On the axiomatics of nearness and compression

Extract

H. Herrlich ((2), p. 193) defines, for a given nearness ξ, a property ξ̃ (which by analogy with the case of filters on a proximity space may be called compression) and remarks that ξ is determined by ξ̃. He continues: ‘Consequently, each of the axioms (N_i) can be translated into a condition concerning ξ̃, thus providing an axiomatization of the concept of collections of sets containing arbitrary small members.’ This last remark seems slightly misleading; it might reasonably be taken to mean that the reciprocity between ξ and ξ̃ is a mere set-theoretic tautology, independent of the nearness axioms themselves. (Compare, for example, the relation between the ideas of ‘closure-point of a set’ and ‘neighbourhood of a point’ in the axiomatics of topology.) This is not in fact the case; however, one cannot select from Herrlich's axioms a subset which is necessary and sufficient for ξ to be determined by ξ̃. Moreover, the relation between ξ and ξ̃ appears to be asymmetrical. We shall exhibit, in terms of stacks ((4), p. 36), an elegant set of mutually independent axioms, first for the more general ‘Čech nearness’ discussed by Naimpally in (3) and then for a Herrlich nearness (called in (3) a LO-nearness, with a compatible topology as there defined); these axioms make the reciprocity between nearness and compression, and its relationship with the axioms, explicit and obvious.