On isometric embeddings of separable metric spaces
isometrically saturated \(\omega\)-saturated class
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.
scientific article; zbMATH DE number 6361142