Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-03T02:32:48.373Z Has data issue: false hasContentIssue false

On the axiomatics of nearness and compression

Published online by Cambridge University Press:  24 October 2008

A. J. Ward
Affiliation:
Emmanuel College, Cambridge

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 (Ni) 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Császár, Á; Fondements de la Topologie Générale (Paris, 1960).Google Scholar
(2)Herrlich, H. A.concept of nearness. General Topology and its Applications 4 (1974), 191212.CrossRefGoogle Scholar
(3)Naimpally, S. A.Reflective functors via nearness. Fundamenta Mathematicae 85 (1974), 245255.CrossRefGoogle Scholar
(4)Thron, W. J.Proximity structures and grills. Math. Ann. 206 (1973), 3562.CrossRefGoogle Scholar