Transfinite Schreier families and cardinal invariants
Abstract
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 ω1 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 ω1 exactly when the fundamental sequences may be chosen so that every compact family of finite subsets of {2,3,…} 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.
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BibTeX
@misc{beanland2026transfinite,
title = {{Transfinite Schreier families and cardinal invariants}},
author = {Kevin Beanland and Christian Rosendal},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28900},
eprint = {2608.28900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28900},
}
RIS
TY - UNPB
TI - Transfinite Schreier families and cardinal invariants
AU - Kevin Beanland
AU - Christian Rosendal
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28900
UR - https://arxiv.org/abs/2608.28900
AB - 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,…} 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.
ER -