Unions of chains of subgroups of a topologucal group

Abstract

We consider the following problem: If a topological group G is the union of an increasing chain of subgroups and certain cardinal invariants of the subgroups in the chain are known, what can be said about G? We prove that if the index of boundedness of each subgroup is strictly less than λ for some infinite cardinal λ, then the index of boundedness of G is at most λ. We also prove that if both the index of boundedness and the pseudocharacter of each subgroup in the chain are at most λ and G is countably compact, then │G│≤2 ^λ. Finally, we show that the last assertion is not valid in general, not even for pseudocompact groups.