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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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