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COARSE AMENABILITY AND DISCRETENESS

Part of: Classical Topics in Algebraic Topology Special properties

Published online by Cambridge University Press:  21 October 2015

JERZY DYDAK
JERZY DYDAK*
Affiliation:
University of Tennessee, Knoxville, TN, USA email [email protected]
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Abstract

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This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

Keywords

absolute extensorsasymptotic dimensioncoarse geometrycoarse amenabilityLipschitz mapsProperty A

MSC classification

Primary: 54F45: Dimension theory
Secondary: 55M10: Dimension theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 65 - 77
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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