Decomposition of Cellular Balleans
  A ballean is a set endowed with some family of its subsets which are called the balls. We postulate the properties of the family of balls in such a way that the balleans can be considered as the asymptotic counterparts of the uniform topological spaces. The isomorphisms in the category of balleans are called asymorphisms. Every metric space can be considered as a ballean. The ultrametric spaces are prototypes for the cellular balleans. We prove some general theorem about decomposition of a homogeneous cellular ballean in a direct product of a pointed family of sets. Applying this theorem we show that the balleans of two uncountable groups of the same regular cardinality are asymorphic. 
Ihor Protasov and Anastasiia Tsvietkova
math.GR math.GN