On star-K-\(\mathcal{I}\)-Hurewicz property (Q2067290)

Fix a proper admissable ideal \(\mathcal I\) of the natural numbers \(\mathbb N\). A topological space \(X\) has the star-K-\(\mathcal I\)-Hurewicz property if for each sequence \(\langle\mathcal U_n\rangle\) of open covers of \(X\) there is a sequence \(\langle K_n\rangle\) of compact subsets of \(X\) such that for each \(x\in X\) we have \(\{n\in\mathbb N\mid x\notin\mathrm{St}(K_n,\mathcal U_n)\}\in\mathcal I\). The behaviour of this property under subspaces, products, images and Alexander duplicate is considered. In general the results are negative but not always. For example it need not be preserved by regular-closed \(G_\delta\) subspaces but is preserved by open \(F_\sigma\) subspaces; it is preserved under countable unions hence under the product with a \(\sigma\)-compact space but need not be preserved under the product with a Lindelöf space.