### 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 |
---|---|

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 |

### Fingerprint

### 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

### Cite this

_{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

**C _{0}-coarse geometry of complements of Z-sets in the hilbert cube.** / Cuchillo-Ibáñez, E.; Dydak, J.; Koyama, Akira; Morón, M. A.

Research output: Contribution to journal › Article

_{0}-coarse geometry of complements of Z-sets in the hilbert cube',

*Transactions of the American Mathematical Society*, vol. 360, no. 10, pp. 5229-5246. https://doi.org/10.1090/S0002-9947-08-04603-5

_{0}-coarse geometry of complements of Z-sets in the hilbert cube. Transactions of the American Mathematical Society. 2008 Oct;360(10):5229-5246. https://doi.org/10.1090/S0002-9947-08-04603-5

}

TY - JOUR

T1 - C0-coarse geometry of complements of Z-sets in the hilbert cube

AU - Cuchillo-Ibáñez, E.

AU - Dydak, J.

AU - Koyama, Akira

AU - Morón, M. A.

PY - 2008/10

Y1 - 2008/10

N2 - 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 C0-coarse category of their complements. The C0-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.

AB - 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 C0-coarse category of their complements. The C0-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.

KW - ANR-space

KW - Asymptotic dimension

KW - C-coarse morphism

KW - C-coarse structure

KW - Compact Z-set

KW - Covering dimension

KW - Higson-Roe compactification and corona

KW - Uniformly continuous map

UR - http://www.scopus.com/inward/record.url?scp=77951471739&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951471739&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-08-04603-5

DO - 10.1090/S0002-9947-08-04603-5

M3 - Article

AN - SCOPUS:77951471739

VL - 360

SP - 5229

EP - 5246

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -