A Kuratowski-Mrówka theorem in approach theory

In this paper we give a number of arguments why, in approach theory, the notion of compactness which from the intrinsic categorical point of view seems most satisfying is 0-compactness, i.e., measure of compactness equal to zero. It was already known from [R. Lowen, Kuratowski's measure of noncompactness revisited, Quart. J. Math. Oxford 39 (1988) 235-254] that measure of compactness has good properties and good interpretations for both topological and metric approach spaces. Here, introducing notions of closed and proper mappings in approach theory, which satisfy all the intrinsic categorical axioms put forth in [Clementino et al., A functional approach to topology, in: M.C. Pedicchio, W. Tholen (Eds.) Categorical Foundations Special Topics in Order, Topology, Algebra, and Sheaf Theory, Cambridge University Press, 2003], we prove fundamental results concerning these concepts, also linked to 0-compactness, and we give a Kuratowski-Mrówka-type characterization of 0-compactness.