## Vaught’s conjecture on analytic sets

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- by Greg Hjorth
- J. Amer. Math. Soc.
**14**(2001), 125-143 - DOI: https://doi.org/10.1090/S0894-0347-00-00349-0
- Published electronically: September 18, 2000
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## Abstract:

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.## References

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**Classification and orbit equivalence relations,**Mathematical Surveys and Monographs, 75, American Mathematical Society, Providence, RI, 2000.

## Bibliographic Information

**Greg Hjorth**- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
- Email: greg@math.ucla.edu
- 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: https://doi.org/10.1090/S0894-0347-00-00349-0
- MathSciNet review: 1800351