Cardinal functions, bornologies and strong Whitney convergence (Q6050063)

A family \(\mathcal{B}\) of nonempty subsets of a set \(X\), which covers \(X\), is called \emph{bornology} if it is closed under finite unions and all nonempty subsets of any element of \(\mathcal{B}\) belong to \(\mathcal{B}\). Let \(X\) be a metric space, \(C(X)\) be the set of all real-valued continuous functions on \(X\), and \(\mathcal{B}\) be a bornology on \(X.\) The following four topologies on \(C(X)\), that depend on \(\mathcal{B}\), have been defined and studied by different authors: the topology \(\tau_{\mathcal{B}}\) of uniform convergence on \(\mathcal{B}\); the topology \(\tau_{\mathcal{B}}^S\) of strong uniform convergence on \(\mathcal{B}\); the topology \(\tau^{W}_{\mathcal{B}}\) of Whitney convergence on \(\mathcal{B}\); and the topology \(\tau^{SW}_{\mathcal{B}}\) of strong Whitney convergence on \(\mathcal{B}\). In most of this paper the authors study the cardinal functions character, density, tightness, weight, and network weight for \(C(X)\) endowed with the topologies \(\tau_{\mathcal{B}}^W\) and \(\tau_{\mathcal{B}}^{SW}\) and express them using other cardinal invariants. Since these spaces are topological groups, they also study their index of narrowness. At the end of the paper the authors find relationships between the characters and tightnesses for each of the two pairs of topological spaces \((C(X),\tau_{\mathcal{B}}^S)\) and \((C(X),\tau_{\mathcal{B}}^{SW})\), and \((C(X),\tau_{\mathcal{B}})\) and \((C(X),\tau_{\mathcal{B}}^{W})\), and also show that for each pair the cardinal functions density, weight, network weight and index of narrowness coincide.