On isometric embeddings of separable metric spaces
type: Item
instance of: scholarly article
title: On isometric embeddings of separable metric spaces
zbMATH Open document ID: 1302.54056
published in: Topology and its Applications
publication date: 2014-10-24T00:00:00Z
reviewed by: Ivan Ivanšić
Mathematics Subject Classification ID: 54C25
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 54E45
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 6361142
zbMATH Keywords: seaparable metric spaces
zbMATH Keywords: isometric embedding
zbMATH Keywords: isometrically saturated \(\omega\)-saturated class
zbMATH Keywords: property of \(f\)-distances
zbMATH Keywords: \(f\)-uniform class of spaces
author: S. D. Iliadis
MaRDI profile type: Publication
full work available at URL: j.topol.2014.08.019
OpenAlex ID: W2027982288
cites work: Q4654006
cites work: A separable complete metric space of dimension \(n\) containing isometrically all compact metric spaces of dimension \(n\)
cites work: On isometric embeddings of compact metric spaces of a countable dimension
cites work: Countable Dimensional Universal Sets
cites work: On the countable sum of zero-dimensional metric spaces
cites work: Q4744459
cites work: On universal spaces for certain classes of infinite dimensional spaces
cites work: A universal separable metric locally finite-dimensional space
DOI: 10.1016/J.TOPOL.2014.08.019
review text: The paper treats eight classes of separable metric spaces \(\mathcal{S}(i), i\in\{1,2,...,8\}\). E.g., \(\mathcal{S}(3)\) is the class of all separable metric countable-dimensional spaces. It introduces a property called \(f\)-distances, where \(f:\omega \to \mathbb R\) satisfies \(f(n) \leq 1/2^n\) for every \(n\in \omega\). A space \(X\) has the property of \(f\)-distances if it has a base whose elements have the property of \(f\)-distances, i.e. for every element \(U\) of the base and every \(n\in \omega\) it holds: NEWLINE\[NEWLINE {\rho}_X(Cl_X(U),(X \setminus U) \setminus O_{1/2^n}(Bd_X(U))) \geq f(n). NEWLINE\]NEWLINE A collection \(S\) of separable metric spaces is called \(f\)-uniform if each element of \(S\) has the property of \(f\)-distances. Simply described, the obtained result is this: If we start with an \(f\)-uniform collection \(S\) of elements of \(\mathcal{S}(i), i\in \{1,2,...,8\}\) then there exists an element \(T\) of \(\mathcal{S}(i)\) having the property of \(g\)-distances, \(g(n)=f(n+2)\), and furthermore, \(T\) contains isometrically each element of started collection \(S\).NEWLINENEWLINEProofs are based on previous work of the author and the coauthor Naidoo.
description: scientific article; zbMATH DE number 6361142