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COARSE AMENABILITY AND DISCRETENESS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 100 / Issue 1 / February 2016
- Published online by Cambridge University Press:
- 21 October 2015, pp. 65-77
- Print publication:
- February 2016
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- Article
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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.
