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$ R$-sets and category


Author: Rana Barua
Journal: Trans. Amer. Math. Soc. 286 (1984), 125-158
MSC: Primary 04A15; Secondary 03D55, 03E15, 28A05, 54H05
DOI: https://doi.org/10.1090/S0002-9947-1984-0756034-6
MathSciNet review: 756034
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Abstract: We prove some category theoretic results for $ R$-sets much in the spirit of Vaught and Burgess. Since the proofs entail many results on $ R$-sets and the $ R$-operator, we have studied them in some detail and have formulated many results appropriate for our purpose in, perhaps, a more unified manner than is available in the literature. Our main theorem is the following: Any $ R$-set in the product of two Polish spaces can be approximated, in category, uniformly over all sections, by sets generated by rectangles with one side an $ R$-set and the other a Borel set. In fact, we prove a levelwise version of this result. For $ C$-sets, this has been proved by V. V. Srivatsa.


References [Enhancements On Off] (What's this?)

  • [1] P. Aczel, Quantifiers, games and inductive definitions, Proc. Third Scandinavian Logic Sympos. (S. Kanger, Ed.), North-Holland, Amsterdam, 1975. MR 0424515 (54:12477)
  • [2] J. P. Burgess, Classical hierarchies from a modern standpoint, Part I: $ C$-sets, Fund. Math. 105 (1983), 81-95. MR 699874 (87a:04002a)
  • [3] -, Classical hierarchies from a modern standpoint, Part II: $ R$-sets, Fund. Math. 105 (1983), 97-105.
  • [4] -, What are $ R$-sets?, Patras Logic Symposion (G. Metakides, Ed.), North-Holland, Amsterdam, 1982.
  • [5] J. P. Burgess and D. E. Miller, Remarks on invariant descriptive set theory, Fund. Math. 90 (1975), 53-75. MR 0403975 (53:7784)
  • [6] J. P. Burgess and R. Lockhart, Classical hierarchies from a modern standpoint, Part III: $ BP$-sets, Fund. Math. 105 (1983), 107-118. MR 699876 (87a:04002c)
  • [7] P. G. Hinman, Hierarchies of effective descriptive set theory, Trans. Amer. Math. Soc. 142 (1969), 111-140. MR 0265161 (42:74)
  • [8] -, Recursion-theoretic hierarchies, Springer-Verlag, Berlin and New York, 1978. MR 499205 (82b:03084)
  • [9] -, The finite levels of the hierarchy of effective $ R$-sets, Fund. Math. 74 (1973), 1-10.
  • [10] L. Kantorovitch and E. Livenson, Memoir on the analytical operations and projective sets. I, II, Fund. Math. 18 (1932), 214-279; 20 (1933), 54-97.
  • [11] A. S. Kechris, Forcing in analysis, Higher Set Theory (Proc. Oberwolfach, Germany, 1977), Springer-Verlag, Berlin and New York, 1978. MR 520191 (80c:03051)
  • [12] A. A. Lyapunov, $ R$-sets, Trudy Mat. Inst. Steklov. 40 (1953). (Russian) MR 0064102 (16:226a)
  • [13] -, On the classification of $ R$-sets, Mat. Sb. 74 (1953), 255-262. (Russian)
  • [14] Y. N. Moschovakis, Elementary induction on abstract structure, North-Holland, Amsterdam, 1974. MR 0398810 (53:2661)
  • [15] -, Descriptive set theory, North-Holland, Amsterdam, 1980. MR 561709 (82e:03002)
  • [16] B. V. Rao, Remarks on analytic sets, Fund. Math. 66 (1970), 237-239. MR 0274689 (43:451)
  • [17] V. V. Srivatsa, Measure and category approximations for $ C$-sets, Trans. Amer. Math. Soc. 278 (1983), 495-505. MR 701507 (85e:04004)
  • [18] -, Existence of measurable selectors and parametrizations for $ {G_\delta }$-valued multifunctions, Fund. Math. (to appear).
  • [19] R. L. Vaught, Invariant sets in topology and logic, Fund. Math. 82 (1974), 269-294. MR 0363912 (51:167)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0756034-6
Keywords: Positive analytical operations, $ \delta - s$ operation, inductive definability, game, $ R$-operator, $ R$-set
Article copyright: © Copyright 1984 American Mathematical Society

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