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An alternative definition of coarse structures. (English) Zbl 1145.54032
Authors’ summary: J. Roe [Lectures on Coarse Geometry. University Lecture Series, vol. 31, Amer. Math. Soc., Providence, RI (2003; Zbl 1042.53027)] introduced coarse structures for arbitrary sets \(X\) by considering subsets of \(X\times X\). 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.

MSC:
54F45 Dimension theory in general topology
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54E35 Metric spaces, metrizability
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D40 Remainders in general topology
20H15 Other geometric groups, including crystallographic groups
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References:
[1] Bell, G.; Dranishnikov, A., A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory · Zbl 1117.20032
[2] N. Brodskiy, J. Dydak, M. Levin, A. Mitra, Hurewicz theorem for Assouad-Nagata dimension, math.MG/0605416, J. London Math. Soc., in press · Zbl 1154.54020
[3] N. Brodskiy, J. Dydak, A. Mitra, Švarc-Milnor lemma: A proof by definition, math.GT/0603487, Topology Proc. 31 (2007), in press · Zbl 1141.22301
[4] Cencelj, M.; Dydak, J.; Smrekar, J.; Vavpetič, A., Sublinear Higson corona and Lipschitz extensions · Zbl 1236.54031
[5] Dranishnikov, A.N.; Smith, J., On asymptotic assouad – nagata dimension · Zbl 1116.54020
[6] Grave, B., Asymptotic dimension of coarse spaces, available at · Zbl 1111.20038
[7] Gromov, M., Asymptotic invariants for infinite groups, (), 1-295
[8] Roe, J., Lectures on coarse geometry, University lecture series, vol. 31, (2003), Amer. Math. Soc. Providence, RI · Zbl 1042.53027
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