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  1. Embedding Locales and Formal Topologies Into Positive Topologies.Francesco Ciraulo & Giovanni Sambin - 2018 - Archive for Mathematical Logic 57 (7-8):755-768.
    A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to (...)
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  • The Associated Sheaf Functor Theorem in Algebraic Set Theory.Nicola Gambino - 2008 - Annals of Pure and Applied Logic 156 (1):68-77.
    We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites.
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  • Quotient Topologies in Constructive Set Theory and Type Theory.Hajime Ishihara & Erik Palmgren - 2006 - Annals of Pure and Applied Logic 141 (1):257-265.
    The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the predicative methods available in constructive type theory and constructive set theory.
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