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2006/2007 Half of an inseparable pair.
Arnold W. Miller
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Real Anal. Exchange 32(1): 179-194 (2006/2007).


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.


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Arnold W. Miller. "Half of an inseparable pair.." Real Anal. Exchange 32 (1) 179 - 194, 2006/2007.


Published: 2006/2007
First available in Project Euclid: 17 July 2007

zbMATH: 1121.03059
MathSciNet: MR2329229

Primary: 03E15 , 03E35 , 03E60

Keywords: analytic sets , Borel sets , self-constructible reals , separation principle

Rights: Copyright © 2006 Michigan State University Press

Vol.32 • No. 1 • 2006/2007
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