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).
Let [[A.sub.1].bar] = ([A.sub.1], [d.sub.A]) and [[A.sub.2].bar] ([A.sub.2], [d.sub.A]) be two induced subobjects of the bounded metric space [A.bar] = (A, [d.sub.A]).
By allowing [infinity] as a distance we have in effect restricted to bounded metric spaces. While the modest generalization of allowing [infinity] as a similarity value does not pose a serious restriction (databases are usually built from finite, and therefore bounded sets of data), it makes the set [0, [infinity], when ordered by the usual [greater than or equal to] relation, a complete lattice [L.sub.[0,[infinity]] as required in sets with similarity.
The Egli-Milner ordering from reflexive sets and the Hausdorff metric from bounded metric spaces can be generalized to sets with similarity as follows:
While in the special case of reflexive sets, the Egli-Milner relation tells us only when a table or an answer is interchangeable with another one, in the special case of bounded metric spaces, the Hausdorff metric allows a more fine-grained control of relaxation.