H-star-Lindelöf property and weaker versions (Q6541999)

The authors introduce Hurewicz like selective versions of the star-Lindelöf property: a topological space \(X\) is called H-star-Lindelöf (almost H-star-Lindelöf) if for every open cover \(\mathcal U\) of \(X\) and every sequence \((D_n)\) of dense subsets of \(X\) there is a sequence \((F_n)\) of finite sets such that \(F_n \subset D_n\) for every \(n\), and every \(x\in X\) belongs to \(St(F_n, \mathcal U)\) (\(\overline{St(F_n,\mathcal U)}\)) for all but finitely many \(n\). Also, they introduce weakly H-star-Lindelöf spaces. Several properties of these classes of spaces are established and their relations with other classes of spaces are investigated. Characterizations of these properties in hyperspaces of \(X\) endowed with the hit-and-miss topologies in terms of properties of \(X\) are given.