Journal of Symbolic Logic

Sublocales in formal topology

Steven Vickers

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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.

Article information

Source
J. Symbolic Logic Volume 72, Issue 2 (2007), 463-482.

Dates
First available in Project Euclid: 30 July 2007

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1185803619

Digital Object Identifier
doi:10.2178/jsl/1185803619

Mathematical Reviews number (MathSciNet)
MR2320286

Zentralblatt MATH identifier
1132.03033

Subjects
Primary: 03F65: Other constructive mathematics [See also 03D45]
Secondary: 03B15: Higher-order logic and type theory 54B05: Subspaces

Keywords
formal topology locale sublocale inductively generated open closed weakly closed compact fitted powerlocale

Citation

Vickers, Steven. Sublocales in formal topology. Journal of Symbolic Logic 72 (2007), no. 2, 463--482. doi:10.2178/jsl/1185803619. http://projecteuclid.org/euclid.jsl/1185803619.


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