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\(C\)- and \(C^{\ast}\)-quotients in pointfree topology. (English) Zbl 1012.54025

The study of \(C\)-embedded and \(C^*\)-embedded subspaces is an important topic in traditional point-set topology. The objective of this article is to establish the basic properties of the corresponding notions in the category of locales; however, since the authors insist on working in the algebraic category of frames rather than its topological dual, the category of locales, the objects of their study appear as quotients rather than embeddings. The actual results will not be greatly surprising to traditional topologists (once they have got into the habit of ‘thinking backwards’), but the elegance and simplicity of the proofs may well surprise those who have not previously encountered locale theory. (However, such readers may ultimately be left wondering what all the fuss was about, since there are few if any explicit examples of non-spatial locales in the article.) The article also contains much useful information on the archimedean \(f\)-rings which occur as rings of continuous real-valued functions on locales, and applications to topics such as \(F\)-locales and \(P\)-locales.

MSC:

54C45 \(C\)- and \(C^*\)-embedding
06D22 Frames, locales
06F25 Ordered rings, algebras, modules
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54G10 \(P\)-spaces
18B30 Categories of topological spaces and continuous mappings (MSC2010)
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