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Pseudometrizable bornological convergence is Attouch-Wets convergence. (English) Zbl 1173.54002
Let 𝒮 be an ideal of subsets of a metric space (X,d). A net of subsets (A λ ) λ of X is called 𝒮-convergent to a subset A of X if for each S𝒮 and each ε>0, we have eventually ASA λ ε and A λ SA ε . Here A ε is the ε enlargement of A. Attouch-Wets convergence is an example of 𝒮-convergence, for suitable 𝒮. The authors identify necessary and sufficient conditions for 𝒮-convergence to be admissible and topological on the power set of X. They prove for example that for an ideal 𝒮 in a metric space (X,d) the conditions 𝒮-convergence is topological and 𝒮 is stable under small enlargements are equivalent. From this it follows that if (X,d) is not discrete, and 𝒮 is the ideal of nowhere dense subsets of X, then 𝒮-convergence is not topological on 𝒫(X). Another interesting result is that if 𝒮 is a bornology in (X,d), then the statements (𝒮,d)-convergence is compatible with a pseudometrizable topology, 𝒮 is stable under small enlargements and has a countable base, and there exists an equivalent metric ϱ for X such that (𝒮,d)-convergence on 𝒫(X) is Attouch-Wets convergence with respect to ϱ, are equivalent.
MSC:
54B20Hyperspaces (general topology)
46A17Bornologies and related structures; Mackey convergence, etc.
54E35Metric spaces, metrizability