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Sublocales in formal topology

Published online by Cambridge University Press:  12 March 2014

Steven Vickers
Steven Vickers*
Affiliation:
School of Computer Science, The University of Birmingham, Birmingham, B15 2TT, UK, E-mail: s.j.vickers@cs.bham.ac.uk, URL: http://www.cs.bham.ac.uk/~sjv
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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.

Keywords

formal topologylocalesublocaleinductively generatedopenclosedweakly closedcompactfittedpowerlocale
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 2 , June 2007 , pp. 463 - 482
Copyright
Copyright © Association for Symbolic Logic 2007

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