Universal valued Abelian groups

作者：Niemiec Piotr^*

来源：Advances in Mathematics, 2013, 235: 398-449.

DOI：10.1016/j.aim.2012.12.005

• Valued Abelian group
• Abelian group of finite exponent
• Polish group
• Universal Polish Abelian group
• Urysohn universal metric space
• Extending continuous homomorphisms
• Universal disposition property
• Topological pseudovector group

The counterparts of the Urysohn universal space in the category of metric spaces and the Gurarii space in the 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)%26apos;s is homeomorphic to the Hilbert space l(2). Those of G(r)(N)%26apos;s which are Urysohn as metric spaces are recognized. %26apos;Linear-like%26apos; 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%26apos; structure which extends the linear structure of the given space.

• 出版日期2013-3-1