## 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

Zbl 0564.05005
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### References:

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