Two classes of metric spaces
real-valued uniformly continuous functions
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 present a common framework where both classes of metric spaces can be studied. More precisely, the authors characterize these two classes of metric spaces in terms of uniform partitions. Along their framework of study, the authors are able to show that every small-determined metric space is \(B\)-simple. In addition, a new kind of metric spaces, namely, those spaces for which the Bourbaki-bounded subsets are exactly the bounded subsets, appears in a natural way.
scientific article; zbMATH DE number 6570460