Productively countably tight spaces of the form \(C_k(X)\) (Q2834138)

For a given bornology \(\mathcal{B}\) on a Tychonoff space \(X\) the space \(C_{\mathcal{B}}(X)\) is defined as the set of all real valued functions on \(X\) endowed with the topology of uniform convergence on elements of \(\mathcal{B}\). The authors use topological games to prove that for a bornology \(\mathcal{B}\) with a compact base, \(C_{\mathcal{B}(X)}\) is productively \(\kappa\)-thight if and only if \(l_{\mathcal{B}}(X)\leq \kappa\), where \(l_{\mathcal{B}}(X)\) is the \(\mathcal{B}\)-Lindelöf degree of \(X\). This immediately implies that for a bornology \(\mathcal{B}\) with a compact base, \(C_{\mathcal{B}(X)}\) is productively countably tight if and only if every \(\mathcal{B}\)-cover of \(X\) that consists of \(G_{\delta}\) sets has a countable \(\mathcal{B}\)-subcover.