Topology and its Applications ( IF 0.6 ) Pub Date : 2021-07-22 , DOI: 10.1016/j.topol.2021.107798 Adam Kwela 1
G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal () as minimal ordinal such that there is with and , where is the filter dual to the ideal . Moreover, they introduced ideals , for all , and conjectured that if and only if contains an isomorphic copy of (). To define in the case of limit ordinals , G. Debs and J. Saint Raymond introduced inductive limits of ideals.
We show that the above conjecture is false in the case of by constructing an ideal of rank ω such that . However, we show that is equivalent to . We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.
中文翻译:
理想的归纳极限
G. Debs 和 J. Saint Raymond 在 2009 年定义了解析理想的 Borel 分离秩 () 作为最小序数 使得有 和 和 , 在哪里 是理想的双重过滤器 . 此外,他们引入了理想, 对全部 ,并推测 当且仅当 包含一个同构的副本 ()。界定 在极限序数的情况下 , G. Debs 和 J. Saint Raymond 介绍了理想的归纳极限。
我们证明上述猜想在以下情况下是错误的 通过构建理想 秩ω使得. 然而,我们证明 相当于 . 我们讨论(由上述结果表明)对极限序数的原始猜想的可能修改。