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Nonstandard methods in large-scale topology. (English) Zbl 1415.54033

The aim of the paper under review is to extend the techniques from nonstandard analysis, which was developed by A. Robinson [Non-standard analysis. Amsterdam: Elsevier (1966; Zbl 0151.00803)], to bornological spaces and coarse spaces. In the first step, in order to clarify the connection between bornological and coarse structures, the author introduces the notion of prebornology, which is a generalization of bornology. He then generalizes the nonstandard notions of galaxy and finite closeness to prebornological spaces, and uses those notions to obtain the nonstandard characterizations of some large scale notions that include bornological maps and proper maps. In the second step, the author extends the nonstandard notions of galaxy and finite closeness to coarse spaces, and characterizes some large scale notions that include bornologous maps. He also characterizes some notions involving both small scale and large scale such as locally compactness for topological spaces with prebornologies, uniformly locally boundedness for uniform spaces with coarse structures. In particular, he characterizes an important notion in the study of large scale geometry, slowly oscillating functions. As an application of the nonstandard characterizations, he reproves that the class of Higson functions forms a \(C^\ast\)-algebra.

MSC:

54J05 Nonstandard topology
46A08 Barrelled spaces, bornological spaces
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citations:

Zbl 0151.00803
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References:

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