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Pointless metric spaces

Published online by Cambridge University Press:  12 March 2014

Giangiacomo Gerla
Giangiacomo Gerla*
Affiliation:
Dipartimento di Matematica ed Applicazioni, Università di Napoli, 80134 Naples, Italy
*
Istituto di Matematica, Università della Basilicata, 35100 Potenza, Italy.
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Extract

In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).

In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].

In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 1 , March 1990 , pp. 207 - 219
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

[1]Gerla, G. and Volpe, R., Geometry without points, American Mathematical Monthly, vol. 92 (1985), pp. 707–711.CrossRefGoogle Scholar
[2]Gerla, G., Some approaches to pointless geometry, Handbook of incidence geometry (Buekenhout, F. and Kantor, W., editors), North-Holland, Amsterdam (to appear).Google Scholar
[3]Gerla, G., Distances, diameters and verisimilitude of theories (unpublished).Google Scholar
[4]Negoita, C. V. and Ralescu, D. A., Applications of fuzzy sets to systems analysis, Birkhäuser, Basel, 1975.CrossRefGoogle Scholar
[5]Weihrauch, K. and Schreiber, U., Embedding metric spaces into cpo's, Theoretical Computer Science, vol. 16 (1981), pp. 5–24.CrossRefGoogle Scholar