A concept for resemblance in large scale geometry (Q6085222)

The notion of proximity studied in [\textit{V. A. Efremovič}, Doklady Akad. Nauk SSSR (N.S.) 76, 341--343 (1951)] is the axiomatisation of nearness for two subsets of an underlying set. In the previous paper [Rocky Mt. J. Math. 46, No. 4, 1231--1262 (2016; Zbl 1355.53019)], \textit{Sh. Kalantari} and \textit{B. Honari} introduced the notion of asymptotic resemblance (AS.R) as the large-scale counterpart of proximity. Formally, an AS.R on a set \(X\) is a binary relation \(\lambda\) on the power set \(\mathcal{P}(X)\) with some properties. In [ General Topology Appl. 4, 191--212 (1974; Zbl 0288.54004)], \textit{H. Herrlich} introduces the notion of nearness structure, which axiomatises the concept of nearness structure for a family of subsets. The paper under review aims to introduce and study the large scale counterpart of Herrlich's nearness, called large scale resemblance (LS.R). In Section 3, an LS.R on a set \(X\) is defined as a subset \(\mathfrak{C}\) of \(\mathcal{P}(\mathcal{P}(X))\) with some properties. In section 4, the relationship between LS.R spaces and Herrlich's near spaces is explored. In section 5, the notion of asymptotic dimension is generalised to the class of LS.R spaces. In section 6, the author proves that (i) the category of A-LS.R spaces \(\mathbf{A}\) is isomorphic to the category of AS.R spaces and (ii) \(\mathbf{A}\) is a reflexive full subcategory of large scale regular LS.R spaces \(\mathbf{R}\).