A new inductive approach for counting dimension in large scale
type: Item
instance of: scholarly article
title: A new inductive approach for counting dimension in large scale
zbMATH Open document ID: 1393.51006
author: Sh. Kalantari
published in: Archiv der Mathematik
publication date: 2018-04-11T00:00:00Z
review text: 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 \(\mathrm{asDg}_\lambda X = \mathrm{asDg}_{\lambda'}Y \).
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 18B30
zbMATH DE Number: 6858691
zbMATH Keywords: asymptotic dimension
zbMATH Keywords: asymptotic resemblance
zbMATH Keywords: coarse structure
zbMATH Keywords: large scale dimensiongrad
zbMATH Keywords: large scale geometry
MaRDI profile type: Publication
full work available at URL: s00013-017-1140-2
OpenAlex ID: W2781754958
cites work: Q4532495
cites work: Universal spaces for asymptotic dimension
cites work: Asymptotic dimension of coarse spaces.
cites work: Q3140233
cites work: Asymptotic resemblance
cites work: On large scale inductive dimension of asymptotic resemblance spaces
cites work: Q5606492
cites work: Q4437801
cites work: On asymptotic dimension of countable Abelian groups.
DOI: 10.1007/S00013-017-1140-2
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description: scientific article; zbMATH DE number 6858691