On strong measure zero subsets of $^{κ}2$
On strong measure zero subsets of $^{κ}2$
复制标题
强测度 $^{κ}2$ 的零子集
DOI：
10.4064/fm170-3-1
复制
发表时间：
2001
影响因子：
0.6
通讯作者：
S. Shelah
中科院分区：
文献类型：
--
作者：
Aapo Halko;S. Shelah
我们研究了广义康托空间2和广义贝尔空间κ作为经典康托空间和贝尔空间的类似物。我们赋予κ一个拓扑，其中点η的基本邻域是集合{ν:(∀j < i)(ν(j) = η(j))}，其中i < κ。我们定义了2的强测度零集的概念。对于后继k = k证明了2的强测度零集的理想是bκ-可加性的，其中bκ是k中最小无界族的大小，并且证明了2的广义Borel猜想是假的。此外，对于正则不可数κ，具有Baire性质的2的子集族在Suslin操作下不闭合。这些结果回答了b[2]中提出的问题。
We study the generalized Cantor space 2 and the generalized Baire space κ as analogues of the classical Cantor and Baire spaces. We equip κ with the topology where a basic neighborhood of a point η is the set {ν : (∀j < i)(ν(j) = η(j))}, where i < κ. We define the concept of a strong measure zero set of 2. We prove for successor κ = κ that the ideal of strong measure zero sets of 2 is bκ-additive, where bκ is the size of the smallest unbounded family in κ, and that the generalized Borel conjecture for 2 is false. Moreover, for regular uncountable κ, the family of subsets of 2 with the property of Baire is not closed under the Suslin operation. These results answer problems posed in [2].