An alternative definition of coarse structures

Details An alternative definition of coarse structures An alternative definition of coarse structures

2006-05-20

arxiv.org/abs/math/0605562v1

Mathematics

Metric Geometry

13 pages

Scientific paper

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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