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Compactification of frames. (English) Zbl 0722.54018

The author surveys the theory of (regular) compactifications of locales, stressing the aspect of locale theory as the constructive (choice-free) counterpart of topology. The main “new” result (actually the localic translation of a result established for spaces by the same author over twenty years ago) is that compactifications of a given locale L correspond bijectively to certain binary “strong inclusion” relations on (the frame corresponding to) L. The author also shows that a locale has a smallest compactification iff it is locally compact and regular, iff it is isomorphic to an open sublocale of a compact regular locale.

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54B30 Categorical methods in general topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
06B35 Continuous lattices and posets, applications
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References:

[1] Banaschewski, Frames and compactifications. Contributions to Extension Theory of Topological Structures Proc. Symposium Berlin 1967 (1969)
[2] Banaschewski, Continuous Lattices. Proceedings Bremen 1979 871 pp 1– (1981)
[3] Banaschewski, Stone-Čech compactification of locales., Houston J. Math. 6 pp 301– (1980) · Zbl 0549.54017
[4] B. Banaschewski
[5] B. Banaschewski
[6] C. H. Dowker Dona Papert 1967
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[8] Johnstone, Stone spaces (1983)
[9] Johnstone, The point of pointless topology., Bull. Amer. Math. Soc. 8 pp 41– (1983) · Zbl 0499.54002
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