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Partitions and filters. (English) Zbl 0588.04011

The purpose of this paper is to study the properties of those filters that can be associated with the dual Galvin-Prikry theorem of T. J. Carlson and S. G. Simpson [Adv. Math. 53, 265-290 (1984; Zbl 0564.05005)]. Given \(X,Y\in (\omega)^{\omega}\) (the collection of all infinite partitions of \(\omega)\), write \(X\leq Y\) if X is coarser than Y, and denote by \(X\cap Y\) the finest partition Z that is coarser than both X and Y. Identify each partition of \(\omega\) with the associated equivalence relation, and put a topology on \((\omega)^{\omega}\) by restricting the product topology on \(2^{\omega \times \omega}\), 2 itself considered discrete. It is shown that assuming the continuum hypothesis, there exists an \(F\subset (\omega)^{\omega}\) such that: i) if \(X\in F\) and \(X\leq Y\), then \(Y\in F\); ii) if X,Y\(\in F\), then \(X\cap Y\in F\); and iii) for every Borel subset A of \((\omega)^{\omega}\), there is an \(X\in F\) such that the set \(\{Y\in (\omega)^{\omega}:\) \(Y\leq X\}\) is either included in A or else disjoint from A.

MSC:

03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
06B10 Lattice ideals, congruence relations

Citations:

Zbl 0564.05005
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References:

[1] DOI: 10.1016/0003-4843(77)90006-7 · Zbl 0369.02041
[2] DOI: 10.1016/0003-4843(70)90009-4 · Zbl 0222.02075
[3] Set theory (1978)
[4] DOI: 10.1016/0001-8708(84)90026-4 · Zbl 0564.05005
[5] DOI: 10.1016/0003-4843(70)90005-7 · Zbl 0231.02067
[6] Borel sets and Ramsey’s theorem 38 pp 193– (1973)
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