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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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