Exponential density vs exponential domination (Q2043687)

\textit{G. Gruenhage} et al. [Topology Appl. 282, Article ID 107306, 10 p. (2020; Zbl 1468.54027)] discussed spaces with exponential \(\kappa\)-domination. In this paper the authors introduce a weaker property called exponential \(\kappa\)-density and show that it behaves nicely under standard operations. A space \(X\) has exponential \(\kappa\)-density if for any family \(\mathcal{U}\) of non-empty open subsets of \(X\) with \(\vert \mathcal{U}\vert \leq 2^\kappa\), there exists a set \(A\subset X\) such that \(\vert A\vert \leq\kappa\) and \(U\cap A\neq\emptyset\) for any \(U\in\mathcal{U}\). It is proved that exponential \(\kappa\)-density is preserved by continuous images, open subspaces, arbitrary products and extensions. At the end of the paper, some interesting open questions are raised.