Nonstandard methods in large-scale topology

TL;DR

This paper introduces nonstandard techniques for large-scale topology, generalizing bornology, and providing new characterizations of large-scale concepts using nonstandard analysis, with applications to Higson functions.

Contribution

It develops nonstandard characterizations for bornological and coarse spaces, extending existing notions and applying them to large-scale topology and functional analysis.

Findings

Nonstandard characterizations of large-scale notions

Generalization of bornology to prebornology

Proof that Higson functions form a C*-algebra

Abstract

We develop some nonstandard techniques for bornological and coarse spaces. We first generalise the notion of bornology to prebornology, which better fits to coarse spaces. We then give nonstandard characterisations of some basic large-scale notions in terms of galaxies and finite closeness relations, concepts that have been developed for metric spaces. Some hybrid notions that involve both small-scale and large-scale are also discussed. Finally we illustrate an application of our nonstandard characterisations to prove some elementary facts in large-scale topology and functional analysis, e.g., the fact that the class of Higson functions forms a $C^{\ast}$-algebra.