Real Analysis Exchange

Transitive properties of the ideal S2.

Jan Kraszewski

Full-text: Open access


In this paper we compute transitive cardinal coefficients of the $\sigma$-ideal $\mathbb{S}_2$, the least nontrivial productive $\sigma$-ideal of subsets of the Cantor space $2^\omega$. We also apply transitive operations to $\mathbb{S}_2$. In particular, we show that $\sigma$-ideal of strongly $\mathbb{S}_2$ sets is equal to $\mathbb{B}2$, one of Mycielski ideals.

Article information

Real Anal. Exchange, Volume 29, Number 2 (2003), 629-639.

First available in Project Euclid: 7 June 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 03E17: Cardinal characteristics of the continuum 03E02: Partition relations

Cantor space ideals transitive cardinal coefficients partitions


Kraszewski, Jan. Transitive properties of the ideal S 2 . Real Anal. Exchange 29 (2003), no. 2, 629--639.

Export citation


  • T. Bartoszyński, A note on duality between measure and category, Proc. Amer. Math. Soc., 128 (2000) 2745–2748.
  • T. Bartoszyński, H. Judah, Set Theory: On the structure of the real line, A. K. Peters, Wellesley, Massachusetts, 1995.
  • A. Blass, Reductions between cardinal characteristics of the continuum, in: Set theory (Boise, ID, 1992–1994), Contemp. Math. 192, Amer. Math. Soc., Providence, RI (1996) 31–49.
  • T. J. Carlson, Strong measure zero and strongly meager sets, Proc. Amer. Math. Soc., 118 (1993), 577–586.
  • J. Cichoń, J. Kraszewski, On some new ideals on the Cantor and Baire spaces, Proc. Amer. Math. Soc., 126 (1998), 1549–1555.
  • J. Cichoń, A. Krawczyk, B. Majcher-Iwanow, B. Węglorz, Dualization of the van Douwen diagram, J. Symbolic Logic, 65 (2000,) 959–968.
  • J. Cichoń, A. Rosłanowski, J. Steprans, B. Węglorz, Combinatorial properties of the ideal $\b2$, J. Symbolic Logic, 58 (1993), 42–54.
  • A. Kamburelis, B. Węglorz, Splittings, Arch. Math. Logic, 35 (1996,) 263–277.
  • J. Kraszewski, Properties of ideals on generalized Cantor spaces, J. Symbolic Logic, 66 (2001), 1303–1320.
  • K. Kunen, Random and Cohen reals, in: Handbook of Set-theoretical Topology, North-Holland, Amsterdam, (1984), 889–911.
  • M. Kysiak, On Erdős-Sierpiński duality for Lebesgue measure and Baire category, Master's thesis, Warsaw, 2000 (in Polish).
  • B. Majcher-Iwanow, Cardinal invariants of the lattice of partitions, Commentat. Math. Univ. Carol., 41 (2000), 543–558.
  • V. I. Malychin, Topological properties of Cohen generic extension, Trans. Mosc. Math. Soc., 52 (1990), 1–32.
  • P. Matet, Partitions and filters, J. Symbolic Logic, 51 (1986), 12-21.
  • J. Mycielski, Some new ideals of subsets on the real line, Colloq. Math., 20 (1969), 71–76.
  • J. Pawlikowski, Powers of transitive bases of measure and category, Proc. Amer. Math. Soc., 93 (1985), 719–729.
  • A. Rosłanowski, On game ideals, Colloq. Math., 59 (1990), 159–168.
  • F. Rothberger, Eine \"Aquivalenz zwischen der Kontinuumhypothese under der Existenz der Lusinschen und Sierpińskischen Mengen, Fund. Math., 30 (1938), 215–217.
  • W. Seredyński, Some operations related with translation, Colloq. Math., 57 (1989), 203–219.