and continuity in metric spaces.

• Continuity in terms of preimages.

• Pointwise and uniform convergence of sequences of functions.

• Completeness and the Contraction Mapping Theorem.

• Compactness in metric spaces, the Heine-Borel and Bolzano-Weierstrass theorems, existence of global extrema.

• Connectedness and the Intermediate Value Theorem.

## Assessment

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

None

### Additional assessment information

There will be five formative assignments with marked work returned in the seminars. At least one of them will contain a longer written part, done in LaTeX.

### Reassessment

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

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy