2021/03/31 by Adam Kwela, Kwela, Adam
Mathematics · #03E05 #03E15 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2103.17169
openalex publication_date 2021/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal I (rk(I)) as minimal ordinal α<ω1 such that there is S∈\bfΣ01+α with I⊆ S and I^⋆∩ S=∅, where I^⋆ is the filter dual to the ideal I (actually, the authors use the dual notion of filters instead of ideals). Moreover, they introduced ideals Finα, for all α<ω1, and conjectured that rk(I)≥α if and only if I contains an isomorphic copy of Finα (Finα\sqsubseteqI). To define Finα in the case of limit ordinals 0<α<ω1, G. Debs and J. Saint Raymond introduced inductive limits of ideals. We show that the above conjecture is false in the case of α=ω by constructing an ideal Fin'ω of rank ω such that Finω\not\sqsubseteqFin'ω. However, we show that Fin'ω\sqsubseteqI is equivalent to ∀n∈ωFinn\sqsubseteqI. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.