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Samuel compactification and completion of uniform frames

Published online by Cambridge University Press:  24 October 2008

B. Banaschewski  and
A. Pultr
B. Banaschewski
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, CanadaL8S 4K1
A. Pultr
Affiliation:
Charles University, Malostranské nam. 25, Prague, Czechoslovakia
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Extract

The aim of this paper is twofold: first, to construct the compact regular coreflection of uniform frames, that is, the frame counterpart of the Samuel compactification of uniform spaces (Samuel [10]), and then to use this for a new description of the completion of a uniform frame, as an alternative to those previously given by Isbell [6] on the one hand and Kříž [8] on the other. In addition, we present a few further results, as well as new proofs of known ones, that are naturally connected with completions and arise particularly easily from our approach to them. Most prominently among these, we identify the uniform space of minimal Cauchy filters of a uniform frame as the spectrum of its completion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 1 , July 1990 , pp. 63 - 78
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Banaschewski, B.. Zur Begründung der Theorie der topologischen Räume. Diplomarbeit, Universität Hamburg (1952).Google Scholar
[2]Banaschewski, B.. Über nulldimensionale Räme. Math. Nachr. 13 (1955), 129–140.Google Scholar
[3]Banaschewski, B. and Mulvey, C. J.. Stone–Čech compactification of locales. Houston J. Math. 6 (1980), 301–312.Google Scholar
[4]Boukbaki, N.. General Topology, part 1 (Hermann and Addison-Wesley, 1966).Google Scholar
[5]Isbell, J. R.. Uniform. Spaces. Math. Surveys no. 12 (American Mathematical Society, 1964).Google Scholar
[6]Isbell, J. R.. Atomless parts of spaces. Math. Scand. 31 (1972), 5–32.CrossRefGoogle Scholar
[7]Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3 (Cambridge University Press, 1982).Google Scholar
[8]Kříž, I.. A direct description of uniform completion in locales and a characterization of LT groups. Cahiers Topologie et Géom. Différentielle Catégoriques 27 (1986), 19–34.Google Scholar
[9]Pultr, A.. Pointless uniformities I. Complete regularity. Comm. Math. Univ. Carolinae 25 (1984), 91–104.Google Scholar
[10]Samuel, P.. Ultrafilters and compactifications. Trans. Amer. Math. Soc. 64 (1948), 100–132.CrossRefGoogle Scholar