Potentially compact Abelian groups.(Russian)Zbl 0704.22003

A group G is said to be potentially compact if, for every ultrafilter on G, there is a Hausdorff group topology on G such that the ultrafilter converges. A subgroup G of a topological group H is called a topological serving subgroup if, for every $$h\in H$$, there is $$g\in G$$ such that $$[Cl<g+h>]\cap G=\{0\}$$. Reduced p-groups and free groups of rank at least 2 are potentially compact, while free cyclic groups are not. In general, a group G is potentially compact iff it is a topological serving subgroup of its Bohr compactification. A periodic group is potentially compact iff G is reduced and the number of p-Sylow subgroups is finite.
Reviewer: D.L.Grant

MSC:

 22A05 Structure of general topological groups 54D30 Compactness 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) 20K45 Topological methods for abelian groups
Full Text: