Vickers, Steven (2007) Sublocales in formal topology. Journal of Symbolic Logic, 72 (2). pp. 463482. ISSN 00224812
 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 setindexed 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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