Bornological convergence and shields.

Let I be an ideal of subsets of a metric space (X,d), and for E⊆X, let E^ε denote the ε-enlargement of E. A net of subsets A_i of X is called (I^-)-convergent (respectively, (I^+)-convergent) to a subset A of X if for each S∈I and each ε>0, we have eventually A∩S⊆(A_i)^ε (respectively, A_i∩S⊆A^ε). The purpose of this article is to give simple necessary and sufficient conditions for the lower and upper I-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 the union of I 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 westudy in detail, and which plays an interesting role in the upper semicontinuity of multifunctions.