Bornological Convergence and Shields

Abstract

Let ${{\fancyscript{S}}}$ be an ideal of subsets of a metric space ${{\langle X, d \rangle}}$ , and for ${{E \subseteq X}}$ , let ${{E^{\varepsilon}}}$ denote the ${{\varepsilon}}$ -enlargement of E. A net of subsets ${\langle A_{i} \rangle}$ of X is called ${{\fancyscript{S}^{-}}}$ -convergent (resp. ${{\fancyscript{S}^{+}}}$ -convergent) to a subset A of X if for each ${{S \in \fancyscript{S}}}$ and each ${{\varepsilon > 0}}$ , we have eventually ${{A \cap S \subseteq A^{\varepsilon}_{i}}}$ (resp ${{A_{i}S \cap A \subseteq A^{\varepsilon})}}$ . The purpose of this article is to give simple necessary and sufficient conditions for the lower and upper ${{\fancyscript{S}}}$ -convergences to be topological on the power set of X and on the closed subsets of X. In the first environment, the condition for upper convergence is stronger than that for lower convergence, while in the second more restrictive environment, it is stronger if and only if ${{\cup\fancyscript{S}}}$ is an open subset of X. In our analysis there arises a pregnant new idea – that of one set serving to shield a fixed subset from closed sets – that we study in detail, and which plays an interesting role in the upper semicontinuity of multifunctions.