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Haar nonmeasurable partitions of compact groups. (English) Zbl 0887.22004
The paper deals with some aspects concerning the behaviour of compact groups. Mention must be made of the following results: Theorem 7. Let \(G\) be a nonmetrizable compact group. Then \(G\) can be partitioned into \(|G|\)-many pairwise disjoint dense homogeneously invariant \(\omega\)-bounded subsets each of which is of cardinality \(|G|\) and Haar nonmeasurable with full Haar outer measure. Theorem 8. Every infinite compact group \(G\) can be partitioned into \(|G|\) pairwise disjoint dense nonmeasurable sets of full Haar outer measure.
At the end of the paper we find an important corollary, namely: Let \(A\) be an infinite discrete topological space. Then its Stone-Čech compactification \(\beta A\) can be partitioned into a collection of size \(|\beta A|\) pairwise disjoint \(\omega\)-bounded sets each of which is itself of size \(|\beta A|\).

MSC:
22C05 Compact groups
22A05 Structure of general topological groups
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54B05 Subspaces in general topology
54B10 Product spaces in general topology
54B15 Quotient spaces, decompositions in general topology
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