New Directions in Descriptive Set Theory
New Directions in Descriptive Set Theory
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描述集合论的新方向
DOI：
10.2307/421088
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发表时间：
1999
影响因子：
0.6
通讯作者：
A. Kechris
中科院分区：
文献类型：
--
作者：
A. Kechris
§1。 （可分开的）希尔伯特空间，更通常是所有分开的Banach空间，Cantor Space2ℕ，Baire Spaceℕℕ，无限对称群S∞，统一组（希尔伯特空间），根据其定义的复杂性和这些层次结构的每个级别的集合结构，在我们的开头分析了我们的定义的复杂性和集合的结构，以层次结构分类。在波兰空间中设置Borel集合，从开放式设置开始，并在完成和可数工会的操作下关闭，以及相应的Borel层次结构（集合）这是通过Borel集合开始的投影集，并在完成和投影的操作下结束，以及相应的投射层次结构（集合）。描述性集理论，但我将在这里限制自己的borel和投影集，实际上只是那些处于投射层次结构的第一层的bore bore共线（）集。
§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝ n , ℂ n , (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc. In this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy ( sets). After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy ( sets). There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets.