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 $\ddot{I}‰$ 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 $\mathcal{I}{\le }_{\square }\mathcal{J}$ 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.