选择原理与不可数情形下的Scheepers图
Selection principles and the Scheepers diagram in an uncountable setting
- Department of Mathematics, Mirza Ghalib College（米尔扎·加利布学院数学系）
- Department of Mathematics, University of Gour Banga（古尔班加大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究不可数情形下的选择原理，确定了Scheepers图（直至三个未解决蕴含），证明弱紧致κ下κ-Hurewicz猜想在GCH下不成立，并给出相关尺度集与K_κ-空间的新例子。
AI中文摘要：
我们继续研究在不可数情形下的选择原理，该研究始于文献[18]，并在文献[1]中得到了进一步发展。我们研究了文献[1]中提出的若干问题，这些问题涉及Scheepers图、κ-Hurewicz猜想以及关于κ-Menger和K_κ-空间的相关问题。我们确定了不可数情形下的Scheepers图，直至三个未解决蕴含关系，并计算了图中出现的若干性质的关键基数。对于弱紧致κ，我们证明了存在一个ZFC模型，在其中κ-Hurewicz猜想不成立；特别地，该猜想在κ处的GCH下不可能成立。此外，任何使得κ弱紧致且κ-Hurewicz猜想成立的模型必须满足2^{mathfrak{b}_κ}=2^κ。我们还研究了作为K_κ-空间的mathfrak{b}_κ-尺度集的存在性，并推导出此类例子所必需的一个树论条件。在完美子树性质的额外假设下，不存在作为K_κ-空间的mathfrak{b}_κ-尺度集。因此，在此假设下，每个mathfrak{b}_κ-尺度集都提供了另一个κ-Menger空间而非K_κ-空间的例子。
英文摘要：
We continue the study of selection principles in the uncountable setting initiated in [18] and further developed in [1]. We investigate several problems posed in [1] concerning the Scheepers diagram, the $κ$-Hurewicz Conjecture, and related questions on $κ$-Menger and $K_κ$-spaces. We determine the Scheepers diagram in the uncountable setting up to three unresolved implications and compute the critical cardinalities of several properties occurring in the diagram. For weakly compact $κ$, we show that there is a model of ZFC in which the $κ$-Hurewicz Conjecture fails; in particular, it cannot hold under GCH at $κ$. Moreover, any model in which $κ$ is weakly compact and the $κ$-Hurewicz Conjecture holds must satisfy $2^{\mathfrak{b}_κ} = 2^κ$. We also study the existence of $\mathfrak{b}_κ$-scale sets which are $K_κ$-spaces and derive a necessary tree-theoretic condition for such an example. Under the additional assumption of the Perfect Subtree Property, no $\mathfrak{b}_κ$-scale set is a $K_κ$-space. Consequently, under this assumption, every $\mathfrak{b}_κ$-scale set yields an alternative example of a $κ$-Menger space which is not a $K_κ$-space.
