Relative version of star-Hurewicz property
In covering properties, the Hurewicz property has been widely studied and rich results have been obtained. A subspace \(A\) of a topological space \(X\) is said to have the \textit{relatively star-Hurewicz property} in \(X\) if for each sequence \(\{\mathcal{U}_n: n\in\omega\}\) of open covers of \(X\) there is a sequence \(\{\mathcal{V}_n: n\in\omega\}\) such that for each \(n\in\omega\), \(\mathcal{V}_n\) is a finite subset of \(\mathcal{U}_n\) and for each \(x\in A\), belongs to {St}\((\bigcup\mathcal{V}_n, \mathcal{U}_n)\) for all but finitely many \(n\). \par The authors obtain some interesting properties of the relatively star-Hurewicz property. For example, it is proved that (1) for a topological space \(X\), \(RSH(X)\) is an admissible \(\sigma\)-ideal in \(X\); (2) for a space \(X\), \(A\) is a relatively star-Hurewicz subspace of \(X\) if and only if \(X\) satisfies \(U^*_{\text{fin}}(\mathcal{O}, \mathcal{O}^{gp}_A)\); (3) let \(f\) be a continuous function from a space \(X\) onto \(Y\); if \(A\in RSH(X)\), then \(f(A)\in RSH(Y)\).
scientific article; zbMATH DE number 7377530
scientific article