Real Analysis Exchange

Half of an inseparable pair.

Arnold W. Miller

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A classical theorem of Luzin is that the separation principle holds for the $\mathbf{\Pi}^0_\propto$ sets but fails for the $\mathbf{\Sigma}^0_\propto$ sets. We show that for every $\mathbf{\Sigma}^0_\propto$ set $A$ which is not $\mathbf{\Pi}^0_\propto$ there exists a $\mathbf{\Sigma}^0_\propto$ set $B$ which is disjoint from $A$ but cannot be separated from A by a $\mathbf{\Delta}^0_\propto$ set $C$. Assuming $\mathbf{Pi}^1_1$-determancy it follows from a theorem of Steel that a similar result holds for $\mathbf{Pi}^1_1$ sets. On the other hand assuming V=L there is a proper $\mathbf{Pi}^1_1$ set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell.

Article information

Real Anal. Exchange, Volume 32, Number 1 (2006), 179-194.

First available in Project Euclid: 17 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05] 03E35: Consistency and independence results 03E60: Determinacy principles

separation principle Borel sets analytic sets self-constructible reals


Miller, Arnold W. Half of an inseparable pair. Real Anal. Exchange 32 (2006), no. 1, 179--194.

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