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June 2007 Sublocales in formal topology
Steven Vickers
J. Symbolic Logic 72(2): 463-482 (June 2007). DOI: 10.2178/jsl/1185803619

Abstract

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.

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Steven Vickers. "Sublocales in formal topology." J. Symbolic Logic 72 (2) 463 - 482, June 2007. https://doi.org/10.2178/jsl/1185803619

Information

Published: June 2007
First available in Project Euclid: 30 July 2007

zbMATH: 1132.03033
MathSciNet: MR2320286
Digital Object Identifier: 10.2178/jsl/1185803619

Subjects:
Primary: 03F65
Secondary: 03B15, 54B05

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.72 • No. 2 • June 2007
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