Coarse proximity and proximity at infinity (Q1625530)

A proximity structure is a small-scale notion that axiomatizes the notion of nearness. The purpose of the paper under review is to define a coarse proximity structure, which means a large-scale analog of small-scale proximity, and to study its fundamental properties. First, the notion of coarse proximity is defined for metric spaces. It is a relation on the power set of a metric space \(X\) that captures the closeness at infinity. Then, the notion of coarse proximity relation \({\mathbf b}\) is defined for any sets \(X\) with bornologies \({\mathcal B}\) (a notion of a subset being bounded). A triple \((X, {\mathcal B}, {\mathbf b})\) of a set \(X\), bornology \({\mathcal B}\) and coarse proximity relation \({\mathbf b}\) is called a coarse proximity space. The notion of coarse neighborhood in a coarse proximity space \((X, {\mathcal B}, {\mathbf b})\) is introduced, and its basic properties are obtained. For metric spaces, this notion coincides with the notion of asymptotic neighborhood in the sense of \textit{G. Bell} and \textit{A. Dranishnikov} [Topology Appl. 155, No. 12, 1265--1296 (2008; Zbl 1149.54017)]. An equivalence relation \(\phi\) on the power set of a set \(X\), called weak asymptotic resemblance, is defined. Each coarse proximity space \((X, {\mathcal B}, {\mathbf b})\) induces a weak asymptotic resemblance. In the case of metric spaces, it is shown that two sets are \(\phi\)-related if and only if they have finite Hausdorff distance, and weak asymptotic resemblance coincides with asymptotic resemblance in the sense of \textit{Sh. Kalantari} and \textit{B. Honari} [Rocky Mt. J. Math. 46, No. 4, 1231--1262 (2016; Zbl 1355.53019)]. The notion of coarse proximity map between coarse proximity spaces is defined, and it is shown that coarse proximity spaces and closeness classes form a category, where closeness between coarse proximity maps is defined in terms of the induced weak asymptotic resemblance. Finally, for each metric space \((X, d)\), a proximity relation \(\delta\) is defined on the hyperspace at infinity \({\mathcal H}_\infty(X)\) of \(X\), which is defined as the set of all unbounded sets. This induces a proximity \({\boldsymbol \delta}\) on the set \({\mathbf B}X\) of all \(\phi\)-equivalence classes on \({\mathcal H}_\infty(X)\), where \(\phi\) is the weak asymptotic resemblance induced by the coarse proximity induced by \(d\). The proximity space \(({\mathbf B}X, {\boldsymbol \delta})\) is called the proximity space at infinity. It is also shown that this construction is functorial.