Constructive uniformities of pseudometrics and Bishop topologies

Petrakis, Iosif

doi:10.4115/jla.2019.11.ft2

Cited by 7 publications

(5 citation statements)

References 16 publications

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“…Next we present the normal Chu representation of the category of Bishop spaces. The notion of Bishop space is a constructive, function-theoretic alternative to the set-based notion of topological space, which was introduced by Bishop in [6], revived by Bridges in [9] and elaborated by the author in [20]- [22] and [25]- [29]. For the sake of completeness we give next all necessary definitions related to the proof of a strict Chu representation of the category of Bishop spaces.…”

Section: Normal Chu Representationsmentioning

confidence: 99%

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Chu representations of categories related to constructive mathematics

Petrakis¹

2021

Preprint

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If C is a closed symmetric monoidal category, the Chu category ChupC, γq over C and an object γ of it was defined by Chu in [5], as a ˚-autonomous category generated from C. In [6] Bishop introduced the category P pXq of complemented subsets of a set X, in order to overcome the problems generated by the use of negation in constructive measure theory. In [34] Shulman mentions that Bishop's complemented subsets correspond roughly to the Chu construction. In this paper we explain this correspondence by showing that there is a Chu representation (a full embedding) of P pXq into ChupSet, X ˆXq. A Chu representation of the category of Bishop spaces into ChupSet, Rq is shown, as the constructive analogue to the standard Chu representation of the category of topological spaces into ChupSet, 2q. In order to represent the category of predicates (with objects pairs pX, Aq, where A is a subset of X, and the category of complemented predicates (with objects pairs pX, Aq, where A is a complemented subset of X, we generalise the Chu construction by defining the Chu category over a cartesian closed category C and an endofunctor on C. Finally, we introduce the antiparallel Grothendieck construction over a product category and a contravariant Set-valued functor on it of which the Chu construction is a special case, in case C is a locally small, cartesian closed category.

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Section: Normal Chu Representationsmentioning

confidence: 99%

“…This representation of Bis is the constructive analogue of the aforementioned Chu representation of Top. The notion of a Bishop space is Bishop's constructive, function-theoretic alternative to the classical, set-based notion of a topological space (see [20]- [22] and [25]- [29]).…”

Section: Introductionmentioning

confidence: 99%

Chu representations of categories related to constructive mathematics

Petrakis¹

2021

Preprint

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“…In a series of papers (see [26]- [31], [35,36], and [41,42]) we have elaborated the theory of Bishop spaces. A motivation for the study of Bishop spaces, an approach within the second way, was the realisation that the classical duality between open and closed subsets cannot be preserved constructively.…”

Section: Introductionmentioning

confidence: 99%

Algebras of Complemented Subsets

Petrakis¹,

Misselbeck-Wessel²

2022

Revolutions and Revelations in Computability

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We introduce cs-topologies, or topologies of complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simulataneously an open set, the first-component of an open complemented subset, and its complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a mudulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of an open subset, given a base for the cs-topology, makes possible to define the notion of uniform continuity of functions between csb-spaces, that is cs-spaces with a given base. In this way, the notions of pointwise and uniform continuity of functions between metric spaces are directly generalised to the notions of pointwise and uniform continuity between csb-topological spaces. Constructions and facts from the classical theory of topologies of open subsets are translated to our constructive theory of topologies of open complemented subsets.

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“…BISH * is the system corresponding to[3] and to the constructive topology of Bishop spaces (see[10][11][12] and[17]), while BISH is the system corresponding to[4].…”

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confidence: 99%

Strong negation in the theory of computable functionals TCF

Nils¹,

Petrakis²

2022

Preprint

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We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and Gödel's system T, by defining simultaneously the strong negation A N of a formula A and the strong negation P N of a predicate P in TCF. As a special case of the latter, we get the strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of the double negation elimination for strong negation in TCF, and we study the so-called tight formulas of TCF i.e., formulas implied from the weak negation of their strong negation. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics.

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Constructive uniformities of pseudometrics and Bishop topologies

Cited by 7 publications

References 16 publications

Chu representations of categories related to constructive mathematics

Chu representations of categories related to constructive mathematics

Algebras of Complemented Subsets

Strong negation in the theory of computable functionals TCF

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