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# Surfaces

Surfaces are a special class of topological spaces that crop up in many places in the...

Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this unit, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the classification theorem of compact surfaces.

By the end of this unit you should be able to:

- explain the terms surface, surface in space, disc-like neighbourhood and half-disc-like neighbourhood;
- explain the terms n-fold torus, torus with n holes, Möbius band and Klein bottle;
- explain what is meant by the boundary of a surface, and determine the boundary number of a given surface with boundary;
- construct certain compact surfaces from a polygon by identifying edges;
- explain how a surface in space can be regarded as a topological space;
- explain the terms orientable and non-orientable with regard to surfaces;
- describe the projective plane, and explain why it is non-orientable;
- outline the connections between the projective plane and the Möbius band;
- explain the terms subdivision and Euler characteristic;
- draw a subdivision of a given surface and calculate its Euler characteristic;
- appreciate that the Euler characteristics of all subdivisions of a given surface are the same;
- state and use the Classification Theorem for surfaces;
- explain the term identification topology;
- explain why an object obtained by identifying pairs of edges of a polygon is a surface.

- Duration 20 hours
- Updated Thursday 18th August 2011
- Advanced level
- Posted under Mathematics

## Contents

- Current section: Introduction
- Learning outcomes
- 1 Topological spaces and homeomorphism
- 2 Examples of surfaces
- 3 The orientability of surfaces
- 4 The Euler characteristic
- 5 Edge identifications
- Acknowledgements

# Surfaces

## Introduction

This unit is concerned with a special class of topological spaces called *surfaces*. Common examples of surfaces are the sphere and the cylinder; less common, though probably still familiar, are the torus and the Möbius band. Other surfaces, such as the projective plane and the Klein bottle, may be unfamiliar, but they crop up in many places in mathematics. Our aim is to *classify* surfaces – that is, to produce criteria that allow us to determine whether two given surfaces are homeomorphic.

This unit is from our archive and is an adapted extract from Topology (M338) which is no longer taught by The Open University. If you want to study formally with us, you may wish to explore other courses we offer in this subject area [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .

## Archive content

This is an extract from an Open University course which is no longer available to new students. If you found this interesting you could explore more free Mathematics course units or view the range of currently available OU Mathematics courses.