Regularity Properties for Dominating Projective Sets
Regularity Properties for Dominating Projective Sets
复制标题
主射影集的正则性质
DOI：
10.1016/0168-0072(94)00027-z
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
O. Spinas
中科院分区：
文献类型：
--
作者：
J. Brendle;G. Hjorth;O. Spinas
我们证明了Baire空间中的每个支配解析集都有一个支配闭子集。这改进了Spinas [15]的一个定理，即每个支配解析集包含一致树的分支，即具有以下性质的超完美树：对于每个splitnode，所有后继splitnode具有相同的长度。在文献[15]中，Baire空间的一个子集称为u-正则的，如果它不是支配的或者它包含一致树的分支，并且证明了Baire空间的21-Kσ-正则性蕴含21-u-正则性。这里我们证明了这些性质实际上是等价的。由于解析u-正则性的证明使用了一个博弈论证，很明显（投射）确定性意味着所有（投射）集合的u-正则性。在这里，我们表明，一个不可接近的基数是足以构建一个模型的投影u-正则性，即它在Solovay的模型。最后，我们证明了均匀树强迫等价于Laver强迫。
We show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, and it was proved that Σ21-Kσ-regularity implies Σ21-u-regularity. Here we show that these properties are in fact equivalent. Since the proof of analytic u-regularity uses a game argument it was clear that (projective) determinacy implies u-regularity of all (projective) sets. Here we show that an inaccessible cardinal is enough to construct a model for projective u-regularity, namely it holds in Solovay's model. Finally we show that forcing with uniform trees is equivalent to Laver forcing.