Title: Half of an inseparable pair
Author: Arnold W. Miller
Abstract:
A classical theorem of Luzin is that the separation principle holds for
the $\bPi^0_\al$ sets but fails for the $\bSig^0_\al$ sets. We show that
for every $\bSig^0_\al$ set $A$ which is not $\bPi^0_\al$ there exists a
$\bSig^0_\al$ set $B$ which is disjoint from $A$ but cannot be separated
from A by a $\bDel^0_\al$ set $C$. Assuming $\bPi^1_1$-determancy it
follows from a theorem of Steel that a similar result holds for $\bPi^1_1$
sets. On the other hand assuming V=L there is a proper $\bPi^1_1$ set
which is not half of a Borel inseparable pair. These results answer
questions raised by F.Dashiell.
LaTeX2e 16 pages
http://www.math.wisc.edu/~miller/
miller@math.wisc.edu