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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ C_{0}$-coarse geometry of complements of Z-sets in the Hilbert cube


Authors: E. Cuchillo-Ibáñez, J. Dydak, A. Koyama and M. A. Morón
Journal: Trans. Amer. Math. Soc. 360 (2008), 5229-5246
MSC (2000): Primary 18B30, 54D35, 54E15; Secondary 54C55, 54E35, 54F45
Published electronically: May 20, 2008
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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.


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Additional Information

E. Cuchillo-Ibáñez
Affiliation: Departamento Matemática Aplicada, E.T.S.I. Montes, Universidad Politécnica, 28040 Madrid, Spain
Email: eduardo.cuchillo@upm.es

J. Dydak
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: dydak@math.utk.edu

A. Koyama
Affiliation: Department of Mathematics, Shizuoka University, Shizuoka, Japan
Email: sakoyam@ipc.shizuoka.ac.jp

M. A. Morón
Affiliation: Departamento Geometría y Topología, Facultad de Cc.Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email: ma\_moron@mat.ucm.es

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04603-5
PII: S 0002-9947(08)04603-5
Keywords: Covering dimension, asymptotic dimension, $C_{0}$-coarse structure, ANR-space, $C_{0}$-coarse morphism, uniformly continuous map, compact Z-set, Higson-Roe compactification and corona.
Received by editor(s): July 17, 2006
Published electronically: May 20, 2008
Additional Notes: The first and fourth named authors were supported by the MEC, MTM2006-0825.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.