Two classes of metric spaces
type: Item
instance of: scholarly article
title: Two classes of metric spaces
zbMATH Open document ID: 1350.54022
author: M. I. Garrido
author: Ana Soledad Meroño
published in: Applied General Topology
publication date: 2016-04-19T00:00:00Z
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 46A17
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 6570460
zbMATH Keywords: metric spaces
zbMATH Keywords: real-valued uniformly continuous functions
zbMATH Keywords: real-valued Lipschitz functions
zbMATH Keywords: bornologies
zbMATH Keywords: Bourbaki-boundedness
zbMATH Keywords: countable uniform partitions
zbMATH Keywords: small-determined spaces
zbMATH Keywords: \(B\)-simple spaces
MaRDI profile type: Publication
full work available at URL: agt.2016.4401
OpenAlex ID: W2335715253
cites work: Uniform continuity of continuous functions of metric spaces
cites work: Lipschitz-type functions on metric spaces
cites work: Q4914057
cites work: New types of completeness in metric spaces
cites work: Q5341504
cites work: Q4383386
cites work: Constructing Metrics with the Heine-Borel Property
cites work: Techniques and examples in U-embedding
cites work: Q5589832
DOI: 10.4995/AGT.2016.4401
review text: Let \((X,d)\) be a metric space. A subset \(B \subseteq X\) is called \textit{\(d\)-bounded} when it has finite \(d\)-diameter, i.e., diam\(_d(B) = \sup \{ d(x, y): x, y \in B \} < \infty\). Recall that an \textit{\(\varepsilon\)-chain} in \(X\) of length \(m\) from \(x\) to \(y\), is any collection of points \(u_0,\dots, u_m \in X\) such that, \(u_0 = x, u_m = y\) and \(d(u_{i-1}, u_i) < \varepsilon\), for \(i = 1,\dots,m\). A subset \(B\) in \((X, d)\) is called \textit{Bourbaki-bounded} in \(X\) if for every \(\varepsilon >0\) there exist finitely many points \(x_1,\dots,x_n\) and some \(m \in \mathbb N\) such that \(B \subseteq \bigcup_{i=1}^n B^m_d(x_i, \varepsilon)\), where \(B^m_d(x_i, \varepsilon)\) denotes the set of points \(y \in X\) that can be joined to \(x_i\) by means of an \(\varepsilon\)-chain in \(X\) of length \(m\). Let \({\mathbb B}_d(X)\) (resp. \({\mathbb B}{\mathbb B}_d(X)\)) denote the family of all \(d\)-bounded (resp. Bourbaki-bounded) subsets in \((X,d)\). Recall that a real-valued function \(f\) on \((X, d)\) is said to be \textit{Lipschitz in the small} if there exist \(\delta >0\) and \(K >0\) such that \(|f(x)-f(y)| < Kd(x, y)\) whenever \(d(x, y) < \delta\). The metric space \((X, d)\) is said to be \textit{small-determined} whenever every real-valued Lipschitz in the small function is Lipschitz. Furthermore, \((X, d)\) is said to be \textit{\(B\)-simple} when there exists some metric \(\varrho\), uniformly equivalent to \(d\), such that \({\mathbb B}{\mathbb B}_d(X) = {\mathbb B}_\varrho(X)\).NEWLINENEWLINEThe class of small-determined metric spaces was firstly studied by \textit{M. I. Garrido} and \textit{J. A. Jaramillo} [J. Math. Anal. Appl. 340, No. 1, 282--290 (2008; Zbl 1139.46025)], and the class of \(B\)-simple metric spaces was firstly studied by \textit{J. Hejcman} [Commentat. Math. Univ. Carol. 38, No. 1, 149--156 (1997; Zbl 0886.54025)], respectively, from different motivations. In the paper under review, the authors pre…
reviewed by: Jiling Cao
description: scientific article; zbMATH DE number 6570460