Vaught’s conjecture on analytic sets

Hjorth, Greg

doi:10.1090/S0894-0347-00-00349-0

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.

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