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1996 | 149 | 3 | 191-203
Tytuł artykułu

Ramsey, Lebesgue, and Marczewski sets and the Baire property

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We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented.

 THEOREM. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets.

 THEOREM. In the Ellentuck topology on $[ω]^ω$, $(s)_0$ is a proper subset of the hereditary ideal associated with (s).

 We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not $B_r$-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology.
Twórcy
  • Department of Mathematics, S.E. Oklahoma State University, Durant, Oklahoma 74701, U.S.A., reardon@marcie.sosu.edu
Bibliografia
  • [Br] J. B. Brown, The Ramsey sets and related sigma algebras and ideals, Fund. Math. 136 (1990), 179-185.
  • [BrCo]J. B. Brown and G. V. Cox, Classical theory of totally imperfect spaces, Real Anal. Exchange 7 (1982), 1-39.
  • [Bu] C. Burstin, Eigenschaften messbaren und nichtmessbaren Mengen, Wien Ber. 123 (1914), 1525-1551.
  • [C] P. Corazza, Ramsey sets, the Ramsey ideal, and other classes over ℝ, J. Symbolic Logic 57 (1992), 1441-1468.
  • [E] E. Ellentuck, A new proof that analytic sets are Ramsey, ibid. 39 (1974), 163-165.
  • [GP] F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, ibid. 38 (1973), 193-198.
  • [GW] C. Goffman and D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116-121.
  • [GNN] C. Goffman, C. Neugebauer and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497-505.
  • [K] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966.
  • [L] A. Louveau, Une démonstration topologique de théorèmes de Silver et Mathias, Bull. Sci. Math. (2) 98 (1974), 97-102.
  • [M] E. Marczewski (Szpilrajn), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, Fund. Math. 24 (1935), 17-34.
  • [O] J. C. Oxtoby, Measure and Category, Springer, Amsterdam, 1971.
  • [P] S. Plewik, On completely Ramsey sets, Fund. Math. 127 (1986), 127-132.
  • [Sc] S. Scheinberg, Topologies which generate a complete measure algebra, Adv. in Math. 7 (1971), 231-239.
  • [Si] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60-64.
  • [T] F. Tall, The density topology, Pacific J. Math. 62 (1976), 275-284.
  • [W] J. T. Walsh, Marczewski sets, measure, and the Baire property. II, Proc. Amer. Math. Soc. 106 (4) (1989), 1027-1030.
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