Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry ISO(U) contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space F there exists another finite metric space F̄ and isometric imbedding j of F to F̄ 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 F̄ and each partial isometry of F can be extended up to global isometry in F̄ . The fact that theorem G, is true was announced in 2005 by author without proof, and was proved by S.Solecki in [17] (see also [10, 11]) based on the previous complicate results of other authors. The theorem St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, 191023 St. Petersburg, Russia. E-mail: [email protected] The paper partially supported by grants of RFBR 05-01-00899 and NSh 4329.2006.1