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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.

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Dedicated to Professor S. Naimpally on the occasion of his 70 th birthday. Abstract.Bertacchi and Costantini obtained some conditions equivalent to the existence of continuous selections for the Wijsman hyperspace of ultrametric Polish spaces. We introduce a new class of hypertopologies, the macro-topologies. Both the Wijsman topology and the Vietoris topology belong to this class. We show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. In the setting of Polish spaces, these conditions are substantially weaker than the ones given by Bertacchi and Costantini. In particular, we conclude that Polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the Wijsman topology, just as it is for [0,1]. This also solves a problem implicitely raised in Bertacchi and Costantini's paper.2000 AMS Classification: 54B20, 54A10, 54D15, 54E35.