Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem
type: Item
instance of: scholarly article
title: Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem
zbMATH Open document ID: 1246.54029
author: Hisao Kato
author: Masahiro Matsumoto
published in: Journal of the Mathematical Society of Japan
publication date: 2011-09-27T00:00:00Z
review text: There are various ways in which one can characterize covering dimension; one such way is by counting the number of open sets of a certain diameter needed to cover the whole space. E.g., the unit interval can be covered by roughly \(n\) open sets of diameter~\(1/n\), whereas the unit square requires roughly \(n^2\) such sets. This suggests that the quotient of the logarithm of the number of sets and the logarithm of the diameter may be related to the dimension of the space and, indeed, the Pontrjagin-Schnirelmann theorem gives an asymptotic formula for the covering dimension of a compact metrizable space that involves all possible diameters and all compatible metrics on the space. The authors derive similar formulas for the dimension of separable metrizable spaces using normal sequences of finite open covers. Given a sequence \(U=\langle\mathcal{U}_i:i\in\mathbb{N}\rangle\) of open covers one defines two numbers \(d_3(U)=\liminf_{i\to\infty}{1\over i}\log_3|\mathcal{U}_i|\) and \(d_2(U)=\liminf_{i\to\infty}{1\over i}\log_2|\mathcal{U}_i|\). The authors establish that \(\dim X\) is equal to the minimum value of \(d_3(U)\) taken over all sequences~\(U\) that are developments and for which \(\mathcal{U}_{i+1}\) is always a star-refinement of~\(\mathcal{U}_i\); a similar result with \(d_2(U)\) is obtained when one considers developments where \(\mathcal{U}_{i+1}\)~is always a barycentric refinement of~\(\mathcal{U}_i\).
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 5950727
zbMATH Keywords: covering dimension
zbMATH Keywords: normal sequence of open covers
zbMATH Keywords: box-counting dimension
zbMATH Keywords: Pontrjagin-Schnirelmann theorem
reviewed by: K. P. Hart
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06330919
OpenAlex ID: W2151316334
cites work: Q3994456
cites work: Bruijning-Nagata and Hashimoto-Hattori characteristics of covering dimension revisited
cites work: A characterization of dimension of topological spaces by totally bounded pseudometrics
cites work: A characterization of covering dimension by use of \(\Delta_k(X)\)
cites work: Q4855460
cites work: Q4431283
cites work: On Nagata's star-index \(*_{k}(X)\)
cites work: Q5794159
cites work: Q3667577
cites work: Open problems left in my wake of research
cites work: Sur une propriété métrique de la dimension
DOI: 10.2969/JMSJ/06330919
Recommended article: Addendum to: Characterizations of topological dimension by use of normal sequences of finite open covers and Pontrjagin-Schnirelmann theorem
Recommended article: Covering dimension and normality in \(L\)-topological spaces
Recommended article: On the normal cohomological dimension of open subsets of completely regular spaces
Recommended article: Q3814395
Recommended article: Q3692449
Recommended article: Q3217247
Recommended article: Topologization of entire sequences spaces
Recommended article: Q5472761
Recommended article: Covering dimension and ideal topological spaces
Recommended article: The dimension Dind of finite topological \(T_0\)-spaces
description: scientific article; zbMATH DE number 5950727