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Unions of chains of subgroups of a topological group. (English) Zbl 1018.54004
Let a topological group $$G$$ be the union of a chain of subgroups, and suppose that certain cardinal characteristics of the subgroups in the chain are bounded by an infinite cardinal $$\lambda$$. Which effect does this have on cardinal characteristics of the group $$G$$? The author shows that if the index of boundedness of each subgroup in the chain is strictly less than than $$\lambda$$, then the index of boundedness of $$G$$ is not greater than $$\lambda$$ [the result was obtained a bit earlier by R. D. Kopperman, M. W. Mislove, S. A. Morris, P. Nickolas, V. Pestov and S. Svetlichny, Houston J. Math. 22, 307-328 (1996; Zbl 0892.22001)]. This fact is applied in the article to show that if both the index of boundedness and the pseudocharacter of each subgroup in the chain are at most $$\lambda$$ and $$G$$ is countably compact, then $$|G|\leq 2^\lambda$$. The author also presents an example showing that, in the latter theorem, countable compactness cannot be relaxed to pseudocompactness.

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54H11 Topological groups (topological aspects) 54D30 Compactness 54D45 Local compactness, $$\sigma$$-compactness 22D05 General properties and structure of locally compact groups 54B10 Product spaces in general topology 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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