
Journal of Convex Analysis 15 (2008), No. 2, 439453 Copyright Heldermann Verlag 2008 Pseudometrizable Bornological Convergence is AttouchWets Convergence Gerald Beer Dept. of Mathematics, California State University, 5151 State University Drive, Los Angeles, CA 90032, U.S.A. gbeer@cslanet.calstatela.edu Sandro Levi Dip. di Matematica e Applicazioni, Università di MilanoBicocca, Via Cozzi 53, 20125 Milano, Italy sandro.levi@unimib.it [Abstractpdf] 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 $\mathcal{S}$\textit{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 AttouchWets convergence with respect to an equivalent metric. Keywords: Bornological convergence, AttouchWets convergence, bounded Hausdorff convergence, hyperspace, bornology. MSC: 54B20; 46A17, 54E35 [ Fulltextpdf (178 KB)] for subscribers only. 