论文标题
Luzin-Novikov定理中的图形的Borel类中没有绑定
There is no bound on Borel classes of the graphs in the Luzin-Novikov theorem
论文作者
论文摘要
我们表明，对于[1，ω_1）$中的每个序列$α\ $ n ose os oble集$ f \ subset 2^ω\ timesω^ω$ $，使得每个$ x \ in 2^ω$ in 2^ω$ in office $ \ \ {y \ inω^ω; （x，y）\ in f \} $是两点集，$ f $不能被数量涵盖，许多图$ b（n）\ subset 2^ω\ timesω\ timesω^ω$ in 2^ω$的变量$ x \ in 2^ω$的函数的函数，每个$ x $ in 2^ω$，因此每个$ b（n）$都在附加的borel class $ $ \ boldsymbol \ boldsymbol c $ neaddive b（n）中。这排除了具有luzin-novikov定理的定量版本的可能性。构造是对哈灵顿的方法的修改，后者发明了它，以表明存在$π^0_1 $设置为$ω^ω$，其中包含一个非Arithmetic Singleton。通过另一个相同方法的应用程序，我们将封闭设置不包括Borel Set上的Saint Raymond Theorem的定量版本，其中包含$σ$ -COMPACT部分。
We show that for every ordinal $α\in [1, ω_1)$ there is a closed set $F \subset 2^ω\times ω^ω$ such that for every $x \in 2^ω$ the section $\{y\in ω^ω; (x,y) \in F\}$ is a two-point set and $F$ cannot be covered by countably many graphs $B(n) \subset 2^ω\times ω^ω$ of functions of the variable $x \in 2^ω$ such that each $B(n)$ is in the additive Borel class $\boldsymbol Σ^0_α$. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable $Π^0_1$ set in $ω^ω$ containing a non-arithmetic singleton. By another application of the same method we get closed sets excluding a quantitative version of the Saint Raymond theorem on Borel sets with $σ$-compact sections.
