Strong uniform continuity

Let S be an ideal of subsets of a metric space < X, d >. This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of S which reduces to ordinary continuity when B consists of the finite subsets of X and agrees with uniform continuity on members of B when B is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions. (C) 2008 Elsevier Inc. All rights reserved.