Cicho\'n's maximum with cardinals of the closed null ideal

TL;DR

This paper demonstrates the consistency of a new extension of Cichoń's maximum by adding the uniformity and covering numbers of the closed null ideal, resulting in a distinct and ordered cardinal structure.

Contribution

It introduces a model where the uniformity and covering of the closed null ideal are added to Cichoń's maximum with unique values, expanding the known cardinal characteristics.

Findings

Established a consistent cardinal configuration extending Cichoń's maximum.

Separated the uniformity and covering of the closed null ideal from other cardinal invariants.

Demonstrated the distinctness of these invariants within the continuum.

Abstract

Let $\mathcal{E}$ denote the $\sigma$-ideal generated by closed null sets on the reals. We show that the uniformity and the covering of $\mathcal{E}$ can be added to Cicho\'n's maximum with distinct values. More specifically, it is consistent that $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{E})<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})<\mathrm{cov}(\mathcal{E})<\mathfrak{d}<\mathrm{non}(\mathcal{N})<\mathrm{cof}(\mathcal{N})<2^{\aleph_0}$ holds.