On set-star-\(K\)-Hurewicz spaces (Q2135299)

A space \(X\) is said to be set-star-\(K\)-Hurewicz if for each \(A\subset X\) and each sequence \((\mathcal U_n : n \in \mathbb N)\) of collections of open subsets of \(X\) covering \(\overline{A}\) there are compact sets \(K_n\) in \(X\), \(n\in \mathbb N\), such that each \(x\in A\) belongs to \(St(K_n,\mathcal U_n)\) for all but finitely many \(n\). Relationships with other classes of star-Hurewicz-type spaces are considered. For example, there is a Hausdorff star-\(K\)-Hurewicz space which is not set-star-\(K\)-Hurewicz, and there is a Tychonoff set-star-\(K\)-Hurewicz space which is not set-star-Hurewicz. Several properties of this class of spaces are established.