Universal valued Abelian groups

TL;DR

This paper constructs and analyzes universal valued Abelian groups analogous to the Urysohn and Gurarii spaces, exploring their geometry, topology, and linear structures, and establishing their uniqueness and homeomorphism to Hilbert spaces.

Contribution

It introduces universal separable valued Abelian groups of fixed exponents, proves their uniqueness, and investigates their geometric and linear properties, extending the theory of universal spaces.

Findings

G_r(N) groups are homeomorphic to Hilbert space l^2

Certain G_r(N) groups are Urysohn as metric spaces

Separable metrizable topological vector spaces can be embedded into G_r(0) with linear-like structures

Abstract

The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii 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.