不可分割的一对的一半
Half of an inseparable pair
摘要：
卢津的一个经典定理是分离原理对Pi^0_alpha集合成立，但对Sigma^0_alpha集合不成立。我们证明了对于每一个不为Pi^0_alpha的Sigma^0_alpha集合A，都存在一个Sigma^0_alpha集合B，它与A不相交，但不能被Delta^0_alpha集合C与A分开。假设Pi^1_1-确定性，从斯蒂尔的一个定理可以得出一个类似的结果对Pi^1_1集也成立。另一方面，假设V=L，则存在一个不是波莱尔不可分对的一半的真Pi^1_1集。这些结果回答了F. Dashiell提出的问题。最新版本：www.math.wisc.edu/~miller/
英文摘要：
A classical theorem of Luzin is that the separation principle holds for the Pi^0_alpha sets but fails for the Sigma^0_alpha sets. We show that for every Sigma^0_alpha set A which is not Pi^0_alpha there exists a Sigma^0_alpha set B which is disjoint from A but cannot be separated from A by a Delta^0_alpha set C. Assuming Pi^1_1-determancy it follows from a theorem of Steel that a similar result holds for Pi^1_1 sets. On the other hand assuming V=L there is a proper Pi^1_1 set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell. Latest version at: www.math.wisc.edu/~miller/
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