点态函数空间可分商空间的一致失败
A consistent failure of separable quotients for pointwise function spaces
- Czech Academy of Sciences（捷克科学院）
- Institute of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机科学研究所）
- Faculty of Mathematics and Computer Science Adam Mickiewicz University（亚当·密茨凯维奇大学数学与计算机科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
在Jensen菱形原则下构造紧致零维空间K，使C_p(K)无无限维Hausdorff可分线性商，并证明可分商与可度量商存在性等价。
AI中文摘要：
假设Jensen的菱形原则，我们构造了一个无限紧致零维空间$K$，使得$C_p(K)$没有无限维Hausdorff可分线性商空间。空间$K$是可分的且稠密的，具有权重$\aleph_1$和基数$2^{\aleph_1}$，并且是一个Efimov空间。我们将$K$构造为以可数序数为指标的可分度量紧空间的逆向极限。在每个非平凡的后继步骤中，投影在选定的闭集上具有两点纤维，在其他地方具有单点纤维；这改变了所选有限支撑测度序列的弱星极限。我们还证明了，对于紧空间$X$，$C_p(X)$存在无限维可分商空间等价于存在无限维可度量商空间。
英文摘要：
Assuming Jensen's diamond principle, we construct an infinite compact zero-dimensional space $K$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. The space $K$ is separable and crowded, has weight $\aleph_1$ and cardinality $2^{\aleph_1}$, and is an Efimov space. We construct $K$ as an inverse limit of compact metrisable spaces indexed by the countable ordinals. At each nontrivial successor step, the projection has two-point fibres over a chosen closed set and singleton fibres elsewhere; this changes the weak-star limit of a selected sequence of finitely supported measures. We also prove that, for compact $X$, the existence of an infinite-dimensional separable quotient of $C_p(X)$ is equivalent to the existence of an infinite-dimensional metrisable quotient.
