Let B 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 B 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. © 2008 Elsevier Inc. All rights reserved.
Beer, G., Levi, S. (2009). Strong uniform continuity. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 350(2), 568-589 [10.1016/j.jmaa.2008.03.058].
Strong uniform continuity
LEVI, SANDRO
2009
Abstract
Let B 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 B 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. © 2008 Elsevier Inc. All rights reserved.
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