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Strong decomposability of ultrafilters on cardinals with countable cofinality. (English) Zbl 0549.04005

For an infinite cardinal \(\kappa\), \(U(\kappa)\) is the space of uniform ultrafilters on \(\kappa\). W. W. Comfort and N. Hindman [Math. Z. 149, 189-199 (1976; Zbl 0314.54039)] asked whether every point in \(U(\kappa)\) is a \(\kappa^+\)-point meaning: there exists a pwd. family \({\mathcal O}\) of open sets with \(| {\mathcal O}| =\kappa^+\) and \(x\in\bar 0\) for every 0\(\in {\mathcal O}.\)
The answer is affirmative in case \(\kappa\) is regular [cf. B. Balcar and P. Vojtáš, Proc. Am. Math. Soc. 79, 465-470 (1980; Zbl 0443.04005) and B. Balcar and P. Simon, Logic Colloquium ’80, Stud. Logic Found. Math. 108, 1-10 (1982; Zbl 0531.54004)]. The author shows that for \(\kappa\) uncountable and of countable cofinality the answer is also affirmative. The proof occupies the whole paper and witnesses once more the author’s ability to manipulate filters and almost disjoint families. The problem for singular cardinals of uncountable cofinality is - essentialy - still open.
Reviewer: K.P.Hart

MSC:

03E05 Other combinatorial set theory
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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