2026/08/03 by Xing-Yu Hu, Zhang-Yi Luo
Mathematics · #math.LO #math.GN #msc:03E17 #msc:54A20 #msc:03E05 #msc:03E15
14 pages
arxiv created 2026/08/03 · arxiv updated 2026/08/05
For each countable ordinal α≥ 2, Filipów, Kowalczuk and Kwela introduced an ideal convα on the countable compact ordinal space ωα+1. Kowalczuk later proved that, for each countable limit ordinal λ, the ideal conv<λ is the greatest lower bound of \convβ:β<λ\ in the Katětov order. At the first uncountable level, let A⊆[2,ω1) be uncountable and let D be a countable dense subset of XA=∏α∈ A(ωα+1). The coordinate ideal Conv(A,D) on D consists of those B⊆ D with πα[B]\inconvα for every α∈ A. For a pair D⊆ E of countable dense sets, call α non-small if πα[E∖ D]\notinconvα. In ZFC, if at most countably many coordinates are non-small, then Conv(A,D)≡KConv(A,E). Under CH this countability bound is sharp: for every A⊆[3,ω1) with |A|=ℵ1, there are countable dense sets D⊆ D^*⊆ XA such that Conv(A,D^*)≤KConv(A,D) but Conv(A,D)\not≤KConv(A,D^*), and in particular Conv(A,D) and Conv(A,D^*) are not Katětov equivalent. The non-reduction is obtained, under CH, by diagonalizing along ω1 coordinates against the elements of ωω that code retractions D^*→ D.