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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2018

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Universal sets for ideals

Cieślak A., Michalski M.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 2

157--166

We consider the notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classical ideals like the null subsets of 2ω and the meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for E, the σ-ideal generated by closed null subsets of 2ω, and for some ideals connected with forcing notions: the Kσ subsets of ωω and the Laver ideal. We also consider Fubini products of ideals and show that there are Σ03 universal sets for N[symbol]M and M[symbol]N.