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Universal sets for ideals

Volume 66 / 2018

Aleksander Cieślak, Marcin Michalski Bulletin Polish Acad. Sci. Math. 66 (2018), 157-166 MSC: Primary 54H05; Secondary 03E57. DOI: 10.4064/ba8146-5-2018 Published online: 30 November 2018

Abstract

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 $2^\omega $ 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 $\mathcal {E}$, the $\sigma $-ideal generated by closed null subsets of $2^\omega $, and for some ideals connected with forcing notions: the $\mathcal K_\sigma $ subsets of $\omega ^{\omega }$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $\Sigma ^0_3$ universal sets for $\mathcal N\otimes \mathcal M$ and $\mathcal M\otimes \mathcal N$.

Authors

  • Aleksander CieślakDepartment of Computer Science
    Faculty of Fundamental Problems of Technology
    Wrocław University of Science and Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • Marcin MichalskiDepartment of Computer Science
    Faculty of Fundamental Problems of Technology
    Wrocław University of Science and Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail

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