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December 2004 A structural investigation on formal topology: coreflection of formal covers and exponentiability
Maria Emilia Maietti, Silvio Valentini
J. Symbolic Logic 69(4): 967-1005 (December 2004). DOI: 10.2178/jsl/1102022209

Abstract

We present and study the category of formal topologies and some of its variants. Two main results are proven. The first is that, for any inductively generated formal cover, there exists a formal topology whose cover extends in the minimal way the given one. This result is obtained by enhancing the method for the inductive generation of the cover relation by adding a coinductive generation of the positivity predicate. Categorically, this result can be rephrased by saying that inductively generated formal topologies are coreflective into inductively generated formal covers. The second result is that unary formal covers are exponentiable in the category of inductively generated formal covers and hence, thanks to the coreflection, unary formal topologies are exponentiable in the category of inductively generated formal topologies. From a localic point of view the exponentiability of unary formal topologies means that algebraic dcpos are exponentiable in the category of open locales. But, the coreflection theorem states that open locales are coreflective in locales and hence, as a consequence of well-known impredicative results on exponentiable locales, it allows to prove that locally compact open locales are exponentiable in the category of open locales.

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Maria Emilia Maietti. Silvio Valentini. "A structural investigation on formal topology: coreflection of formal covers and exponentiability." J. Symbolic Logic 69 (4) 967 - 1005, December 2004. https://doi.org/10.2178/jsl/1102022209

Information

Published: December 2004
First available in Project Euclid: 2 December 2004

zbMATH: 1071.03040
MathSciNet: MR2135653
Digital Object Identifier: 10.2178/jsl/1102022209

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.69 • No. 4 • December 2004
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