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Transactions of the American Mathematical Society

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On generalizations of Lavrentieff's theorem for Polish group actions

Authors: Longyun Ding and Su Gao
Journal: Trans. Amer. Math. Soc. 359 (2007), 417-426
MSC (2000): Primary 54H05, 22F05
Published electronically: August 24, 2006
MathSciNet review: 2247897
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that for every Polish group $ G$ that is not locally compact there is a continuous action $ a$ of $ G$ on a $ \boldsymbol{\Pi}^1_1$-complete subset $ A$ of a Polish space $ X$ such that $ a$ cannot be extended to any superset of $ A$ in $ X$. This answers a question posed by Becker and Kechris and shows that an earlier theorem of them is optimal in several aspects.

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

  • 1. Howard Becker and Alexander 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
  • 2. Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597
  • 3. A. S. Kechris, A. Louveau and W. H. Woodin, The structure of $ \sigma$-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), no. 1, 263-288.MR 0879573 (88f:03042)
  • 4. M. Lavrentieff, Contribution à la théorie des ensembles homéomorphes, Fund. Math. 6 (1924), 149-160.

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Additional Information

Longyun Ding
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

Su Gao
Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76210

Received by editor(s): December 13, 2004
Published electronically: August 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.