from The Open University
Alternatively you can skip the navigation by pressing 'Enter'.
The world’s busiest railway 2015 – Mumbai Railway: Episode 4Saturday, 3rd October 2015 16:55 - BBC TwoThis final episode considers the challenges faced by the historic Chhatrapati Shivaji Terminus - and asks what the... Read more: The world’s busiest railway 2015 – Mumbai Railway: Episode 4
The Bottom Line: Autumn 2015: Art and the Business of TasteSaturday, 3rd October 2015 17:30 - BBC Radio 4
The Great British Year: WinterMonday, 5th October 2015 21:00 - BBC Four
Canals: The Making of a Nation: HeritageTuesday, 6th October 2015 20:00 - BBC Four
The Bottom Line: Autumn 2015: Art and the Business of TasteAvailable for over a yearHow do you value something like a painting? What makes one artist worth more than another? Who decides what is in... Read more: The Bottom Line: Autumn 2015: Art and the Business of Taste
BBC Inside Science: Preserving global diversity: Kew specialAvailable for over a year
Countdown To Life: The Extraordinary Making Of You: The Final PushAvailable until Saturday, 31st October 2015 00:15
The ascent of woman: PowerAvailable until Friday, 30th October 2015 02:30
Who will lead Britain out of the European UnionPerhaps it's not surprising that campaigners against the European Union don't want to work... Read more: Who will lead Britain out of the European Union
OpenLearn Live: 2nd October 2015Two mathematicians who threw shapes, Brexit, canals -a collection of free learning through... Read more: OpenLearn Live: 2nd October 2015
Learning to teachThis free course, which comprises four study units, is aimed at people who are considering... Try: Learning to teach now
English: skills for learningThis course is for anybody who is thinking of studying for a university degree and would like to... Try: English: skills for learning now
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.
- 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
Study this free course
Enrol to access the full course, get recognition for the skills you learn and track your progress. Make your learning visible!
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 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 courses or view the range of currently available OU Mathematics courses.
Copyright & revisions
Originally published: Thursday, 18th August 2011
- 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 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.