Tukey 归约与基数特性

In this project, we will continue to investigate the structural problems in the theory of cardinal invariants, especially the classification problem.among some important partial orders under the Tukey reduction. Particularly, we will focus on the following questions: （1）Tukey reducibility.between F_σ ideal on natural numbers and G_δ σ-ideal of compact subsets in a topological space. (2) Is there a Tukey reduction from F_σ ideal to the famous ideal of density zero? (3) Whether the image of summable ideal or the image ideal of density zero under D-operation is Tukey reducible to the sigma-ideal of compact Lebesque measure zero? (4) Is there a Menger filter of character d which is not a Hurewicz filter? Through the project, we will determine the Tukey reducibility among some important ideals and find further important applications of the Tukey reduction and the theory of cardinal invariants to other related mathematical fields.

本申请项目将继续深入研究基数不变量理论中的结构性问题，特别是研究在Tukey 归约关系下一些重要偏序的分类问题。具体地，我们将研究：（1）自然数上的理想和拓扑空间上紧集的σ-理想之间的Tukey关系问题，特别是确定自然数上F_σ的理想和拓扑空间上紧集G_δ的σ-理想的Tukey关系。（2）是否存在非凡的F_σ理想Tukey 归约到零密度理想Z_0。（3） 可和理想的D-算子像是否可Tukey归约到紧Lebesque零测集的σ-理想？以及零密度理想的D-算子像是否可Tukey归约到紧Lebesque零测集的σ-理想？（4）ZFC中是否存在特征是d的Menger 滤子但它不是Hurewicz的？.通过对上述问题的探讨，解决一些重要特殊理想的相应有向偏序结构的Tukey 归约关系问题，发现Tukey 归约及基数不变量理论在相关数学领域的重要应用。

我们深入研究了F-sigma理想的Tukey规约，证明了每个非平凡的F-sigma理想不能Tukey规约到紧集的G-delta理想。通过引入flat及gradually flat 理想的概念证明了flat理想的结构二分定理。用Tukey规约给出了关于gradually flat 理想的几个刻画，同时证明了gradually flat 理想正是能够Tukey规约到零密度理想的flat 理想。这些结果发表在J. of Symbolic Logic上，论文审稿人认为这篇论文的结果是Tukey规约研究的重要贡献对进一步深入研究Tukey 规约的结构关系理论具有重要理论价值。. 解决了由Blass, Di Nasso, Forti提出的公开问题，证明了存在rapid non-interval P-point 及rapid non-weakly Ramsey interval P-point。此项结果发表在Topology and its Application，对深入了解超滤的结构有重要理论意义。. 证明了the product of arbitrary two Scheepers-bounded metrizable group is Scheepers-bounded 等价于著名的集论组合原理NCF，从而表明这类拓扑群的可乘性独立于ZFC系统。此项结果已在线发表在Topology and its Application，对最终攻克o-bounded的可度量拓扑群可乘性问题有重要参考价值。. 利用理想之间Katetov规约我们成功解决了由理想生成的拓扑的三个基本问题。此项结果发表在Topology and its Application，对进一步研究这种拓扑的性质有重要理论价值。