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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
##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 22A05 Structure of general topological groups
##### Keywords:
weak Lindelöf number; pseudocharacter; boundedness number
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