Dominating and unbounded free sets

  title={Dominating and unbounded free sets},
  author={Slawomir Solecki and Otmar Spinas},
  journal={Journal of Symbolic Logic},
  pages={75 - 80}
Abstract We prove that every analytic set in ωω × ωω with σ-bounded sections has a not σ-bounded closed free set. We show that this result is sharp. There exists a closed set with bounded sections which has no dominating analytic free set. and there exists a closed set with non-dominating sections which does not have a not σ-bounded analytic free set. Under projective determinacy analytic can be replaced in the above results by projective. 
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    The Bulletin of Symbolic Logic
  • 2013
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SEREDYNSKI, Infinite free sets for small measure set mappings
  • Proceedings of the American Mathematical Society, vol
  • 1987
SAINT RAYMOND, Approximation des sous-ensembles analytiques par linterieur
  • Comptes Rendus I'Academie des Science Paris, Serie A, vol
  • 1975
KECHRIS, On a notion of smallness for subsets of the Baire space, Transactions of the American Mathematical Society, vol
  • 1977