Mathematical Notes

, Volume 91, Issue 5–6, pp 789–799 | Cite as

Effective compactness and sigma-compactness

  • V. G. Kanovei
  • V. A. Lyubetsky


Using the Gandy-Harrington topology and other methods of effective descriptive set theory, we prove several theorems about compact and σ-compact sets. In particular, it is proved that any Δ 1 1 -set A in the Baire space N either is an at most countable union of compact Δ 1 1 -sets (and hence is σ-compact) or contains a relatively closed subset homeomorphic to N (in this case, of course, A cannot be σ-compact).


effective descriptive set theory effectively compact σ-compact the Baire space Gandy-Harrington topology Δ11-set 


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  1. 1.
    Handbook of Mathematical Logic, Part B: Set Theory, Ed. by J. Barwise (North-Holland, Amsterdam, 1978).Google Scholar
  2. 2.
    V. G. Kanovei, “Luzin’s projective hierarchy: State of the art,” in Handbook of Mathematical Logic, PartB: Set Theory, Ed. by J. Barwise (Nauka, Moscow, 1982), Suppl. to Russian transl., pp. 273–364 [in Russian].Google Scholar
  3. 3.
    V. G. Kanovei, “The development of the descriptive theory of sets under the influence of the work of Luzin,” UspekhiMat. Nauk 40(3), 117–155 (1985) [RussianMath. Surveys 40 (3), 135–180 (1985)].MathSciNetGoogle Scholar
  4. 4.
    V. G. Kanovei and V. A. Lyubetsky, “On some classical problems of descriptive set theory,” Uspekhi Mat. Nauk 58(5), 3–88 (2003) [Russian Math. Surveys 58 (5), 839–927 (2003)].MathSciNetGoogle Scholar
  5. 5.
    V. G. Kanovei and V. A. Lyubetsky, Modern Set Theory: Borel and Projective Sets (MTsNMO, Moscow, 2010) [in Russian].Google Scholar
  6. 6.
    V. Kanovei, Borel Equivalence Relations: Structure and Classification, in Univ. Lecture Ser. (Amer. Math. Soc., Providence, R. I., 2008), Vol. 44.Google Scholar
  7. 7.
    L. A. Harrington, A. S. Kechris, and A. Louveau, “A Glimm-Effros dichotomy for Borel equivalence relations,” J. Amer.Math. Soc. 3(4), 903–928 (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    G. Hjorth, “Actions by the classical Banach spaces,” J. Symbolic Logic 65(1), 392–420 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    V. G. Kanovei, “Topologies generated by effectively Suslin sets, and their applications in descriptive set theory,” UspekhiMat. Nauk 51(3), 17–52 (1996) [Russian Math. Surveys 51 (3), 385–417 (1996)].MathSciNetGoogle Scholar
  10. 10.
    W. Hurewicz, “Relativ perfekte Teile von Punktmengen und Mengen (A),” Fundam. Math. 12, 78–109 (1928).zbMATHGoogle Scholar
  11. 11.
    J. Saint-Raymond, “Approximation des sous-ensembles analytiques par l’intérieur,” C. R. Acad. Sci. Paris Sér. A 281(2–3), 85–87 (1975).MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. S. Kechris, Classical Descriptive Set Theory, in Grad. Texts in Math. (Springer-Verlag, New York, 1995), Vol. 156.Google Scholar

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© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Institute for Problems of Information TransmissionMoscowRussia

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