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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.

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