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Transactions of the American Mathematical Society

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Luzin gaps


Author: Ilijas Farah
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2197-2239
MSC (2000): Primary 03E50, 03E65, 06E05
DOI: https://doi.org/10.1090/S0002-9947-04-03565-2
Published electronically: February 2, 2004
MathSciNet review: 2048515
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Abstract: We isolate a class of $F_{\sigma\delta}$ ideals on $\mathbb{N} $ that includes all analytic P-ideals and all $F_\sigma$ ideals, and introduce `Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients $\mathcal{P}(\mathbb{N} )/\mathcal{I}$. We also prove that the ideals $\operatorname{NWD}(\mathbb{Q} )$ and $\operatorname{NULL}(\mathbb{Q} )$have the Radon-Nikodým property, and (using OCA$_\infty$) a uniformization result for $\mathcal{K}$-coherent families of continuous partial functions.


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  • 1. U. Abraham, M. Rubin, and S. Shelah, On the consistency of some partition theorems for continuous colorings, and the structure of $\aleph_1$-dense real order types, Annals of Pure and Applied Logic 29 (1985), 123-206. MR 87d:03132
  • 2. T. Bartoszynski and H. Judah, Set theory: on the structure of the real line, A.K. Peters, 1995. MR 96k:03002
  • 3. M. Bell, Compact ccc non-separable spaces of small weight, Topology Proceedings 5 (1980), 11-25. MR 83f:54027
  • 4. J.R.P. Christensen, V. Kanovei, and M. Reeken, On Borel orderable groups, Topology Appl. 109 (2001), 285-299. MR 2001m:03092
  • 5. W.W. Comfort and S. Negrepontis, Theory of ultrafilters, Springer, 1974. MR 53:53135
  • 6. H.G. Dales and W.H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series, vol. 115, Cambridge University Press, 1987. MR 90d:03101
  • 7. A. Dow and K.P. Hart, The measure algebra does not always embed, Fundamenta Mathematicae 163 (2000), 163-176. MR 2001g:03089
  • 8. T. Downarowicz and A. Iwanik, Quasi-uniform convergence in compact dynamical systems, Studia Mathematica 89 (1988), 11-25. MR 89j:54042
  • 9. I. Farah, Cauchy nets and open colorings, Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 146-152, 50th anniversary of the Mathematical Institute, Serbian Academy of Sciences and Arts (Belgrade, 1996). MR 2000e:54021
  • 10. -, Completely additive liftings, The Bulletin of Symbolic Logic 4 (1998), 37-54. MR 99h:03025
  • 11. -, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Memoirs of the American Mathematical Society, vol. 148, no. 702, 2000, 177 pp. MR 2001c:03076
  • 12. -, Approximate homomorphisms II: Group homomorphisms, Combinatorica 20 (2000), 47-60. MR 2001h:28007
  • 13. -, Liftings of homomorphisms between quotient structures and Ulam stability, Logic Colloquium '98 (S. Buss, P. Hájek, and P. Pudlák, eds.), Lecture notes in logic, vol. 13, A.K. Peters, 2000, pp. 173-196. MR 2001e:03085
  • 14. -, Analytic Hausdorff gaps, Set Theory, DIMACS Series, vol. 58, American Mathematical Society, 2002, pp. 65-72. MR 2003e:03092
  • 15. -, Rigidity conjectures, Proceedings of Logic Colloquium 2000, to appear, available at http://www.math.yorku.ca/$\sim$ifarah.
  • 16. I. Farah and S. Solecki, Two $F_{\sigma\delta}$ ideals, Proceedings of the American Mathematical Society 131 (2003), 1971-1975.
  • 17. D.H. Fremlin, Measure theory, vol. 4, Torres-Fremlin, 2003.
  • 18. -, Notes on FARAH P99, preprint, University of Essex, June 1999.
  • 19. D.H. Fremlin and K. Kunen, Essentially unbounded chains in compact sets, Mathematical Proceedings of the Cambridge Philosophical Society 1-9 (1991), 149-160. MR 91i:03094
  • 20. F. Hausdorff, Summen von $\aleph_1$ Mengen, Fundamenta Mathematicae 26 (1936), 241-255.
  • 21. S.-A. Jalali-Naini, The monotone subsets of cantor space, filters and descriptive set theory, Ph.D. thesis, Oxford, 1976.
  • 22. W. Just, A modification of Shelah's oracle c.c. with applications, Transactions of the American Mathematical Society 329 (1992), 325-341. MR 92j:03047
  • 23. -, A weak version of AT from OCA, Mathematical Science Research Institute Publications 26 (1992), 281-291. MR 94g:03099
  • 24. W. Just and A. Krawczyk, On certain Boolean algebras $\mathcal{P}(\omega)/I$, Transactions of the American Mathematical Society 285 (1984), 411-429. MR 86f:04003
  • 25. V. Kanovei and M. Reeken, On Ulam's problem of approximation of non-exact homomorphisms, preprint, 1999.
  • 26. -, On Ulam's problem concerning the stability of approximate homomorphisms, Tr. Mat. Inst. Steklova 231 (2000), 249-283; English transl., Proc. Steklov Inst. Math. 2000, no. 4 (231), 238-270. MR 2002d:03082
  • 27. -, New Radon-Nikodym ideals, Mathematika 47 (2002), 219-227. MR 2003g:03071
  • 28. A.S. Kechris, Classical descriptive set theory, Graduate texts in mathematics, vol. 156, Springer, 1995. MR 96e:03057
  • 29. K. Kunen, $\langle \kappa,\lambda^*\rangle $-gaps under MA, preprint, 1976.
  • 30. -, Set theory: An introduction to independence proofs, North-Holland, 1980. MR 82f:03001
  • 31. C. Laflamme, Forcing with filters and complete combinatorics, Annals of Pure and Applied Logic 42 (1989), 125-163. MR 90d:03104
  • 32. {\cyr N.} {\cyr Luzin}, {\itcyr O chastyah naturalp1nogo ryada}, {\cyr Izv. AN SSSR, seriya mat.} 11, {\cyr N05} (1947), 714-722. MR 9:82c
  • 33. K. Mazur, $F_\sigma$-ideals and $\omega_1\omega_1^*$-gaps in the Boolean algebra $\mathcal{P}(\omega)/I$, Fundamenta Mathematicae 138 (1991), 103-111. MR 92g:06019
  • 34. S. Shelah, Proper forcing, Lecture Notes in Mathematics 940, Springer, 1982. MR 84h:03002
  • 35. S. Shelah and J. Steprans, PFA implies all automorphisms are trivial, Proceedings of the American Mathematical Society 104 (1988), 1220-1225. MR 89e:03080
  • 36. S. Solecki, Analytic ideals and their applications, Annals of Pure and Applied Logic 99 (1999), 51-72. MR 2000g:03112
  • 37. M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Mathematica 67 (1980), 13-43. MR 82e:28009
  • 38. S. Todorcevic, Partition problems in topology, Contemporary mathematics, vol. 84, American Mathematical Society, Providence, Rhode Island, 1989. MR 90d:04001
  • 39. -, Analytic gaps, Fundamenta Mathematicae 150 (1996), 55-67. MR 98j:03070
  • 40. -, Definable ideals and gaps in their quotients, Set Theory: Techniques and Applications (C.A. DiPrisco et al., eds.), Kluwer Academic Press, 1997, pp. 213-226. MR 99f:03066
  • 41. -, Gaps in analytic quotients, Fundamenta Mathematicae 156 (1998), 85-97. MR 99i:03059
  • 42. S. Todorcevic and I. Farah, Some applications of the method of forcing, Mathematical Institute, Belgrade and Yenisei, Moscow, 1995. MR 99f:03001
  • 43. B. Velickovic, Definable automorphisms of $\mathcal{P}(\omega) /\operatorname{Fin}$, Proceedings of the American Mathematical Society 96 (1986), 130-135. MR 87m:03070
  • 44. -, OCA and automorphisms of $\mathcal{P}(\omega) /\operatorname{Fin}$, Topology and its Applications 49 (1992), 1-12. MR 94a:03080

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Additional Information

Ilijas Farah
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada, M3J 1P3 – and – Matematicki Institut, Kneza Mihaila 35, Belgrade
Email: ifarah@mathstat.yorku.ca

DOI: https://doi.org/10.1090/S0002-9947-04-03565-2
Received by editor(s): October 15, 2001
Published electronically: February 2, 2004
Additional Notes: The author acknowledges support received from the National Science Foundation (USA) via grant DMS-0196153, PSC-CUNY grant #62785-00-31, the York University start-up grant, and the NSERC (Canada)
Article copyright: © Copyright 2004 American Mathematical Society

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