Journal of Mathematical Sciences

, Volume 158, Issue 5, pp 708–712 | Cite as

Borel reducibility as an additive property of domains

  • V. G. Kanovei
  • V. A. Lyubetsky

We prove that under certain requirements, if E and F are Borel equivalence relations, \( X = {\bigcup {_{n} } }X_{n} \), is a countable union of Borel sets, and E ↾ X n , is Borel reducible to F for all n, then E ↾ X is Borel reducible to F. Thus the property of Borel reducibility to F is countably additive as a property of domains. Bibliography: 19 titles.


Russia Equivalence Relation Wide Class Information Transmission Additive Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Barwise (ed.), Handbook of Mathematical Logic. Part II. Set Theory [Russian translation, with a supplement by V. G. Kanovei], Nauka, Moscow (1982).Google Scholar
  2. 2.
    A. M. Vershik, “Theory of orbits,” in: Dynamical Systems- 2, Vol. 2, Itogi Nauki i Tekhniki, VINITI, Moscow (1985), pp. 89–105.Google Scholar
  3. 3.
    V. Kanovei, “Supplement. Luzin’s projective hierarchy: the current state of the theory,” in: Handbook of Mathematical Logic. Part II. Set Theory [Russian translation], J. Barwise (ed.), Nauka, Moscow (1982), pp. 273–364.Google Scholar
  4. 4.
    V. Kanovei, “Topologies generated by effectively Suslin sets and their applications in descriptive set theory,” Uspekhi Mat. Nauk, 51, No. 3, 385–417 (1996).zbMATHMathSciNetGoogle Scholar
  5. 5.
    V. G. Kanovei and V. A. Lyubetsky, “On some classical problems in descriptive set theory,” Uspekhi Mat. Nauk, 58, No. 5, 839–927 (2003).MathSciNetGoogle Scholar
  6. 6.
    V. G. Kanovei and V. A. Lyubetsky, Modern Set Theory: Foundations of Descriptive Dynamics [in Russian], Nauka, Moscow (2007).Google Scholar
  7. 7.
    V. Kanovei and M. Reeken, “Some new results on the Borel irreducibility of equivalence relations,” Izv. Ross. Akad. Nauk Ser. Mat., 67, 59–82 (2003).MathSciNetGoogle Scholar
  8. 8.
    E. Shchegolkov, “On the uniformization of certain B-sets,” Dokl. Akad. Nauk SSSR, 59, 1065–1068 (1948).Google Scholar
  9. 9.
    J. Burgess and D. Miller, “Remarks on invariant descriptive set theory,” Fund. Math., 90, No. 1, 53–75 (1975).zbMATHMathSciNetGoogle Scholar
  10. 10.
    R. Dougherty, S. Jackson, and A. S. Kechris, “The structure of hyperfinite Borel equivalence relations,” Trans. Amer. Math. Soc., 341, No. 1, 193–225 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Su Gao, “Equivalence relations and classical Banach spaces,” in: Mathematical Logic in Asia. Proceedings of the 9th Asian Logic Conference, Novosibirsk, Russia (2005), pp. 70–89.Google Scholar
  12. 12.
    L. A. Harrington, A. S. Kechris, and A. Louveau, “A Glimm-Effros dichotomy for Borel equivalence relation,” J. Amer. Moth. Soc., 3, No. 4, 903–928 (1990).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    G. Hjorth, “Actions by the classical Banach spaces,” J. Symbolic Logic, 65, No. 1, 392–420 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    G. Hjorth and A. S. Kechris, “Recent developments in the theory of Borel reducibility,” Fund. Math., 170, No. 1–2, 21–52 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Hjorth and A. S. Kechris, Rigidity Theorems for Actions of Product Groups and Countoble Borel Equivalence Relations, Mem. Amer. Math. Soc., 177, No. 833 (2005).Google Scholar
  16. 16.
    A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York (1995).zbMATHGoogle Scholar
  17. 17.
    A. S. Kechris and A. Louveau, “The classification of hypersmooth Borel equivalence relations,” J. Amer. Math. Soc., 10, No. 1, 215–242 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    S. M. Srivastava, “Selection and representation theorems for o—compact valued multifunctions,” Proc. Amer. Math. Soc., 83, No. 4, 775–780 (1981).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Thomas, “Some applications of superrigidity to Borel equivalence relations,” in: Set Theory (Piscataway, New Jersey, 1999), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 58, Amer. Math. Soc., Providence, Rhode Island (2002), pp. 129–134.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsMoscowRussia

Personalised recommendations