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Inductively generated formal topologies

Thierry Coquand, Giovanni Sambin, Jan Smith, Silvio Valentini
2003 Annals of Pure and Applied Logic  
Here we give a systematic exposition of one of the main tools in formal topology: inductive generation.  ...  In fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively.  ...  instead of restricting to consider only inductively generated formal topologies.  ... 
doi:10.1016/s0168-0072(03)00052-6 fatcat:4x4uwyxq6zgzbljubv4womnabq

Exponentiation of Scott Formal Topologies

Maria Emilia Maietti, Silvio Valentini
2004 Electronical Notes in Theoretical Computer Science  
We prove that Scott formal topologies are exponentiable in the category of inductively generated formal topologies.  ...  Acknowledgements This work is based on so many results in formal topology that we feel to thank all of the people that contributed to it, especially Paolo Virgili, Giovanni Sambin, Silvia Gebellato, Giovanni  ...  We also acknowledge fruitful discussions with Martín Escardó on the general topic of this paper.  ... 
doi:10.1016/j.entcs.2004.08.005 fatcat:ptrc7f7ifrdl5jpyfjuh4qtmvm

General Recursion and Formal Topology

Claudio Sacerdoti Coen, Silvio Valentini
2010 Electronic Proceedings in Theoretical Computer Science  
In this work we propose a new approach to this problem based on the use of inductively generated formal topologies.  ...  It is well known that general recursion cannot be expressed within Martin-Loef's type theory and various approaches have been proposed to overcome this problem still maintaining the termination of the  ...  Definition 2.1 (Inductively generated formal topology).  ... 
doi:10.4204/eptcs.43.5 fatcat:vjrvnxqycbet5pp4oregqfrhb4

Spatiality for formal topologies

NICOLA GAMBINO, PETER SCHUSTER
2007 Mathematical Structures in Computer Science  
We define what it means for a formal topology to be spatial, and investigate properties related to spatiality both in general and in examples.  ...  The formal points of an inductively generated formal topology can be characterised purely in terms of the covering system that generates the formal topology.  ...  Principles related to spatiality Inductively generated formal topologies As for locales (Johnstone, 1982; MacLane and Moerdijk, 1994; Simmons, 2004) , definitions by 'generators and relations' provide  ... 
doi:10.1017/s0960129506005810 fatcat:3ensmyux4ngzdpf6kttzyuo6gi

Why topology in the minimalist foundation must be pointfree

Maria Emilia Maietti, Giovanni Sambin
2013 Logic and Logical Philosophy  
We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our two-level minimalist foundation, the choice for a pointfree approach to topology  ...  The formal topology of real numbers R ≡ (Q × Q, ¡ R , Pos R ) is an inductively generated formal topology defined as follows.  ...  Formal covers, as well as formal topologies, can be inductively generated from a set-indexed family of axioms of the form a ¡ U : Definition 4.4.  ... 
doi:10.12775/llp.2013.010 fatcat:djpf6wtw4fd2lc3wpobxmbmsue

Equivalents of the finitary non-deterministic inductive definitions [article]

Ayana Hirata, Hajime Ishihara, Tatsuji Kawai, Takako Nemoto
2019 arXiv   pre-print
We present statements equivalent to some fragments of the principle of non-deterministic inductive definitions (NID) by van den Berg (2013), working in a weak subsystem of constructive set theory CZF.  ...  We show that several statements in constructive topology which were initially proved using NID are equivalent to the elementary and finitary NIDs.  ...  The class of formal topology maps from a set-presented formal topology to an inductively generated formal topology is set-generated. Proof.  ... 
arXiv:1903.05852v3 fatcat:jkjsoadrqjctnmzlbz65cbboru

Sublocales in formal topology

Steven Vickers
2007 Journal of Symbolic Logic (JSL)  
They generate a boolean algebra amongst the sublocales. In the case of an inductively generated formal topology, the collection of inductively generated sublocales has coframe structure.  ...  Compact fitted sublocales of an inductively generated formal topology are in bijection with "upper powerpoints", arising from the impredicative theory of the upper powerlocale.  ...  Another general case is that of inductively generated formal topologies (Section 4).  ... 
doi:10.2178/jsl/1185803619 fatcat:jpfsgkf43rcaxad76p4djtwvii

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice [article]

Maria Emilia Maietti, Samuele Maschio, Michael Rathjen
2021 arXiv   pre-print
We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mtt, extended with inductively generated formal topologies, Church's thesis and axiom of choice  ...  A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel's constructive set theory CZF extended with the regular extension axiom.  ...  Other relevant examples of inductively generated formal topologies are the topologies of Cantor and Baire spaces.  ... 
arXiv:1905.11966v5 fatcat:2m2ythcamnh3doezr4tiszdoj4

Geometric theories of patch and Lawson topologies [article]

Tatsuji Kawai
2017 arXiv   pre-print
We present the patch topology of a stably locally compact formal topology by a geometric theory whose models are the points of the given topology that are located, and the Lawson topology of a continuous  ...  Moreover, the construction of Lawson topologies naturally induces a monad on the category of compact regular formal topologies, which is shown to be isomorphic to the Vietoris monad.  ...  Inductively generated formal topologies The notion of inductively generated formal topology by Coquand et al. [6, Theorem 3.1] allows us to define a formal topology by a set of axioms.  ... 
arXiv:1709.06403v1 fatcat:q2zt3ubsxjgwbo4wx3jw7g3vfi

Tychonoff's theorem in the framework of formal topologies

Sara Negri, Silvio Valentini
1997 Journal of Symbolic Logic (JSL)  
The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies.  ...  The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory  ...  of localization and suggesting the definition of product of formal topologies.  ... 
doi:10.2307/2275645 fatcat:cdjkwvklwvhbxfuejpmds6u7ti

On the formal points of the formal topology of the binary tree

Silvio Valentini
2002 Archive for Mathematical Logic  
Formal topology is today an established topic in the development of constructive mathematics and constructive proofs for many classical results of general topology have been obtained by using this approach  ...  Here we analyze one of the main concepts in formal topology, namely, the notion of formal point.  ...  Thus all we need is to be able to inductively generate a ≤-formal topology.  ... 
doi:10.1007/s001530100133 fatcat:ozyw2hrmc5eqfeepsgnmgd2rt4

Constructive characterizations of bar subsets

Silvio Valentini
2007 Annals of Pure and Applied Logic  
, for an inductively generated formal topology.  ...  Moreover, by using a completeness theorem for inductively generated formal topologies, we prove that such characterizations are classically equivalent to the standard one.  ...  Definition 2.9 (Inductive Bar for a Formal Topology). Let A be an inductively generated formal topology and U be a subset of A.  ... 
doi:10.1016/j.apal.2006.10.003 fatcat:smt657jbpbgt3a3ajwt2cdwizq

Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita [chapter]

Andrea Asperti, Maria Emilia Maietti, Claudio Sacerdoti Coen, Giovanni Sambin, Silvio Valentini
2011 Lecture Notes in Computer Science  
The project entitled "Formalization of Formal Topology by means of the interactive theorem prover Matita" is an official bilateral project between the Universities of Padova and Bologna, funded by the  ...  assistant based on the Calculus of Inductive Constructions as its logical foundation.  ...  The code of the formalization developed in the project can be found on-line at the address http://matita.cs.unibo.it/library.shtml, under the section "The Basic Picture".  ... 
doi:10.1007/978-3-642-22673-1_20 fatcat:6ofb3wfh7rgevcnwhb2wl7ax4e

Regular universes and formal spaces

Erik Palmgren
2006 Annals of Pure and Applied Logic  
We present an alternative solution to the problem of inductive generation of covers in formal topology by using a restricted form of type universes.  ...  The technique of regular universes is also used to construct canonical positivity predicates for inductively generated covers.  ...  I thank the referee for suggesting the simplified existence proof of inductively defined covers (Theorem 5.1). The author is supported by a grant from the Swedish Research Council (VR).  ... 
doi:10.1016/j.apal.2005.05.025 fatcat:ilrhijevszeavmsh6kjlawerau

Page 9557 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
The authors also give an example of a formal topology that cannot be inductively generated in this way.  ...  If we have such a transitive closure the corresponding formal topology is said to be inductively generated.  ... 
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