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An alternative definition of coarse structures

J. Dydak·C. S. Hoffland
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Abstract

John Roe \cite{Roe lectures} introduced coarse structures for arbitrary sets $X$ by considering subsets of $X\times X$. That definition, while natural for analysts, is a bit more difficult to digest for topologists and geometers. In this paper we introduce large scale structures on $X$ via the notion of uniformly bounded families and we show their equivalence to coarse structures on $X$. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.

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J. Dydak, C. S. Hoffland. 2006-05-20. An alternative definition of coarse structures. https://arxiv.org/abs/math/0605562

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