Abstract

We consider nearapproaching, a simple generalisation of pseudonearness and approach distances. The central idea in an approach space (X,d) in the sense of Lowen is that of a distance d, which is a function on X×2X to [0,∞]. Of fundamental importance is the fact that such a distance can be defined not only in metric spaces, but also in topological spaces, uniform spaces, and related structures. Pseudonearness structures on a set X consist of pairs (BX,N), where BX is a bornology and N is a nearoperator from BX to P̲(P̲(P̲X)), satisfying certain near axioms. In this framework, bornologies, generalised Kuratowski operators, pseudoproximity structures, and, last but not least, classical nearness spaces can all be unified within a single setting. In our treatment of approaching, we define the corresponding spaces by focusing on functions τ from CX⊂P̲(P̲(X) into the set of functions from BX into the closed interval [0,∞], known as targets, which we compare setwise, where BX is a boundedness structure. Thus, a target can be regarded as a map that measures how close a collection S∈CX is to a bounded set B∈BX.

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