*MATH3079 *Metric Spaces and Topology

## Module Overview

A central idea of analysis is the notion of a limit of a function at a point, which captures the behaviour of a function close to the given point; here great care is required to specify what we mean by close. A typical answer requires some way of measuring distances between points, which on the real line is done with the use of the absolute value distance. Metric spaces provide the template for the most general situations where measuring distances makes sense, and provide a natural background for a more general treatment of analysis. Metric spaces allow us to discuss limiting properties of sequences, as well as continuity properties of functions. These lead to the concepts of open and closed sets, through which we discover a new class of spaces, called topological spaces. In particular, we will concentrate on three important properties of such spaces: compactness, completeness and connectedness. An important unification of these ideas is provided by the Contraction Mapping Theorem, which is a simple statement with far-reaching applications. The module will then move on to the study of more general notions of Topology, and our guiding principle will be the notion of continuity. We will begin with the observation that while in the context of metric spaces the definition of continuity seems to depend on a metric, the key features of a continuous map can be described in terms of open sets. This is the natural setting for Topology, which was developed in the early part of the twentieth century to provide rigorous foundations for Analysis

### Aims and Objectives

#### Module Aims

The module aims to give students an understanding of metric spaces and concepts and more general notions of topology.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

- Recall the defining properties of a metric space, and determine whether a given function defines a metric
- State and prove the Contraction Mapping Theorem
- Recall the definition of a topological space, and be able to verify the axioms in examples
- Apply the Contraction Mapping Theorem to problems in differential equations and numerical analysis
- Determine whether or not a given subset of a metric space is open or closed; determine the interior, closure, and boundary of a given set
- Prove straightforward results concerning open and closed sets
- Understand the construction and basic properties of the Cantor set
- State and prove the Heine-Borel theorem and use it to determine compactness of subsets of R^n
- Understand the concepts of subspace and product topologies
- Recall the definition of homeomorphism
- Recall the definitions of connectedness and compactness
- State Tychonov’s theorem and be able to use it in examples
- Be able to determine if a given space is Hausdorff, connected, path-connected, compact

### Syllabus

• Definition of metric space. Examples: Euclidean metric, taxicab metric, discrete metric. Metrics on spaces of sequences and functions. • Open and closed subsets of metric spaces. Interior, closure, boundary of a set. Equivalent metrics.Compactness, the Cantor set, and the Heine-Borel theorem. • Cauchy sequences and completeness. Key examples: the space of real numbers and the space of continuous functions on a closed interval. • Continuous maps of metric spaces and their properties. • Topological spaces. Basis of a topology. Subspaces, products. • Homeomorphisms • Quotient spaces, constructions and examples. • The Hausdorff condition. • Further properties of compact spaces, continuous maps on compact spaces. Tychonoff’s theorem. • Connected and path-connected spaces. • Contraction Mapping Theorem and its applications. • (If time permits) Elements of coarse geometry

### Learning and Teaching

#### Teaching and learning methods

Lectures, problem sheets, private study

Type | Hours |
---|---|

Independent Study | 150 |

Total study time | 150 |

#### Resources & Reading list

W A Sutherland. Introduction to Metric and Topological Spaces.

M O Searcoid. Metric Spaces.

### Assessment

#### Summative

Method | Percentage contribution |
---|---|

Class Test | 20% |

Exam (2 hours) | 80% |

#### Referral

Method | Percentage contribution |
---|---|

Exam | % |

#### Repeat Information

**Repeat type: Internal & External**

### Linked modules

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

Code | Module |
---|---|

MATH2003 | Group Theory |

MATH2039 | Analysis |

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.