Skip contents

# 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

Except for third party materials and otherwise stated in the acknowledgements section, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence

Please see full copyright statement and terms of use for more details.