## A First Course in Algebraic TopologyThis self-contained introduction to algebraic topology is suitable for a number of topology courses. It consists of about one quarter 'general topology' (without its usual pathologies) and three quarters 'algebraic topology' (centred around the fundamental group, a readily grasped topic which gives a good idea of what algebraic topology is). The book has emerged from courses given at the University of Newcastle-upon-Tyne to senior undergraduates and beginning postgraduates. It has been written at a level which will enable the reader to use it for self-study as well as a course book. The approach is leisurely and a geometric flavour is evident throughout. The many illustrations and over 350 exercises will prove invaluable as a teaching aid. This account will be welcomed by advanced students of pure mathematics at colleges and universities. |

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### Contents

Sets and groups | 1 |

metric spaces | 6 |

Topological spaces | 11 |

Continuous functions | 16 |

Induced topology | 20 |

Quotient topology and groups acting on spaces | 27 |

Product spaces | 39 |

Compact spaces | 44 |

The fundamental group | 124 |

The fundamental group of a circle | 135 |

Covering spaces | 143 |

The fundamental group of a covering space | 151 |

The fundamental group of an orbit space | 154 |

The BorsukUlam and hamsandwich theorems | 157 |

lifting theorems | 162 |

existence theorems | 170 |

Hausdorff spaces | 50 |

Connected spaces | 58 |

The pancake problems | 63 |

Manifolds and surfaces | 68 |

Paths and path connected spaces | 93 |

12A The Jordan curve theorem | 100 |

Homotopy of continuous mappings | 110 |

Multiplication of paths | 118 |

I Generators | 176 |

II Relations | 187 |

III Calculations | 194 |

The fundamental group of a surface | 202 |

I Background and torus knots | 209 |

II Tame knots | 221 |

28A Table of knots | 234 |

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### Common terms and phrases

abelian group base point bijective chapter circle closed path closed subset compact connected and locally connected subsets connected sum continuous function continuous map Corollary covering map deduce Definition denote discrete topology element equivalence classes equivalence relation example Exercises f is continuous finite number follows free group function f fundamental group G-space give given Hausdorff space hence Hint Hn(X homeomorphic homology theory homotopy equivalent I X I integer intuitively isomorphic Jordan curve Klein bottle Lemma Let f locally path connected map f metric space Mobius strip n-manifold non-empty open cover open neighbourhood open sets open subset path f Proof Let Proof Suppose quotient space quotient topology real numbers result S1 X S1 Seifert-Van Kampen theorem simple closed curve simply connected strong deformation retract subspace surface surjective Theorem Let topological space torus knot Ui n U2 unknotted usual topology words X X Y