Topological structure of Urysohn universal spaces

The main aim of the paper is to prove that every nonempty member P of the algebra of subsets of a nontrivial Urysohn space generated by all balls (open and closed) is an l2-manifold of finite homotopy type. An algorithm of finding a polyhedron K such that P and K×l2 are homeomorphic is presented. An alternative proof of the Uspenskij theorem [V.V. Uspenskij, The Urysohn universal metric space is homeomorphic to a Hilbert space, Topology Appl. 139 (2004) 145–149] is given.