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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.

By: The Open University

  • Duration 20 hours
  • Updated Monday 22nd February 2016
  • Advanced level
  • Posted under Mathematics
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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 Mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]

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