论文标题
关于第一个Banach问题，关于绝对$κ$ -Borel的凝结
On the first Banach problem, concerning condensations of absolute $κ$-Borel sets onto compacta
论文作者
论文摘要
一致的是，连续体是任意大的，并且没有绝对的$κ$ -Borel集合$ x $密度$κ$，$ \ aleph_1 <κ<\ Mathfrak {c} $，将其浓缩到紧凑的度量空间上。一致的是，连续体是任意的，并且任何绝对的$κ$ -Borel集合$ x $密度$κ$，$κ\ leq \ leq \ mathfrak {c} $，其中包含重量$κ$的baire baire Space的封闭子空间，凝聚在压缩上。特别是，将Brian的结果应用于模型理论，我们将获得以下意外结果。给定任何$ a \ subseteq \ mathbb {n} $，$ 1 \ in $ in $ in，有一个强迫扩展名，其中每个绝对的$ \ aleph_n $ -borel套件，包含baire baire of aleph_n $的baire baire Space的封闭子空间，compactum in Compactum in Compactum if，并且仅在$ n \ $ n \ in $ n \ in $ n f上。
It is consistent that the continuum be arbitrary large and no absolute $κ$-Borel set $X$ of density $κ$, $\aleph_1<κ<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large and any absolute $κ$-Borel set $X$ of density $κ$, $κ\leq\mathfrak{c}$, containing a closed subspace of the Baire space of weight $κ$, condenses onto a compactum. In particular, applying Brian's results in model theory, we get the following unexpected result. Given any $A\subseteq \mathbb{N}$ with $1\in A$, there is a forcing extension in which every absolute $\aleph_n$-Borel set, containing a closed subspace of the Baire space of weight $\aleph_n$, condenses onto a compactum if, and only if, $n\in A$.
