Cardinal invariants about shrinkability of unbounded sets
The authors study the cardinal number \(\mathfrak b^*\), defined as the minimal cardinal \(\lambda\) with the property that every unbounded family -- in the mod-finite order on \(\omega^\omega\) -- can be shrunk to an unbounded subfamily of size \(\lambda\) or less. This cardinal number was used in the study of the tightness of products of sequential fans. In this paper the definition is extended to cover general ideals of subsets and the resulting invariant is denoted by \(\text{shr}({\mathcal I})\). The main results of the paper give the position of the values of this new invariant for the ideals of meager and null sets in Cichoń's diagram as well as the consistency of various strict inequalities.
scientific article; zbMATH DE number 987926
scientific article