On isometric embeddings of separable metric spaces

In this paper, we consider spaces having the so-called property of f-distances, where f is a positive decreasing function defined on ω such that f(n)≤12n. It is proved that for well-known classes S of separable metric spaces (in [2] they are called isometrically ω-saturated classes of spaces) the following is true: for a given collection S of elements of S with the property of f-distances, there exists an element of S with the property of g-distances containing isometrically each element of S, where g is the function on ω for which g(n)=f(n+2), n∈ω.