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Pseudometrizable bornological convergence is Attouch-Wets convergence. (English) Zbl 1173.54002
Let \(\mathcal S\) be an ideal of subsets of a metric space \((X,d)\). A net of subsets \((A_\lambda)_\lambda\) of \(X\) is called \(\mathcal S\)-convergent to a subset \(A\) of \(X\) if for each \(S\in {\mathcal S}\) and each \(\varepsilon > 0\), we have eventually \(A\cap S\subset A^\varepsilon_\lambda\) and \(A_\lambda\cap S\subset A^\varepsilon\). Here \(A^\varepsilon\) is the \(\varepsilon\) enlargement of \(A\). Attouch-Wets convergence is an example of \({\mathcal S}\)-convergence, for suitable \(\mathcal S\). The authors identify necessary and sufficient conditions for \({\mathcal S}\)-convergence to be admissible and topological on the power set of \(X\). They prove for example that for an ideal \({\mathcal S}\) in a metric space \((X,d)\) the conditions \(\mathcal S\)-convergence is topological and \(\mathcal S\) is stable under small enlargements are equivalent. From this it follows that if \((X,d)\) is not discrete, and \(\mathcal S\) is the ideal of nowhere dense subsets of \(X\), then \(\mathcal S\)-convergence is not topological on \({\mathcal P}(X)\). Another interesting result is that if \(\mathcal S\) is a bornology in \((X,d)\), then the statements \(({\mathcal S},d)\)-convergence is compatible with a pseudometrizable topology, \(\mathcal S\) is stable under small enlargements and has a countable base, and there exists an equivalent metric \(\varrho\) for \(X\) such that \(({\mathcal S},d)\)-convergence on \({\mathcal P}(X)\) is Attouch-Wets convergence with respect to \(\varrho\), are equivalent.

54B20 Hyperspaces in general topology
46A17 Bornologies and related structures; Mackey convergence, etc.
54E35 Metric spaces, metrizability
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