Two inequalities between cardinal invariants
Volume 237 / 2017
Fundamenta Mathematicae 237 (2017), 187-200
MSC: 03E17, 03E55, 03E05, 03E20.
DOI: 10.4064/fm253-7-2016
Published online: 1 December 2016
Abstract
We prove two $\mathrm {ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega $ of asymptotic density $0$. We obtain an upper bound on the $\ast $-covering number, sometimes also called the weak covering number, of this ideal by proving that ${\rm cov }^{\ast }({{\mathcal {Z}}}_{0}) \leq {\mathfrak {d}}$. Next, we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove that, in sharp contrast to the case when $\kappa = \omega $, if $\kappa $ is any regular uncountable cardinal, then ${\mathfrak {s}}_{\kappa } \leq {\mathfrak {b} }_{\kappa }$.
