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Transactions of the American Mathematical Society

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Analytic groups and pushing small sets apart


Author: Jan van Mill
Journal: Trans. Amer. Math. Soc. 361 (2009), 5417-5434
MSC (2000): Primary 54H05, 54E52, 03E15
DOI: https://doi.org/10.1090/S0002-9947-09-04665-0
Published electronically: May 12, 2009
MathSciNet review: 2515817
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Abstract: We say that a space $ X$ has the separation property provided that if $ A$ and $ B$ are subsets of $ X$ with $ A$ countable and $ B$ first category, then there is a homeomorphism $ f\colon X\to X$ such that $ f(A)\cap B=\emptyset$. We prove that a Borel space with this property is Polish. Our main result is that if the homeomorphisms needed in the separation property for the space $ X$ come from the homeomorphisms given by an action of an analytic group, then $ X$ is Polish. Several examples are also presented.


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  • 1. F. D. Ancel, An alternative proof and applications of a theorem of E. G. Effros, Michigan Math. J. 34 (1987), 39-55. MR 873018 (88h:54058)
  • 2. R. D. Anderson, D. W. Curtis, and J. van Mill, A fake topological Hilbert space, Trans. Amer. Math. Soc. 272 (1982), 311-321. MR 656491 (83j:57009)
  • 3. S. Baldwin and R. E. Beaudoin, Countable dense homogeneous spaces under Martin's axiom, Israel J. Math. 65 (1989), 153-164. MR 998668 (90f:54010)
  • 4. H. Becker and A. S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877 (98d:54068)
  • 5. D. P. Bellamy and K. F. Porter, A homogeneous continuum that is non-Effros, Proc. Amer. Math. Soc. 113 (1991), 593-598. MR 1070510 (91m:54016)
  • 6. R. Bennett, Countable dense homogeneous spaces, Fund. Math. 74 (1972), 189-194. MR 0301711 (46:866)
  • 7. L. Ding and S. Gao, On generalizations of Lavrentieff's theorem for Polish group actions, Trans. Amer. Math. Soc. 359 (2007), 417-426 (electronic). MR 2247897 (2007e:54045)
  • 8. E. K. van Douwen, A compact space with a measure that knows which sets are homeomorphic, Adv. Math. 52 (1984), 1-33. MR 742164 (85k:28018)
  • 9. E. G. Effros, Transformation groups and $ C^*$-algebras, Annals of Math. 81 (1965), 38-55. MR 0174987 (30:5175)
  • 10. R. Engelking, General topology, Heldermann Verlag, Berlin, second ed., 1989. MR 1039321 (91c:54001)
  • 11. I. Farah, M. Hrušák, and C. Martınez Ranero, A countable dense homogeneous set of reals of size $ \aleph_1$, Fund. Math. 186 (2005), 71-77. MR 2163103 (2006f:54031)
  • 12. A. Hohti, Another alternative proof of Effros' theorem, Top. Proc. 12 (1987), 295-298. MR 991756 (90e:54090)
  • 13. M. Hrušák and B. Zamora Avilés, Countable dense homogeneity of definable spaces, Proc. Amer. Math. Soc. 133 (2005), 3429-3435. MR 2161169 (2006h:54026)
  • 14. W. Hurewicz, Relativ perfekte Teile von Punktmengen und Mengen (A), Fund. Math. 12 (1928), 78-109.
  • 15. A. S. Kechris, Classical descriptive set theory, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • 16. S. Levi, On Baire cosmic spaces, Proceedings of the Fifth Prague Topological Symposium, Heldermann Verlag, Praha, pp. 450-454. MR 698438 (84d:54052)
  • 17. J. van Mill, Strong local homogeneity does not imply countable dense homogeneity, Proc. Amer. Math. Soc. 84 (1982), 143-148. MR 633296 (83e:54033)
  • 18. J. van Mill, Sierpiński's Technique and subsets of $ \mathbb{R}$, Top. Appl. 44 (1992), 241-261. MR 1173263 (94d:54096)
  • 19. J. van Mill, The infinite-dimensional topology of function spaces, North-Holland Publishing Co., Amsterdam, 2001. MR 1851014 (2002h:57031)
  • 20. J. van Mill, A note on the Effros Theorem, Amer. Math. Monthly. 111 (2004), 801-806. MR 2104051 (2005g:54069)
  • 21. J. van Mill, Homogeneous spaces and transitive actions by analytic groups, Bull. London Math. Soc. 39 (2007), 329-336. MR 2323467 (2008d:54031)
  • 22. J. van Mill, Homogeneous spaces and transitive actions by Polish groups, Israel J. Math. 165 (2008), 133-159. MR 2403618
  • 23. V. V. Uspenskiĭ, Why compact groups are dyadic, General topology and its relations to modern analysis and algebra, VI (Prague, 1986), Res. Exp. Math., vol. 16, Heldermann, Berlin, 1988, pp. 601-610. MR 952642 (89i:22005)

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Additional Information

Jan van Mill
Affiliation: Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081$^{a}$, 1081 HV Amsterdam, The Netherlands
Email: vanmill@few.vu.nl

DOI: https://doi.org/10.1090/S0002-9947-09-04665-0
Keywords: Analytic group, meager set, countable set, Polish space.
Received by editor(s): May 29, 2007
Received by editor(s) in revised form: October 18, 2007
Published electronically: May 12, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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