Universal valued Abelian groups

Abstract

The counterparts of the Urysohn universal space in category of metric spaces and the Gurariǐ space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by G r (N ), is investigated and it is shown that each of G r (N )'s is homeomorphic to the Hilbert space l 2 . Those of G r (N )'s which are Urysohn as metric spaces are recognized. 'Linear-like' structures on G r (N ) are studied and it is proved that every separable metrizable topological vector space may be enlarged to G r (0) with a 'linear-like' structure which extends the linear structure of the given space.