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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 $\left(X,d\right)$. A net of subsets ${\left({A}_{\lambda }\right)}_{\lambda }$ of $X$ is called $𝒮$-convergent to a subset $A$ of $X$ if for each $S\in 𝒮$ and each $\epsilon >0$, we have eventually $A\cap S\subset {A}_{\lambda }^{\epsilon }$ and ${A}_{\lambda }\cap S\subset {A}^{\epsilon }$. Here ${A}^{\epsilon }$ is the $\epsilon$ 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 $\left(X,d\right)$ the conditions $𝒮$-convergence is topological and $𝒮$ is stable under small enlargements are equivalent. From this it follows that if $\left(X,d\right)$ is not discrete, and $𝒮$ is the ideal of nowhere dense subsets of $X$, then $𝒮$-convergence is not topological on $𝒫\left(X\right)$. Another interesting result is that if $𝒮$ is a bornology in $\left(X,d\right)$, then the statements $\left(𝒮,d\right)$-convergence is compatible with a pseudometrizable topology, $𝒮$ is stable under small enlargements and has a countable base, and there exists an equivalent metric $\varrho$ for $X$ such that $\left(𝒮,d\right)$-convergence on $𝒫\left(X\right)$ is Attouch-Wets convergence with respect to $\varrho$, are equivalent.
##### MSC:
 54B20 Hyperspaces (general topology) 46A17 Bornologies and related structures; Mackey convergence, etc. 54E35 Metric spaces, metrizability