Scattered toposes
A topos \(\mathbb X\) is said to be scattered if every Boolean subtopos of \(\mathbb X\) is locally closed (i.e., is an open subtopos of its closure). This is the natural generalization of the notion of `scattered space' from spaces to toposes; in the present paper, the authors show that it is equivalent to the validity in the internal logic of \(\mathbb X\) of a `relativized' version of Kuroda's principle, namely the formula \(\forall x[(p(x)\Rightarrow\forall y p(y))\Rightarrow\forall y p(y)]\Rightarrow\forall x p(x)\), and also to the validity of Löb's principle \([\forall\psi(\psi\vee(\psi\Rightarrow\phi)) \Rightarrow\phi]\Rightarrow\phi\). Some examples and applications are given, in particular an application to a fixed point theorem for `unchanging' maps.
scientific article; zbMATH DE number 1448519