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Inductively generated formal topologies

2003
*
Annals of Pure and Applied Logic
*

Here we give a systematic exposition of one of the main tools in

doi:10.1016/s0168-0072(03)00052-6
fatcat:4x4uwyxq6zgzbljubv4womnabq
*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*. ...##
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Exponentiation of Scott Formal Topologies

2004
*
Electronical Notes in Theoretical Computer Science
*

We prove that Scott

doi:10.1016/j.entcs.2004.08.005
fatcat:ptrc7f7ifrdl5jpyfjuh4qtmvm
*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. ...##
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General Recursion and Formal Topology

2010
*
Electronic Proceedings in Theoretical Computer Science
*

In this work we propose a new approach to this problem based on the use of

doi:10.4204/eptcs.43.5
fatcat:vjrvnxqycbet5pp4oregqfrhb4
*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*). ...##
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Spatiality for formal topologies

2007
*
Mathematical Structures in Computer Science
*

We define what it means for a

doi:10.1017/s0960129506005810
fatcat:3ensmyux4ngzdpf6kttzyuo6gi
*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 ...##
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Why topology in the minimalist foundation must be pointfree

2013
*
Logic and Logical Philosophy
*

We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that

doi:10.12775/llp.2013.010
fatcat:djpf6wtw4fd2lc3wpobxmbmsue
*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. ...##
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Equivalents of the finitary non-deterministic inductive definitions
[article]

2019
*
arXiv
*
pre-print

We present statements equivalent to some fragments of the principle of non-deterministic

arXiv:1903.05852v3
fatcat:jkjsoadrqjctnmzlbz65cbboru
*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. ...##
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Sublocales in formal topology

2007
*
Journal of Symbolic Logic (JSL)
*

They

doi:10.2178/jsl/1185803619
fatcat:jpfsgkf43rcaxad76p4djtwvii
*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). ...##
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A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
[article]

2021
*
arXiv
*
pre-print

We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mtt, extended with

arXiv:1905.11966v5
fatcat:2m2ythcamnh3doezr4tiszdoj4
*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. ...##
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Geometric theories of patch and Lawson topologies
[article]

2017
*
arXiv
*
pre-print

We present the patch

arXiv:1709.06403v1
fatcat:q2zt3ubsxjgwbo4wx3jw7g3vfi
*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. ...##
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Tychonoff's theorem in the framework of formal topologies

1997
*
Journal of Symbolic Logic (JSL)
*

The most important aspect of our proof is that it is based on an

doi:10.2307/2275645
fatcat:cdjkwvklwvhbxfuejpmds6u7ti
*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*. ...##
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On the formal points of the formal topology of the binary tree

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*. ...

##
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Constructive characterizations of bar subsets

2007
*
Annals of Pure and Applied Logic
*

, for an

doi:10.1016/j.apal.2006.10.003
fatcat:smt657jbpbgt3a3ajwt2cdwizq
*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. ...##
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Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita
[chapter]

2011
*
Lecture Notes in Computer Science
*

The project entitled "

doi:10.1007/978-3-642-22673-1_20
fatcat:6ofb3wfh7rgevcnwhb2wl7ax4e
*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". ...##
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Regular universes and formal spaces

2006
*
Annals of Pure and Applied Logic
*

We present an alternative solution to the problem of

doi:10.1016/j.apal.2005.05.025
fatcat:ilrhijevszeavmsh6kjlawerau
*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). ...##
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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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