Metric Spaces - MAT00037H

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  • Department: Mathematics
  • Module co-ordinator: Dr. Stephen Connor
  • Credit value: 10 credits
  • Credit level: H
  • Academic year of delivery: 2019-20

Module summary

Pre-requisites

Pure Mathematics MAT00032I (see link in Pre-requisites section)

or

Pure Mathematics Option 1 MAT00040I

Related modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20

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
University - closed examination
Metric Spaces
2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
University - closed examination
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.

Indicative reading

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



The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.