Completely Regular Bishop Spaces

Petrakis, Iosif

doi:10.1007/978-3-319-20028-6_31

Cited by 10 publications

(10 citation statements)

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“…As in the case of uniform spaces one can show that any Bishop topology F on some X is isomorphic as an algebra and a lattice to a separating topology ρF on ρX . If we define the equivalence relation 11), and if τ = π : X → X/≈, where x → [x] ≈ , then if ρX = X/≈ is endowed with the quotient Bishop topology ρF = {ρf | f ∈ F} = G π , where (ρf )([x] ≈ ) = f (x), for every [x] ≈ ∈ ρX , the following theorem is proved (see also [32]). Proof By definition ρD(F) = (ρX, ρD(F)) and D(ρF) = (ρX, D(ρF)), where since…”

Section: Petrakismentioning

confidence: 99%

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Constructive uniformities of pseudometrics and Bishop topologies

Petrakis

2019

JLA

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We develop the first steps of a constructive theory of uniformities given by pseudometrics and study its relation to the constructive theory of Bishop topologies. Both these concepts are constructive, function-theoretic alternatives to the notion of a topology of open sets. After motivating the constructive study of uniformities of pseudometrics we present their basic theory and we prove a Stone-Čech theorem for them. We introduce the f-uniform spaces and we prove a Tychonoff embedding theorem for them. We study the uniformity of pseudometrics generated by some Bishop topology and the pseudo-compact Bishop topology generated by some uniformity of pseudometrics. Defining the large uniformity on reals we prove a "large" version of the Tychonoff embedding theorem for f-uniform spaces and we show that the notion of morphism between uniform spaces captures Bishop continuity. We work within BISH * , Bishop's informal system of constructive mathematics BISH extended with inductive definitions with rules of countably many premisses.

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Section: Petrakismentioning

confidence: 99%

“…Defining the notion of topological embedding of a Bishop spaces into another, and the notion of a Euclidean Bishop space R I in the obvious way, the same embedding e X : X → R F together with the corresponding -lifting of openness for Bishop morphisms show the Tychonoff embedding theorem for Bishop spaces (see [32]).…”

Section: Petrakismentioning

confidence: 99%

Constructive uniformities of pseudometrics and Bishop topologies

Petrakis

2019

JLA

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“…No negation of some sort is used in the definition of the inequality x ‰ pX,F q x 1 , in the proof of its extensionality, while the last implication above can be seen as completely positive formulation of its tightness. If F is a Bishop topology of functions (see [26,35,36,40,42]), then x ‰ pX,F q x 1 is the canonical inequality of a Bishop space. The inequality a ‰ R b is equivalent to such an inequality induced by functions, as a ‰ R b ô a ‰ pR,BicpRqq b, where BicpRq is the topology of Bishop-continuous functions of type R Ñ R (see [26], Proposition 5.1.2.).…”

Section: Definition 23 (Strong Negation In Bst)mentioning

confidence: 99%

“…Categorical approaches to Bishop sets are found e.g., in the work of Palmgren [21] and Coquand [10]. For all notions and results of BST that we use without explanation or proof we refer to [33,36,39,41]. For all notions and facts from constructive analysis that we use without explanation or proof, we refer to [2,6,7].…”

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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“…The concept of a family of sets indexed by a (discrete) set was asked to be defined in Bishop (1967, Exercise 2, p. 72), and a definition, attributed to Richman, was given in Bishop and Bridges (1985, Exercise 2, p. 78). An elaborate study though, of this concept within BISH, was missing, despite its central character in the measure theory of Bishop (1967), its extensive use in the theory of Bishop spaces (Petrakis 2015a(Petrakis ,b, 2016a(Petrakis ,b, 2019a(Petrakis ,b, 2020a(Petrakis ,b, 2021(Petrakis , to appear, 2022a and in abstract constructive algebra (Mines et al 1988). Actually, in Mines et al (1988) Richman introduced the more general notion of a family of objects of a category indexed by some set, but the categorical component in the resulting mixture of Bishop's set theory and category theory was not explained in constructive terms.…”

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confidence: 99%

Proof-relevance in Bishop-style constructive mathematics

Petrakis

2022

Math. Struct. Comp. Sci.

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Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(\mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $\phi$ of BISH a set ${\texttt{Prf}}(\phi)$ of “proofs” or witnesses of $\phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(\mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.

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Completely Regular Bishop Spaces

Cited by 10 publications

References 7 publications

Constructive uniformities of pseudometrics and Bishop topologies

Constructive uniformities of pseudometrics and Bishop topologies

Algebras of Complemented Subsets

Proof-relevance in Bishop-style constructive mathematics

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