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Mycielski ideal and the perfect set theorem

Author: Miroslav Repicky
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2141-2150
MSC (2000): Primary 03E15; Secondary 03E17, 91A44
Published electronically: January 23, 2004
MathSciNet review: 2053988
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Abstract: We make several observations on the Mycielski ideal and prove a version of the perfect set theorem concerning this ideal for analytic sets: If $A\subseteq{}^\omega2$ is an analytic set all projections of which are uncountable, then there is a perfect set $B\subseteq A$ a projection of which is the whole space. We also prove that (a modification of) an infinite game of Mycielski is determined for analytic sets.

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  • 1. B. Balcar, J. Pelant, and P. Simon, The space of ultrafilters on $N$covered by nowhere dense sets, Fund. Math. 110 (1980), no. 1, 11-24; MR 82c:54003
  • 2. M. Balcerzak and A. Ros\lanowski, Coinitial families of perfect sets, J. Appl. Anal. 1 (1995), no. 2, 181-204; MR 97g:28001
  • 3. J. Cichon, A. Ros\lanowski, J. Steprans, and B. Weglorz, Combinatorial properties of the ideal $\mathfrak{P}_2$, J. Symbolic Logic 58 (1993), no. 1, 42-54; MR 94d:03098
  • 4. W. Debski, J. Kleszcz, and Sz. Plewik, Perfect sets of independent functions, Acta Univ. Carolinae Math. Phys. 33 (1992), no. 2, 31-33; MR 95i:54014
  • 5. E. van Douwen, The integers and topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.), North-Holland, Amsterdam-New York, 1984, pp. 111-167; MR 87f:54008
  • 6. J. Mycielski, Some new ideals of sets on the real line, Colloq. Math. 20 (1969), 71-76; MR 39:2934
  • 7. M. Repický, Permitted trigonometric thin sets and infinite combinatorics, Comment. Math. Univ. Carolinae 42 (2001), no. 4, 609-627; MR 2002k:03071
  • 8. A. Ros\lanowski, On game ideals, Colloq. Math. 59 (1990), no. 2, 159-168; MR 92e:04004

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Additional Information

Miroslav Repicky
Affiliation: Mathematical Institute of Slovak Academy of Sciences, Jesenná 5, 04154 Košice, Slovakia

Keywords: Mycielski ideal, analytic sets, perfect set theorem
Received by editor(s): August 21, 2002
Received by editor(s) in revised form: March 27, 2003
Published electronically: January 23, 2004
Additional Notes: This work was supported by a grant of Slovak Grant Agency VEGA 2/7555/20.
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

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