# Metric Spaces

## Unit aims

To introduce metric and topological spaces and to extend some theorems about convergence and continuity in the case of sequences of real numbers and real-valued functions of one real variable.

## Unit description

This course generalizes some theorems about convergence and continuity of functions from the Level 4 unit Analysis 1, and develops a theory of convergence and uniform convergence and in any *metric space*. Topics will include basic topology (open, closed, compact, connected sets), continuity of functions, completeness, the contraction mapping theorem and applications, compactness and connectedness.

## Relation to other units

This unit is a member of a sequence of analysis units at levels 5, 6 and 7. It is a prerequisite for Measure Theory & Integration, Advanced Topics in Analysis, and Functional Analysis.

## Learning objectives

At the end of the course the student should know and understand the definitions and theorems (and their proofs) and should be able to use the ideas of the course in unseen situations.

## Transferable skills

Assimilation of abstract ideas and reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.

## Syllabus

Metric spaces (definition, examples, open sets, closed sets, interior, closure, limit points, equivalent metrics, product metrics).

Completeness (limits, continuity, Cauchy sequence, complete sets, isometries, completion of a metric space, contraction mapping theorem, existence and uniqueness of ordinary differential equations).

Compactness (definition, examples, continuous functions, uniform continuity, Heine-Borel theorem, Arzela- Ascoli theorem).

Connectedness (definition, examples, **R**^{n}, components, continuous functions, path connectedness).

Introduction to point set topology.

## Reading and References

J.C. Burkill & H. Burkill, A second course in mathematical analysis, Cambridge University Press, Cambridge

I. Kaplansky, Set theory and metric spaces, Chelsea Publishing Company, New York.

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.

W. A. Sutherland, Introduction to metric and topological spaces, Clarendon Press, Oxford.

**Unit code:** MATH20006

**Level of study:** I/5

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturer:** Dr Viveca Erlandsson

## Pre-requisites

MATH10003 Analysis 1A, MATH10006 Analysis 1B, MATH10004 Foundations and Proof

## Co-requisites

None

## Methods of teaching

A standard lecture course of 33 lectures, 3 revision lectures and 9 problem classes.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.