## Cellular Structures in TopologyThis book describes the construction and the properties of CW-complexes. These spaces are important because firstly they are the correct framework for homotopy theory, and secondly most spaces that arise in pure mathematics are of this type. The authors discuss the foundations and also developments, for example, the theory of finite CW-complexes, CW-complexes in relation to the theory of fibrations, and Milnor's work on spaces of the type of CW-complexes. They establish very clearly the relationship between CW-complexes and the theory of simplicial complexes, which is developed in great detail. Exercises are provided throughout the book; some are straightforward, others extend the text in a non-trivial way. For the latter; further reference is given for their solution. Each chapter ends with a section sketching the historical development. An appendix gives basic results from topology, homology and homotopy theory. These features will aid graduate students, who can use the work as a course text. As a contemporary reference work it will be essential reading for the more specialized workers in algebraic topology and homotopy theory. |

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### Contents

Adjunction of ncells | 11 |

CWcomplexes | 22 |

Subcomplexes | 33 |

Finiteness and countability | 40 |

Whitehead complexes | 51 |

Combinatorial complexes | 89 |

Simplicial sets | 132 |

Spaces of the type of CAV complexes | 223 |

Coverings | 248 |

Union spaces of expanding sequences | 273 |

Absolute neighbourhood retracts in the category of metric spaces | 281 |

Dimension and embedding | 299 |

306 | |

List of symbols | 313 |

321 | |

Appendix | 241 |

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### Common terms and phrases

adjoint adjunction of n-cells adjunction space anodyne extension assume barycentric base point based space c-space cellular characteristic map closed cells closed cofibration closed subspace codomain compact composition cone construction contained Corollary corresponding cosimplicial countable covering projection CW-complex CW-structure define deformation retract degeneracy operators denote dimension domain embedding Euclidean complex Example expanding sequence face operators function functor fundamental group geometric realization given homeomorphism homotopy groups homotopy H homotopy inverse homotopy rel implies inclusion induced interior point intersection isomorphism Kan fibration Lemma locally finite map g mapping cylinder Moreover n-ad n-connected n-simplex neighbourhood normal subdivision obtained open sets pair partial map partition of unity path-component path-connected presimplicial set Proof Let Proposition respectively Section simplex simplicial complex simplicial homotopy simplicial map simplicial set simplicial subset SiSets strong deformation retract subcomplex Theorem topology union space unique vertex vertices weak Hausdorff weak homotopy equivalence