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Realcompact spaces and regular σ -frames

Published online by Cambridge University Press:  24 October 2008

Christopher R. A. Gilmour
Christopher R. A. Gilmour
Affiliation:
University of Cape Town, South Africa
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Abstract

We show that there is an adjoint dual equivalence between realcompact Alexandroff spaces and the Alexandroff σ-frames. This gives a corresponding adjoint duality for realcompact Tychonoff spaces. Consequently we characterize lattice theoretically the cozero-sets of a topological space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 1 , July 1984 , pp. 73 - 79
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

[1] Banaschewski, B.. The duality of distributive σ-continuous lattices. In Continuous Lattices Lecture Notes in Math. vol. 871 (Springer-Verlag, 1981), 12–19.Google Scholar
[2] Banaschewski, B.. σ-Frames (in preparation).Google Scholar
[3] Bourbaki, N.. General Topology, part 1 (Addison-Wesley, 1966).Google Scholar
[4] Charalambous, M. G.. Dimension theory for σ-frames. J. London Math. Soc. (2), 8 (1974), 149–160.Google Scholar
[5] Charalambous, M. G.. A new covering dimension function for uniform spaces. J. London Math. Soc. (2), 11 (1975), 137–143.CrossRefGoogle Scholar
[6] Gillman, L. and Jerison, M.. Rings of Continuous Functions. (Van Nostrand, 1960).CrossRefGoogle Scholar
[7] Gilmour, C. R. A.. Special morphisms for zero-set spaces. Bull. Austral. Math. Soc. 13 (1975), 57–68.Google Scholar
[8] Gilmour, C. R. A.. Realcompactifications through zero-set spaces. Quaestiones Math. 6 (1983), 73–95.Google Scholar
[9] Gordon, H.. Rings of functions determined by zero-sets. Pacific J. Math. 36 (1971), 133–157.CrossRefGoogle Scholar
[10] Hager, A. W.. Some nearly fine uniform spaces. Proc. London Math. Soc. (3), 28 (1974), 517–546.Google Scholar
[11] Hager, A. W.. Cozero fields. Confer. Sem. Mat. Univ. Bari 175 (1980), 1–23.Google Scholar
[12] Johnstone, P. T.. Stone Spaces. Cambridge studies in advanced mathematics, vol. 3 (Cambridge University Press, 1982).Google Scholar
[13] Reynolds, G.. Alexandroff algebras and complete regularity. Proc. Amer. Math. Soc. 76 (1979), 322–326.Google Scholar