Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 72, Issue 2 (2007), 463-482.
Sublocales in formal topology
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.
J. Symbolic Logic Volume 72, Issue 2 (2007), 463-482.
First available in Project Euclid: 30 July 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03F65: Other constructive mathematics [See also 03D45]
Secondary: 03B15: Higher-order logic and type theory 54B05: Subspaces
Vickers, Steven. Sublocales in formal topology. J. Symbolic Logic 72 (2007), no. 2, 463--482. doi:10.2178/jsl/1185803619. http://projecteuclid.org/euclid.jsl/1185803619.