Certain Observations on Selection Principles Related to Bornological Covers Using Ideals

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