Pseudometrizable Bornological Convergence is Attouch-Wets Convergence

Let $\mathcal{S}$ be an ideal of subsets of a metric space $\langle X,d \rangle$. A net of subsets $\langle A_\lambda\rangle$ 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 $A \cap S \subseteq A^\varepsilon_\lambda \ \textrm{and} \ A_\lambda \cap S \subseteq A^\varepsilon.$ We identify necessary and sufficient conditions for this convergence to be admissible and topological on the power set of $X$. We show that $\mathcal{S}$-convergence is compatible with a pseudometrizable topology if and only if $\mathcal{S}$ has a countable base and each member of $\mathcal{S}$ has an $\varepsilon$-enlargement that is again in $\mathcal{S}$. Further, in the case that the ideal is a bornology, we show that $\mathcal{S}$-convergence when pseudometrizable is Attouch-Wets convergence with respect to an equivalent metric.