Find characteristic properties of moduli f for which the equalities [W[S.sup.f]] - lim [A.sub.k] = A and [W[S.sup.f]] - lim [A.sub.k] = A are equivalent for all bounded metric spaces (X, [rho]), ([A.sub.k]) [subset] CL(X) and A [member of] CL(X).
In particular f(x) = x/(l + x) produces one of the standard bounded metrics equivalent to an initial metric [rho]: X x X [right arrow] [0, [infinity]) which can be unbounded.
By allowing [infinity] as a distance we have in effect restricted to bounded metric spaces.
Thus the bounded metric space (A, [d.sub.A]) can be transformed to the set with similarity (A, [L.sub.[0,[infinity]], [d.sub.A]) and embedded into sets with similarities.