### Abstract

Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube Q and the C_{0}-coarse category of their complements. The C_{0}-coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natural way some of the topological invariants of Z-sets to the geometry of their complements.

Original language | English |
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Pages (from-to) | 5229-5246 |

Number of pages | 18 |

Journal | Transactions of the American Mathematical Society |

Volume | 360 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2008 Oct |

Externally published | Yes |

### Keywords

- ANR-space
- Asymptotic dimension
- C-coarse morphism
- C-coarse structure
- Compact Z-set
- Covering dimension
- Higson-Roe compactification and corona
- Uniformly continuous map

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Cuchillo-Ibáñez, E., Dydak, J., Koyama, A., & Morón, M. A. (2008). C

_{0}-coarse geometry of complements of Z-sets in the hilbert cube.*Transactions of the American Mathematical Society*,*360*(10), 5229-5246. https://doi.org/10.1090/S0002-9947-08-04603-5