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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Vaught’s conjecture on analytic sets
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by Greg Hjorth PDF
J. Amer. Math. Soc. 14 (2001), 125-143 Request permission


Let $G$ be a Polish group. We characterize when there is a Polish space $X$ with a continuous $G$-action and an analytic set (that is, the Borel image of some Borel set in some Polish space) $A\subset X$ having uncountably many orbits but no perfect set of orbit inequivalent points. Such a Polish $G$-space $X$ and analytic $A$ exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of $G$ onto the infinite symmetric group, $S_\infty$, consisting of all permutations of $\mathbb {N}$ equipped with the topology of pointwise convergence.
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Additional Information
  • Greg Hjorth
  • Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
  • Email:
  • Received by editor(s): June 8, 1998
  • Received by editor(s) in revised form: June 22, 2000
  • Published electronically: September 18, 2000
  • Additional Notes: The author’s research was partially supported by NSF grant DMS 96-22977.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 14 (2001), 125-143
  • MSC (2000): Primary 03E15
  • DOI:
  • MathSciNet review: 1800351