## A Topological AperitifThis is a book of elementary geometric topology, in which geometry, frequently illustrated, guides calculation. The book starts with a wealth of examples, often subtle, of how to be mathematically certain whether two objects are the same from the point of view of topology.After introducing surfaces, such as the Klein bottle, the book explores the properties of polyhedra drawn on these surfaces. Even in the simplest case, of spherical polyhedra, there are good questions to be asked. More refined tools are developed in a chapter on winding number, and an appendix gives a glimpse of knot theory.There are many examples and exercises making this a useful textbook for a first undergraduate course in topology. For much of the book the prerequisites are slight, though, so anyone with curiosity and tenacity will be able to enjoy the book. As well as arousing curiosity, the book gives a firm geometrical foundation for further study."A Topological Aperitif provides a marvellous introduction to the subject, with many different tastes of ideas.Stephen Huggett and David Jordan have excellent credentials for explaining the beauty of this curiously austere but potentially enormously general form of geometry".Professor Sir Roger Penrose OM FRS, Mathematical Institute, Oxford, UK |

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### Contents

Homeomorphic Sets | 1 |

Topological Properties | 17 |

Equivalent Subsets | 29 |

Surfaces and Spaces | 61 |

Polyhedra | 77 |

Winding Number | 103 |

Continuity | 117 |

Knots | 125 |

History | 133 |

Solutions | 139 |

163 | |

165 | |

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### Common terms and phrases

ambient isotopic angle function antiprisms chapter closed and bounded closed disc closed path closure compact completes the proof construct continuous mapping corresponding cross caps cut-pairs cut-points cylinder define definition deformation edge points equivalent subsets Euclidean set Euler number Euler's theorem Example 3.6 Exercise f is continuous faces follows formula four geometrical give given glued Hence hexagons homeomorphic sets idea indicated in Figure integer interval 0,1 inverse isomorphic joining Jones polynomial Klein bottle knot invariant Mobius band n-point n)-circle neighbourhood non-orientable north pole number of components number of edges open disc opposite points identified path-connected plane sets Platonic solids polyhedra pre-image real line real projective plane rectangle regular polygons regular polyhedron rooted trees round sets shown shown in Figure shrunk skein relation spherical polyhedron square stereographic projection Suppose symmetry topological property topological space torus consisting trefoil knot unit circle unknot vertex whereas the complement winding number