Bornological modifications of hyperspace topologies (Q386217)

Given an \textit{ideal} \(\mathcal S\) on a metric space \((X,d)\), define the \textit{bornological convergence} on \(2^X\), the power set of \(X\), as a pretopology on \(2^X\) via the closure operator \( \text{cl}_{\mathcal S}^{H}(\mathcal A)=\{A\in 2^X: \forall \epsilon>0 \;\forall S\in \mathcal S \;\exists B\in \mathcal A \;(A\cap S\subseteq B^{\epsilon} \;\text{and} \;B\cap S\subseteq A^{\epsilon}) \}\), where \(\mathcal A\subseteq 2^X\), and \(A^{\epsilon}\) is the set of points of \(d\)-distance less than \(\epsilon\) from \(A\). One can naturally split this definition to obtain closure operators \(\text{cl}_{\mathcal S}^{H^-}, \text{cl}_{\mathcal S}^{H^+}\) for the lower/upper bornological convergence. These definitions are modeled on the construction of the (lower/upper) Hausdorff metric hyperspace topology \(H^-,H^+\), and \(H\), which are the special cases when \(\mathcal S=2^X\). The author studies the topologizations of these pretopologies; e.g., if \(H^-(\mathcal S), H^+(\mathcal S)\) are the topologization of the lower/upper bornological convergences, then the ideal \(\mathcal S'\) for which \(\text{cl}_{\mathcal S'}^{H^-}\) equals the closure operator associated with \(H^-(\mathcal S)\) is characterized if such an \(\mathcal S'\) exists, and it is shown that such \(\mathcal S'\) may not exist. Contrary to this, there is always an ideal \(\mathcal S'\) for which \(\text{cl}_{\mathcal S'}^{H^+}\) equals the closure operator associated with \(H^+(\mathcal S)\). The author also defines a generalized version of these pretopologies and then studies their topologization; for example, \(\text{cl}_{\mathcal S}^{H^-}\) is generalized via the closure operator \(\text{cl}_{\mathcal S}^{T}(\mathcal A)=\{A\in 2^X: \exists S\in \mathcal S \;(S\subseteq A \;\text{and} \;A\in \text{cl}^T(\mathcal A))\}\), where \(\text{cl}^T\) is the closure operator of a fixed lower hyperspace topology \(T\) on \(2^X\) (i.e. such that if \(\mathcal A\in T\), then \(\mathcal A=\{B\in 2^X: \exists A\in\mathcal A \;(B\supseteq A)\}\)); there is a suitable generalization \(\text{cl}_{\mathcal S}^{T}\) of \(\text{cl}_{\mathcal S}^{H^+}\), when \(T\) is an upper hypertopology. It turns out that a single topology \(\tau(\mathcal S)\) on \(2^X\) -- having base elements of the form \(\{A\in 2^X: S\subseteq A\subseteq X\setminus S'\}\), where \(S,S'\in \mathcal S\) -- can be used to characterize the said topologizations; in particular, \(H^-(\mathcal S)=H^-\land \tau(\mathcal S)\), and \(H^+(\mathcal S)=H^+\land \tau(\mathcal S)\).