Gap functionals, proximities and hyperspace compactification

The aim of this paper is twofold: first we investigate the role of proximity notions in the framework of approach theory [R. Lowen, Approach Spaces. The Missing Link in the Topology-Uniformity-Metric Triad, Clarendon Press, Oxford, 1997] and show how quantified proximity structures can be used to obtain a canonical intrinsic version of the Smirnov compactification in this setting. Second we introduce a hyperspace structure which commutes with this type of Smirnov compactification, solving a ‘dual’ problem to the one treated concerning approach completion in [R. Lowen, M. Sioen, Proximal hypertopologies revisited, Set-Valued Anal. 6 (1998) 1–19].