The Open University since 2006
Alternatively you can skip the navigation by pressing 'Enter'.
Exodus: Our Journey To Europe: EPISODE 2Thursday, 28th July 2016 00:45 - BBC Two (except Scotland and Northern Ireland)In episode 2, we continue Ahmad’s journey. Read more: Exodus: Our Journey To Europe: EPISODE 2
Exodus: Our Journey To Europe: EPISODE 2Available until Saturday, 27th August 2016 01:45In episode 2, we continue Ahmad’s journey. Read more: Exodus: Our Journey To Europe: EPISODE 2
Exodus: Our Journey To Europe: EPISODE 1Available until Friday, 26th August 2016 03:45
Discovery: The Truth About Success And Failure In Medicine: Going Lean: Heath and the Toyota WayAvailable for over a year
Thinking Allowed July 2016: Food bank Britain, Food poverty in EuropeAvailable for over a year
Managing virtual project teamsMany projects are now ‘virtual’, i.e. some or all of the team are located remotely and may be... Try: Managing virtual project teams now
An introduction to music theoryGain an understanding of the basic building blocks of musical theory and notation. This free... Try: An introduction to music theory now
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.
- This site has Copy Reuse Tracking enabled - see our FAQs for more information.
If you enjoyed this, why not follow a feed to find out when we have new things like it? Choose an RSS feed from the list below. (Don't know what to do with RSS feeds?)
Remember, you can also make your own, personal feed by combining tags from around OpenLearn.
All our alternative formats are free for you to download, for more information about the different formats we offer please see our FAQs. The most frequently used are Word (for accessibility), PDF (for print) and ePub and Kindle to download to eReaders*.
*Please note you will need an ePub and Mobi reader for these formats.