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Sublocales in formal topology

Vickers, Steven (2007) Sublocales in formal topology. Journal of Symbolic Logic, 72 (2). pp. 463-482. ISSN 0022-4812

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URL of Published Version: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.jsl/1185803619&page=record

Identification Number/DOI: doi:10.2178/jsl/1185803619

The paper studies how the localic notion of sublocale transfers to formal topology. For any formal topology (not necessarily with positivity predicate) we define a sublocale to be a cover relation that includes that of the formal topology. The family of sublocales has set-indexed joins. For each set of base elements there are corresponding open and closed sublocales, boolean complements of each other. They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure. Overt sublocales and weakly closed sublocales are described, and related via a new notion of “rest closed” sublocale to the binary positivity predicate. Overt, weakly closed sublocales of an inductively generated formal topology are in bijection with “lower powerpoints”, arising from the impredicative theory of the lower powerlocale. Compact sublocales and fitted sublocales are described. Compact fitted sublocales of an inductively generated formal topology are in bijection with “upper powerpoints”, arising from the impredicative theory of the upper powerlocale.

Type of Work:Article
Date:2007 (Publication)
School/Faculty:Schools (1998 to 2008) > School of Computer Science
Department:Computer Science
Subjects:Q Science (General)
QA75 Electronic computers. Computer science
QA Mathematics
Institution:University of Birmingham
Copyright Holders:Association for Symbolic Logic
ID Code:195
Refereed:YES
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