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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: 2024-25
  • See module specification for other years: 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

Additional information

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 2024-25
B	Semester 1 2024-25

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

Special assessment rules

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

Indicative reading

To be confirmed.