A new inductive approach for counting dimension in large scale (Q1742508)

The author considers large scale invariants of metric spaces. The following concept appears in several of the definitions. An asymptotic resemblance relation (AS.R) is an equivalence relation \(\lambda\) on the family of all subsets of a set \(X\) with two additional properties: (i) If \(A_1\lambda B_1\) and \(A_2\lambda B_2\), then \((A_1\cup A_2)\lambda (A_2\cup B_2)\). (ii) If \(A_1, A_2 \neq\emptyset\) and \((A_1\bigcup A_2)\lambda B\), then there are nonempty subsets \(B_1\) and \(B_2\) of \(B\) such that \(A_1\lambda B_1\) and \(A_2\lambda B_2\); \(A_1, A_2, B_1, B_2 \subseteq X\). The pair \((X,\lambda)\) is called an asymptotic resemblance space. Let \((X, d)\) be a proper metric space. A subset \(C\) of \(X\) is called an asymptotic separator between asymptotically disjoint subsets \(A\) and \(B\) of \(X\) if \(\nu C\) is a separator between \(\nu A\) and \(\nu B\) in \(\nu X\). Here \(\nu X\) is the boundary of \(X\). The author introduces a number of concepts, such as asymptotic compactification, asymptotic corona, asymptotic separator, large scale separator, large scale inductive dimension, asymptotic cut, large scale continuum, large scale cut, large scale dimensiongrad. He develops properties of the introduced objects. In particular, he shows that the large scale dimensiongrad is different from the asymptotic dimension. He proves the following theorem. Assume that \((X,\lambda)\) and \((Y,\lambda')\) are two asymptotically equivalent \(AS.R\) spaces. Then \(\mathrm{asDg}_\lambda X = \mathrm{asDg}_{\lambda'}Y \).