Scattered toposes
type: Item
instance of: scholarly article
title: Scattered toposes
zbMATH Open document ID: 0961.03062
DOI: 10.1016/S0168-0072(99)00035-4
author: D. Pataraia
author: Leo Esakia
author: M. Jibladze
published in: Annals of Pure and Applied Logic
publication date: 2001-05-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 54G12
zbMATH DE Number: 1448519
zbMATH Keywords: internal logic
zbMATH Keywords: Kuroda's principle
zbMATH Keywords: Löb's principle
zbMATH Keywords: fixed point theorem
reviewed by: Peter Johnstone
MaRDI profile type: Publication
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full work available at URL: s0168-0072(99)00035-4
OpenAlex ID: W4212818683
description: scientific article; zbMATH DE number 1448519