Sublocales in Formal Topology

Journal of Symbolic Logic 72 (2):463 - 482 (2007)

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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DOI 10.2178/jsl/1185803619
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References found in this work BETA

Inductively Generated Formal Topologies.Thierry Coquand, Giovanni Sambin, Jan Smith & Silvio Valentini - 2003 - Annals of Pure and Applied Logic 124 (1-3):71-106.
Heyting-Valued Interpretations for Constructive Set Theory.Nicola Gambino - 2006 - Annals of Pure and Applied Logic 137 (1):164-188.
The Problem of the Formalization of Constructive Topology.Silvio Valentini - 2004 - Archive for Mathematical Logic 44 (1):115-129.
Compactness in Locales and in Formal Topology.Steven Vickers - 2006 - Annals of Pure and Applied Logic 137 (1):413-438.

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Citations of this work BETA

Positivity Relations on a Locale.Francesco Ciraulo & Steven Vickers - 2016 - Annals of Pure and Applied Logic 167 (9):806-819.
Cosheaves and Connectedness in Formal Topology.Steven Vickers - 2012 - Annals of Pure and Applied Logic 163 (2):157-174.
Locatedness and Overt Sublocales.Bas Spitters - 2010 - Annals of Pure and Applied Logic 162 (1):36-54.
Continuity and Geometric Logic.Steven Vickers - 2014 - Journal of Applied Logic 12 (1):14-27.

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