Certain observations on selection principles related to bornological covers using ideals

Abstract

We study selection principles related to bornological covers using the notion of ideals. We consider ideals $\mathcal I$ and $\mathcal J$ on $\omega$ and standard ideal orderings $KB, K$. Relations between cardinality of a base of a bornology with certain selection principles related to bornological covers are established using cardinal invariants such as modified pseudointersection number, the unbounding number and slaloms numbers. When $\mathcal I \leq_\square \mathcal J$ for ideals $\mathcal I, \mathcal J$ and $\square\in \{1\text{-}1,KB,K\}$, implications among various selection principles related to bornological covers are established. Under the assumption that ideal $\mathcal I$ has a pseudounion we show equivalences among certain selection principles related to bornological covers. Finally, the $\mathcal I\text{-}\mathfrak B^s$-Hurewicz property of $X$ is investigated. We prove that $\mathcal I\text{-}\mathfrak B^s$-Hurewicz property of $X$ coincides with the $\mathfrak B^s$-Hurewicz property of $X$ if $\mathcal I$ has a pseudounion. Implications or equivalences among selection principles, games and $\mathcal I\text{-}\mathfrak B^s$-Hurewicz property which are obtained from our investigations are described in diagrams.