  We prove the equivalence of the two important facts about finite metric
spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and
theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$
contains everywhere dense locally finite subgroup; Theorem G(Globalization):
For each finite metric space $F$ there exists another finite metric space $\bar
F$ and isometric imbedding $j$ of $F$ to $\bar F$ such that isometry $j$
induces the imbedding of the group monomorphism of the group of isometries of
the space $F$ to the group of isometries of space $\bar F$ and each partial
isometry of $F$ can be extended up to global isometry in $\bar F$. The fact
that theorem $G$, is true was announced in 2005 by author without proof, and
was proved by S.Solecki in \cite{Sol} (see also \cite{P,P1}) based on the
previous complicate results of other authors. The theorem is generalization of
the Hrushevski's theorem about the globalization of the partial isomorphisms of
finite graphs. We intend to give a constructive proof in the same spirit for
metric spaces elsewhere. We also give the strengthening of homogeneity of
Urysohn space and in the last paragraph we gave a short survey of the various
constructions of Urysohn space including the new proof of the construction of
shift invariant universal distance matrix from \cite{CV}.
