Relativization of star-\(C\)-Hurewicz property in topological spaces (Q2196430)

A subspace \(A\) of a topological space \(X\) is said to have the relative star-\(C\)-Hurewicz property in \(X\) (in short, RSCH(\(X\))) if for each sequence \(\langle\mathcal U_n\mid n\in\omega\rangle\) of open covers of \(X\) there is a sequence \(\langle A_n\mid n\in\omega\rangle\) of countably compact subsets of \(X\) such that each point of \(A\) belongs to St\((A_n,\mathcal U_n)\) for all but finitely many \(n\). Various properties of RSCH(\(X\)) spaces are discussed. For example a subspace of \(X\) having the star-\(C\)-Hurewicz property has the RSCH(\(X\)) but the converse fails. The collection of RSCH(\(X\)) subsets forms an admissible \(\sigma\)-ideal. In a paracompact Hausdorff space \(X\) RSCH(\(X\)) is equivalent to the relative Hurewicz property. Preservation under functions and products as well as connections with other Hurewicz type properties are also considered.