Uniform Continuity, Uniform Convergence, and Shields

Let B be a bornology in a metric space [X, d], that is, a cover of X by nonempty subsets that also forms an ideal. In Beer and Levi ( J Math Anal Appl 350: 568-589, 2009), the authors introduced the variational notions of strong uniform continuity of a function on B as an alternative to uniform continuity of the restriction of the function to each member of B, and the topology of strong uniform convergence on B as an alternative to the classical topology of uniform convergence on B. Here we continue this study, showing that shields as introduced in Beer, Costantini and Levi ( Bornological Convergence and Shields, Mediterranean J. Math, submitted) play a pivotal role. For example, restricted to continuous functions, the topology of strong uniform convergence on B reduces to the classical topology if and only if the natural closure of the bornology is shielded from closed sets. The paper also further develops the theory of shields and their applications.