发表机构
Washington and Lee University; University of Maryland, College Park（华盛顿与李大学; 马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨超限Schreier层级对可数极限序数基本序列选择的依赖，关联区间端点函数，证明边界数、控制数与基本序列选择的等价条件，结合相关定理与吸收构造得出结论。
AI 中文摘要
我们研究超限Schreier层级对可数极限序数基本序列选择的依赖性，为每个Schreier族关联一个区间端点函数。我们证明：当且仅当可选择基本序列使得这些端点函数在终支配序中构成无界族时，边界数等于ω₁；等价地，此时该层级满足Shiliaev提出的尾覆盖性质，即每个无限紧区间族都与该层级的某个成员有无限交集。我们还证明：当且仅当可选择基本序列使得{2,3,…}的每个有限子集的紧族都包含于某个Schreier族时，控制数等于ω₁；后一等价性结合了Fremlin关于有理数紧子集的共尾性定理，以及紧族的吸收构造。
英文摘要
We study the dependence of the transfinite Schreier hierarchy on the choice of fundamental sequences for the countable limit ordinals. With each Schreier family we associate an interval endpoint function. We prove that the bounding number is equal to $ω_1$ exactly when the fundamental sequences may be chosen so that these endpoint functions form an unbounded family in the eventual domination order. Equivalently, the hierarchy then satisfies the tail-covering property isolated by Shiliaev, or every infinite compact interval family has infinite intersection with some member of the hierarchy. We also prove that the dominating number is equal to $ω_1$ exactly when the fundamental sequences may be chosen so that every compact family of finite subsets of $\{2,3,\ldots\}$ is contained in one Schreier family. The latter equivalence combines Fremlin's cofinality theorem for compact subsets of the rationals with an absorption construction for compact families.
