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Cauchy points of uniform and nearness frames. (English) Zbl 0861.54023

The authors investigate nearness frames, which are the frame-theoretic version of nearness spaces. They begin with a discussion of what an ‘ideal point’ of a nearness frame might be taken to mean, and show that general considerations lead to a unique sensible answer: namely, that points should be identified with regular Cauchy filters in the frame. They then investigate the ‘Cauchy spectrum’ of a nearness frame, formed by the set of all such filters with its canonical nearness space structure, and discuss such questions as the functoriality of the Cauchy spectrum and its relation to completion (it turns out that the Cauchy spectrum of a nearness frame is just the ordinary spectrum of its completion).

MSC:

54E17 Nearness spaces
06B99 Lattices
54H99 Connections of general topology with other structures, applications
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