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Cauchy Points of Metric Locales

Published online by Cambridge University Press:  20 November 2018

B. Banaschewski  and
A. Pultr
B. Banaschewski
Affiliation:
MacMaster University, Hamilton, Ontario
A. Pultr
Affiliation:
Charles University, Prague, Czechoslovakia
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A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 5 , 01 October 1989 , pp. 830 - 854
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Banaschewski, B.and Mulvey, C. J., Stone-Cech compactification of locales, I, Houston J. Math. 6(1980), 301–312.Google Scholar
2. Banaschewski, B.and Pultr, A., Samuel compactification and completion of uniform frames, Math. Proc. Cambridge Phil. Soc, to appear.Google Scholar
3. Caratheodory, C., Ûber die Begrenzung einfach zusammenhangender Gebiete, Math. Annalen 73 (1913).Google Scholar
4. Ehresmann, C., Gattungen von lokalen Structuren, Uber. Deutsch. Math.-Verein 60 (1957), 59- 77.Google Scholar
5. Freudenthal, H., Neaufbau der Endentheorie, Annals of Math. 43 (1942), 261–279.Google Scholar
6. Isbell, J. R., Atomless parts of spaces, Math. Scand. 31 (1972), 5–32.Google Scholar
7. Johnstone, P. T., Stone spaces (Cambridge Univ. Press, Cambridge, 1982).Google Scholar
8. Johnstone, P. T., The point of pointless topolgy, Bull, of the AMS (New Series) 8 (1983), 41–53.Google Scholar
9. Kaufmann, B., Über die Berandung ebener und räumlicher Gebiete (Primendentheorie), Math. Annalen 103 (1930), 70–144.Google Scholar
10. Mac Lane, S., Categories for the working mathematician, Grad. Texts in Math. 5 (Springer- Verlag, 1971).Google Scholar
11. Mulvey, C. J., Banach sheaves, J. Pure Appl. Alg. 17 (1980), 69–83.Google Scholar
12. Pultr, A., Pointless uniformities II. (Dia)metrization, Comment. Math. Univ. Carolinae 25 (1984), 105–120.Google Scholar
13. Pultr, A., Categories of diametric frames, to appear in Math. Proc. Cambridge Phil. Soc.Google Scholar
14. Pultr, A., Remarks on metrizable locales, Proc. 12th Winter School, Suppl. ai Rend, del Circ. Mat. di Palermo, Série Il-no.6 (1984), 247–258.Google Scholar
15. Pultr, A., Diameters in locales: How bad they can be, to appear in Comment. Math. Univ. Carolinae.Google Scholar