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Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this free course, 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.
After studying this course, 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.
- Current section: Introduction
- Learning outcomes
- 1 Topological spaces and homeomorphism
- 2 Examples of surfaces
- 2.1 Surfaces in space
- 2.2 Surfaces in space
- 2.3 Paper-and-glue constructions
- 2.4 Homeomorphic surfaces
- 2.5 Defining surfaces
- 3 The orientability of surfaces
- 4 The Euler characteristic
- 5 Edge identifications
- Keep on learning
Study this free course
Enrol to access the full course, get recognition for the skills you learn, track your progress and on completion gain a statement of participation to demonstrate your learning to others. Make your learning visible!
This course 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 OpenLearn course provides a sample of Level 3 study in
This free course includes adapted extracts from an Open University course which is no longer available to new students. If you found this interesting you could explore more free Mathematics courses or view the range of currently available OU Mathematics courses.
Copyright & revisions
Originally published: Thursday, 18th August 2011
Last updated on: Monday, 22nd February 2016
- Creative-Commons: The Open University is proud to release this free course under a Creative Commons licence. However, any third-party materials featured within it are used with permission and are not ours to give away. These materials are not subject to the Creative Commons licence. See terms and conditions. Full details can be found in the Acknowledgements and our FAQs section.
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