AI 中文总结
在ZF集合论下证明BCT_WO与RS_WO等价，在特拉斯的费弗曼型模型中二者刻画了2^ω的子集性质，研究该模型中2^ω的良序子集理想，得出其与相关拓扑性质的等价关系及该模型的其他结论。
AI 中文摘要
令$\boldsymbol{\text{BCT}_{\text{WO}}}$断言，完备波兰空间的每个良序稠密开子集族都有稠密交；令$\boldsymbol{\text{RS}_{\text{WO}}}$断言，对每个非空可数偏序，存在一个滤子与任意给定的良序稠密子集族的每个元相交。在$\text{ZF}$集合论下，我们证明二者等价。我们证明在特拉斯模型中，它们刻画了那些$A\textit{⊆}2^\boldsymbol{\text{ω}}$，使得每个以$A$为指标的稠密开集族都有稠密交。假设对非空集合的良序族的选择公理成立，我们对每个$Y$是$2^\boldsymbol{\text{ω}}$的满射像时，得到关于具有贫乏竖截面的全关系$R\textit{⊆}X\times Y$的类似刻画。特拉斯研究的费弗曼型模型$\boldsymbol{\text{N}_{\boldsymbol{\text{ℵ}_1}}}$满足这些假设和$\text{DC}$。我们研究该模型中$2^\boldsymbol{\text{ω}}$的良序子集的理想及其与其他理想的关系。我们还证明：$2^\boldsymbol{\text{ω}}$的良序子集恰好是那些在每个有限次幂中都是Rothberger的、具有强测度零、是普遍零测的、是Marczewski零的，或者不包含完美子集的集合。$2^\boldsymbol{\text{ω}}$的良序子集的理想在良序并下是封闭的，且与贫乏理想不可比。与Rothberger性质的等价性不能推广到Hurewicz性质。此外，在该模型中，每个集合都是Marczewski可测的，映射到可分度量空间的任意映射都有连续的完美限制，且每个具有正外测度的集合都包含一个完美子集。
英文摘要
Let $\mathsf{BCT}_{\mathsf{WO}}$ assert that every well-orderable family of dense open subsets of a perfect Polish space has dense intersection, and let $\mathsf{RS}_{\mathsf{WO}}$ assert that for every nonempty countable partial order, there is a filter meeting every member of any given well-orderable family of dense subsets. Over $\mathsf{ZF}$ they are shown to be equivalent. We show that in Truss's model, they characterize those $A\subseteq2^ω$ for which every $A$-indexed family of dense open sets has dense intersection. Assuming choice for well-orderable families of nonempty sets, we obtain an analogous characterization for total relations $R\subseteq X\times Y$ with meager vertical sections, whenever $Y$ is a surjective image of $2^ω$. The Feferman-type model $\mathfrak N_{\aleph_1}$ studied by Truss \cite{Truss1974} satisfies these hypotheses and $\mathsf{DC}$. We study the ideal of well-orderable subsets of $2^ω$ and its relations to other ideals in this model. We show, among other things: the well-orderable subsets of $2^ω$ are exactly the sets which are Rothberger in every finite power, have strong measure zero, are universally null, are Marczewski null, or contain no perfect subset. The ideal of well-orderable subsets of $2^ω$ is closed under well-ordered unions and is incomparable with the meager ideal. The equivalence with the Rothberger property does not extend to the Hurewicz property. Further more, in this model every set is Marczewski measurable, arbitrary maps into separable metric spaces have continuous perfect restrictions, and every set of positive outer measure contains a perfect subset.
