The Higher Closed Null Ideal(s)

Abstract

We generalise the $\sigma$-ideal $\mathcal{E}$ generated by the closed Lebesgue null subsets of the Cantor space ${}^{\omega }2$ to the context of higher Cantor spaces ${}^{\kappa }2$ for an uncountable cardinal $\kappa$ satisfying $\kappa ={\kappa }^{<\kappa }$. We introduce candidates for a higher closed null ideal based on slaloms on partitions of $\kappa$ into sets of size $<\kappa$, and study their combinatorial properties. We also give some relations between the cardinal characteristics of the higher closed null ideals and other cardinal characteristics.

arXiv commentVarious minor corrections