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On the number of compact subsets in topological groups. (English) Zbl 0632.54001
Let \({\mathcal K}\) be the set of all compact subsets of a nondiscrete \(T_ 2\) topological group G, and \(\psi\) (G) the pseudocharacter. The boundedness number bo(G) of G is defined by the smallest infinite cardinal \(\alpha\) such that for any open neighborhood V of the unit of G, there is a subset A of G with \(| A| \leq \alpha\) so that \(V\cdot A=G\). The author proves two theorems, namely, \(\psi\) (G)\(\leq | G| \leq | {\mathcal K}| \leq bo(G)^{\psi (G)}\) and, under GCH, \(| {\mathcal K}|^{\aleph_ 0}=| {\mathcal K}|\) for a pseudocompact G. Finally there are five examples which are useful to readers.
Reviewer: K.Iséki
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
22A05 Structure of general topological groups
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