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Lindelöf locales and realcompactness

Published online by Cambridge University Press:  24 October 2008

J. Madden  and
J. Vermeer
J. Madden
Affiliation:
University of Kansas, Lawrence, KS 66045, U.S.A.
J. Vermeer
Affiliation:
University of Kansas, Lawrence, KS 66045, U.S.A.
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Extract

We show that a locale possesses the localic analogue of the property of realcompactness if and only if it is regular Lindelöf. Thus, the localic version of the Hewitt real-compactification, originally defined by G.Reynolds using σ-frames, is the regular Lindelöf reflection. An immediate consequence is that a space is realcompact if and only if it is the point space of a regular Lindelöf local (3·2). We point out a nice analogy between a theorem of Reynolds and Stone's classical representation theorem for boolean algebras. Finally, we show that the quasi-F cover of a compact Hausdorff space is the Stone–čech compactifications of the smallest dense Lindelöf sublocale.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 473 - 480
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Banaschewski, B. and Mulvey, C. J.. Stone čech compactification of locales. I. Houston J. Math. 6 (1980), 301–312.Google Scholar
[2]Dashiell, F., Hager, A. and Henriksen, M.. Order-Cauchy completions of rings of continuous functions. Canad. J. Math. 32 (1980), 657–385.CrossRefGoogle Scholar
[3]Van Douwen, E. K. and Hao-Xuan, Zhou. The number of cozero sets is an ω-power. (Preprint.)Google Scholar
[4]Dowker, C. H. and Strauss, D.. Sums in the category of frames. Houston J. Math. 3 (1976), 17–32.Google Scholar
[5]Engelking, R.. General Topology. (Warsaw, 1977.)Google Scholar
[6]Gilmour, C. R. A.. Realcompact spaces and regular σ-frames. Math. Proc. Cambridge Philos. Soc. 96 (1984), 73–79.Google Scholar
[7]Henriksen, M., Vermeer, J. and Woods, R. G.. Quase-f covers of Tychonoff spaces. (To appear.)Google Scholar
[8]Isbell, J. R.. Atomless parts of spaces. Math. Scand. 31 (1972), 5–32.Google Scholar
[9]Isbell, J. R.. Review of [10]. Bull. Amer. Math. Soc. (N.S.) 11 (1984), 389–392.CrossRefGoogle Scholar
[10]Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. No. 3 (Cambridge University Press, 1982).Google Scholar
[11]Reynolds, G.. On the spectrum of a real representable ring. In Application of Sheaves, Lecture Notes in Math. vol. 753 (Springer-Verlag, 1979), 595–611.Google Scholar
[12]Vermeer, J.. The smallest basically disconnected preimage of a space. Topology Appl. 17 (1984), 217–232.Google Scholar