In this short note we show that any proof of a general spatiality theorem for inductively generated formal topologies requires full classical logic. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We propose a formal representation of objects , those being mathematical or empirical objects. The powerful framework inside which we represent them in a unique and coherent way is grounded, on the formal side, in a logical approach with a direct mathematical semantics in the well-established field of constructive topology, and, on the philosophical side, in a neo-Kantian perspective emphasizing the knowing subject’s role, which is constructive for the mathematical objects and constitutive for the empirical ones.

In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.

If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale . Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of the binary Tychonoff theorem is given, including a characterization of the finite covers of the product by basic opens. The discussion (...) is then related to the double powerlocale. (shrink)

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.