MINIMAL UNIVERSAL METRIC SPACES

Let M be a class of metric spaces. A metric space Y is minimal M-universal if every X 2 M can be isometrically embedded in Y; but there are no proper subsets of Y; which satisfy this property. We find sucient conditions under which M has a minimal M-universal metric space and sucient conditions under which M has no minimal universal metric spaces. We generalize the notion of minimal M-universal metric space to notion of minimal M-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. Examples of minimal universal metric spaces are constructed for some subclasses of the class of metric spaces X; which possesses the following property. Among every three distinct points of X there is one point lying between the other two points.