Stabilizers of closed sets in the Urysohn space.

*(English)*Zbl 1089.22019The Urysohn space \(U\) is the unique (up to isometry) separable metric space which is universal and ultrahomogeneous (meaning that \(U\) contains an isometric copy of every separable metric space and every isometry between two finite metric subspaces of \(U\) can be extended to an isometry of \(U\) onto itself). The existence of such a space was proved in two posthumously published articles by P. Urysohn himself [C. R. 180, 803-806 (1925; JFM 51.0452.03); Bulletin sc. Math. (2) 51; 43-64, 74-90 (1927; JFM 53.0556.01)]. The paper under review goes one important step further on the development of the rich theory which relatively recently has emerged concerning this space and its group of isometries. S. Gao and S. Kechris [On the classification of Polish metric spaces up to isometry. (Mem. Am. Math. Soc. 766, Am. Math. Soc., Providence, RI) (2003; Zbl 1012.54038)] proved that any Polish group is isomorphic (as a topological group) to the group of isometries of some Polish space, endowed with the topology of convergence on finite (equivalently, compact) subsets. This result and the characteristic properties of the Urysohn space led them to pose the following natural question, which is answered in the positive in the present paper: is every Polish group isomorphic to the subgroup of the isometry group of \(U\) formed by those isometries which map some convenient closed subset of \(U\) onto itself? As a by-product of the above quoted result, Gao and Kechris had provided a first approximation to a positive answer, showing that every Polish group is isomorphic to a countable intersection of such subgroups. The techniques used in both steps of the proof rely on Katětov’s construction of \(U\) by means of succesive one-point metric extensions.

Reviewer: Xabier Dominguez (La Coruña)