On disjoint Borel uniformizations
逻辑
2021-02-09 v1
摘要
Larman showed that any closed subset of the plane with uncountable vertical cross-sections has aleph_1 disjoint Borel uniformizing sets. Here we show that Larman's result is best possible: there exist closed sets with uncountable cross-sections which do not have more than aleph_1 disjoint Borel uniformizations, even if the continuum is much larger than aleph_1. This negatively answers some questions of Mauldin. The proof is based on a result of Stern, stating that certain Borel sets cannot be written as a small union of low-level Borel sets. The proof of the latter result uses Steel's method of forcing with tagged trees; a full presentation of this method, written in terms of Baire category rather than forcing, is given here.
引用
@article{arxiv.math/9610207,
title = {On disjoint Borel uniformizations},
author = {Howard Becker and Randall Dougherty},
journal= {arXiv preprint arXiv:math/9610207},
year = {2021}
}