On a notion of smallness for subsets of the Baire space
On a notion of smallness for subsets of the Baire space
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关于贝尔空间子集的小概念
DOI：
10.1090/s0002-9947-1977-0450070-1
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发表时间：
1977
影响因子：
1.3
通讯作者：
A. Kechris
中科院分区：
文献类型：
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作者：
A. Kechris
我们称一个从ω到ω σ-有界的函数集A <$ω^ω，如果存在一个可数函数列（α_n：n <$ω）<$ω^ω，使得A的每个成员都被该列的一个元素逐点控制。本文研究了关于ω^ω的子集的小性概念的可定义性问题。我们证明了大多数关于ω^ω的可数子集的结构的可定义性结果都有相应的版本，它们在ω^ω的σ-有界子集上成立。例如，我们证明了ω^ω的每个Σ_（2n+1^1）σ-有界子集都有一个Δ_（2n+1）^1“界”{α_m：m <$ω}，并且对任意n ≥ 0，都有最大的σ-有界集合σ_（2n+1）^1和Σ_（2n+2）^1。这里我们需要射影决定性公理，如果n ≥ 1。为了研究σ-有界性的概念，设计了一个简单的博弈，它在这里扮演着类似于可数集理论中标准^*-博弈（见[My]）的角色。在本文的最后一部分中，定义了一类博弈，它推广了^*-和^（**）-（或Banach-Mazur）博弈（见[My]）以及上面提到的博弈。这些对策中的每一个都自然地定义了ω^ω的子集的小性概念，其特殊情况包括可数性、第一类和σ-有界性，并且可以推广本文的所有主要结果。
Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of ω^ω. We show that most of the usual definability results about the structure of countable subsets of ω^ω have corresponding versions which hold about σ-bounded subsets of ω^ω. For example, we show that every Σ_(2n+1^1 σ-bounded subset of ω^ω has a Δ_(2n+1)^1 "bound" {α_m: m Є ω} and also that for any n ≥ 0 there are largest σ-bounded Π_(2n+1)^1 and Σ_(2n+2)^1 sets. We need here the axiom of projective determinacy if n ≥ 1. In order to study the notion of σ-boundedness a simple game is devised which plays here a role similar to that of the standard ^*-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the ^*- and ^(**)-(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of ω^ω whose special cases include countability, being of the first category and σ-boundedness and for which one can generalize all the main results of the present paper.