On boundedly compact metrics and UC metrics
type: Item
instance of: scholarly article
title: On boundedly compact metrics and UC metrics
zbMATH Open document ID: 1461.54052
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2021-03-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1599616822
review text: The author introduces the concepts of independent and strongly independent properties of metric spaces. Let \(M\) be the class of all metric spaces and let \(P_1\), \(P_2\) be some properties of metric spaces. Write \(M_i\) for the subclass of \(M\) of spaces satisfying property \(P_i\), \[ M_i = \{(X, d) \in M \colon (X, d)\text{ has the property } P_i\}, \quad i = 1, 2. \] The properties \(P_1\) and \(P_2\) are independent if \(M_1 \cap M_2\), \(M_1 \setminus M_2\), \(M_2 \setminus M_1\) and \(M \setminus (M_1 \cup M_2)\) are nonvoid subclasses of \(M\). Let \((X, \tau)\) be a metrizable topological space and let \(M(\tau)\) be the set of all metric spaces \((X, d)\) for which \(d \colon X^2 \to \mathbb{R}\) is compatible with \(\tau\). Write \(M_i(\tau) = M_i \cap M(\tau)\), \(i = 1\), \(2\). \textbf{Definition}. The properties \(P_1\) and \(P_2\) are strongly independent if the following implications \[ \bigl(M_1(\tau) \neq \varnothing \neq M_2(\tau)\bigr) \Rightarrow \bigl(M_1(\tau) \cap M_2(\tau) \neq \varnothing\bigr), \] \[ \bigl(M_1(\tau) \neq \varnothing \neq M(\tau) \setminus M_2(\tau)\bigr) \Rightarrow \bigl(M_1(\tau) \cap (M(\tau) \setminus M_2(\tau)) \neq \varnothing\bigr), \] \[ \bigl(M_2(\tau) \neq \varnothing \neq M(\tau) \setminus M_1(\tau)\bigr) \Rightarrow \bigl(M_2(\tau) \cap (M(\tau) \setminus M_1(\tau)) \neq \varnothing\bigr), \] \[ \bigl(M(\tau) \setminus M_1(\tau) \neq \varnothing \neq M(\tau) \setminus M_2(\tau)\bigr) \Rightarrow \bigl(M(\tau) \setminus (M_1(\tau) \cup M_2(\tau)) \neq \varnothing\bigr) \] are valid for every metrizable topological space \((X, \tau)\). Recall that a metric space \((X, d)\) is boundedly compact if every closed bounded \(A \subseteq X\) is compact. Moreover, a metric space is an UC space (= Atsuji space) if every continuous \(f \colon Y \to \mathbb{R}\) is uniformly continuous. It is proved that boundedly compactness and UC-ness form a strongly independent pair of properties. Some other examples of independent…
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E45
Mathematics Subject Classification ID: 54E40
zbMATH DE Number: 7324951
zbMATH Keywords: boundedly compact metric
zbMATH Keywords: UC metric
zbMATH Keywords: Atsuji metric
zbMATH Keywords: strongly independent properties of metrics
author: Gerald Beer
reviewed by: Aleksey A. Dovgoshey
MaRDI profile type: Publication
OpenAlex ID: W3086203965
cites work: Q2960989
cites work: Lineability
cites work: Uniform continuity of continuous functions of metric spaces
cites work: UC Spaces Revisited
cites work: Q4277838
cites work: Between compactness and completeness
cites work: Q4904972
cites work: Q3990242
cites work: Uniform continuity and a new bornology for a metric space
cites work: Q5520521
cites work: Q5341487
cites work: On Uniform Continuity and Compactness in Metric Spaces
cites work: Q3593921
cites work: Q5810344
cites work: On Normal Metrics
cites work: Q5802778
cites work: Spaces whose finest uniformity is metric
cites work: Q4197032
DOI: 10.36045/BBMS/1599616822
description: scientific article; zbMATH DE number 7324951