Surfaces
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Contents

  • Introduction
  • Learning outcomes
  • 1 Topological spaces and homeomorphism
  • 2 Examples of surfaces
    • 2.1 Surfaces in space
    • 2.2 Surfaces in space
      • 2.2.1 Surfaces without boundary
      • 2.2.2 Hollow tubing surfaces
      • 2.2.3 Surfaces with boundary
      • 2.2.4 Boundary numbers
    • 2.3 Paper-and-glue constructions
      • 2.3.1 Cylinder
      • 2.3.2 Möbius band
      • 2.3.3 Torus
      • 2.3.4 Klein bottle
      • 2.3.5 Projective plane
      • 2.3.6 Torus with 1 hole
      • 2.3.7 Two-fold torus
      • 2.3.8 Sphere
    • 2.4 Homeomorphic surfaces
      • 2.4.1 Remarks
    • 2.5 Defining surfaces
  • 3 The orientability of surfaces
    • 3.1 Surfaces with twists
      • 3.1.1 Inserting half-twists
    • 3.2 Orientability
      • 3.2.1 Remarks
    • 3.3 The projective plane
  • 4 The Euler characteristic
    • 4.1 Nets on surfaces
    • 4.2 Subdivisions
    • 4.3 The Euler characteristic
    • 4.4 Historical note on the Euler characteristic
    • 4.5 Some general results
      • 4.5.1 Surfaces with holes
      • 4.5.2 n-fold toruses
    • 4.6 The Classification Theorem
      • 4.6.1 Remarks
  • 5 Edge identifications
    • 5.1 Identifying edges of a polygon
    • 5.2 The identification topology
      • 5.2.1 Proof
    • 5.3 Neighbourhoods
      • 5.3.1 Torus
      • 5.3.2 Klein bottle
      • 5.3.3 Torus with 1 hole
  • Conclusion
  • Acknowledgements

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