key: cord-0046147-o8kivylv
authors: Petrakis, Iosif
title: Functions of Baire Class One over a Bishop Topology
date: 2020-06-24
journal: Beyond the Horizon of Computability
DOI: 10.1007/978-3-030-51466-2_19
sha: a7bb230016a3fb1ba179af2edc674a262ee1b95a
doc_id: 46147
cord_uid: o8kivylv

If [Formula: see text] is a topology of open sets on a set X, a real-valued function on X is of Baire class one over [Formula: see text], if it is the pointwise limit of a sequence of functions in the corresponding ring of continuous functions C(X). If F is a Bishop topology of functions on X, a constructive and function-theoretic alternative to [Formula: see text] introduced by Bishop, we define a real-valued function on X to be of Baire class one over F, if it is the pointwise limit of a sequence of functions in F. We show that the set [Formula: see text] of functions of Baire class one over a given Bishop topology F on a set X is a Bishop topology on X. Consequently, notions and results from the general theory of Bishop spaces are naturally translated to the study of Baire class one-functions. We work within Bishop’s informal system of constructive mathematics [Formula: see text], that is [Formula: see text] extended with inductive definitions with rules of countably many premises.

If T is a topology of open sets on a set X, a function f : X → R is of Baire class one over T , if it is the pointwise limit of a sequence of functions in the corresponding ring of continuous functions C(X). Such functions, which may no longer be in C(X), were introduced by Baire in [2] , suggesting the use of functions, instead of sets, to tackle problems of real analysis. If B 0 (X) = C(X), and if B 1 (X) is the set of all Baire class one-functions, one defines for every ordinal α ≤ Ω, where Ω is the first uncountable ordinal Ω, the set

where, if F(X) is the set of real-valued functions on X, Φ ⊆ F(X), and f n p −→ f denotes that f is the pointwise limit of (f n ) ∞ n=1 , we set

The theory of Baire class-functions is a function-theoretic version of the theory of Baire sets i.e., of sets the characteristic function of which is in some Baire class 1 .

Generalisations of Baire class functions between metrizable spaces are central objects of study in descriptive set theory (see e.g., [13, 14] ), with Baire class one-functions having applications to the theory of Banach spaces (see e.g., [9] ). The theory of Bishop spaces (TBS) is a function-theoretic approach to constructive topology within Bishop's informal system of constructive mathematics BISH. The fundamental notion of a function space, here called a Bishop space, was only introduced by Bishop in [3] , p. 71. The subject was revived much later by Bridges in [5] , where the notion of a Bishop morphism was also defined, and by Ishihara in [11] . In [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] we try to develop TBS.

A Bishop topology of functions F on a set X is a set of real-valued functions defined on X that satisfies the main properties of the set of all Bishop continuous functions from R to R. A function φ : R → R is called (Bishop) continuous, if it is uniformly continuous on every bounded subset B of R i.e., if for every bounded subset 2 B of R and for every ε > 0 there exists ω φ,B (ε) > 0 such that

where the function ω φ,B : R + → R + , ε → ω φ,B (ε), is called a modulus of continuity for φ on B. Their set is denoted by Bic(R), and two functions φ 1 , φ 2 ∈ Bic(R) are equal, if φ 1 (a) = φ 2 (a), for every a ∈ R. The restriction of this notion of continuity to a compact interval [a, b] of R is equivalent to uniform continuity. By using this stronger notion of continuity, rather than the standard pointwise continuity, Bishop managed to avoid the use of fan theorem in the proof of the uniform continuity theorem and to remain "neutral" with respect to classical mathematics (CLASS), intuitionistic mathematics (INT), and intuitionistic computable mathematics (RUSS).

Extending our work [22] , where the Baire sets over a Bishop topology F are studied, here we give an introduction to the constructive theory of Baire class one-functions over a Bishop topology. In analogy to the classical concept, if F is a Bishop topology on a set X, we define a function f : X → R to be of Baire class one over F , if it is the pointwise limit of a sequence of functions in F . Our constructive translation of the fundamentals of the classical theory of Baire class one-functions (see e.g., [10] ) within TBS is summarized by Theorem 1, according to which the set B 1 (F ) of Baire class one-functions over F is a Bishop topology on X that includes F . As we explain in Sect. 5, and based on the examples of Baire class one-functions included in Sect. 4, this result offers a way to study constructively classically discontinuous functions.

We work within BISH * , that is BISH extended with inductive definitions with rules of countably many premises. A formal system for BISH * is Myhill's system CST * , developed in [17] , or CZF with dependence choice 3 (see [6] , p. 12), and some very weak form of Aczel's regular extension axiom (see [1] ).

In this section we include all definitions and facts necessary to the rest of the paper. All proofs not given given here are found in [18] .

If a, b ∈ R, let a∨b := max{a, b} and a∧b := min{a, b}. Hence, |a| = a∨(−a).

A set X is inhabited, if it has an element. We denote by a, or simply by a, the constant function on X with value a ∈ R, and by Const(X) their set.

such that the following conditions hold:

We call F a Bishop topology on X.

It is easy to show that if F is a topology on X, then F = Mor(F, R) i.e., an element of F is a real-valued "continuous" function on X. A Bishop topology F on X is an algebra and a lattice, where f ∨ g and f ∧ g are defined pointwise, and Const(X) ⊆ F ⊆ F(X). If F * (X) denotes the bounded elements of F(X), then F * := F ∩ F * (X) is a Bishop topology on X. If x = X y is the given equality on X, a Bishop topology F on X separates the points of X, or F is completely regular (see [19] for their importance in the theory of Bishop spaces), if

In Proposition 5.1.3. of [18] it is shown that F separates the points of X if and only if the induced by F apartness relation on X

We use the last result in the proof of Proposition 1(iv). An apartness relation on X is a positively defined inequality on X. E.g., if a, b ∈ R, then a = R b :⇔ |a − b| > 0. In Proposition 5.1.2. of [18] we show that a = R b ⇔ a = Bic(R) b.

Turning the definitional clauses (BS 1 ) − (BS 4 ) into inductive rules, the least topology F 0 generated by a set F 0 ⊆ F(X), called a subbase of F 0 , is defined by the following inductive rules:

where the last rule is reduced to the following rule with countably many premisses 

We denote their set by B 1 (F ).

Hence, for every n ≥ m 0 (ε) :

Note that (R, ∨, ∧) is not a distributive lattice, since not even (Q, ∨, ∧) is one. For the properties of a ∧ b and a ∨ c used in the next proof see [7] , p. 52.

Proof. (i) Since b ≤ b∨c, we get a ≤ b∨c. Since also c ≤ b∨c, we get a∨c ≤ b∨c. Since a ∧ c ≤ a ≤ b, and since also a ∧ c ≤ c, we get a ∧ c ≤ 

The proofs of the following two lemmas for B 1 (X) (see [8] ) are constructive.

Proof. Since g k is bounded by M k , for every k ≥ 1, by Lemma 4 there is

Proof. Using dependent choice there is a subsequence (g n k ) ∞ k=1 of (g n ) ∞ n=1 with U X; g, g n k , 1 2 k , for every k ≥ 1. Let h k := g n k+1 − g n k ∈ B 1 (F ). If x ∈ X, then

we get g = h + g n1 ∈ B 1 (F ), as B 1 (F ) is trivially closed under addition.

Clearly, F ⊆ B 1 (F ), and hence Const(X) ⊆ B 1 (F ). Moreover, B 1 (F ) is closed under addition. By Lemma 2 B 1 (F ) is closed under composition with elements of Bic(R), and by Lemma 6 B 1 (F ) is closed under uniform limits.

By Theorem 1, if g 1 , g 2 ∈ B 1 (F ), then g 1 ∨ g 2 , g 1 ∧ g 2 , g 1 · g 2 , and |g 1 | are in B 1 (F ). These facts also follow trivially by the definition of B 1 (F ). The importance of Theorem 1 though, is revealed by the use of the general theory of Bishop spaces in the proof of non-trivial properties of B 1 (F ) that, consequently, depend only on the Bishop space-structure of B 1 (F ). Proof. These facts follow from the corresponding facts on general Bishop spaces. See [18] , p. 41, for (i), Theorem 5.4.8. in [18] for (ii), [26] for a proof of (iii), Proposition 5.3.3.(ii) in [18] for (iv), Theorem 5.4.9. in [18] for (v), and the Urysohn extension theorem for general Bishop spaces in [20] for (vi).

Corollary 1, except from case (iii), are classically shown in [8] specifically for B 1 (X). Notice that in [20] we avoid quantification over the powerset of Y in the formulation of the Urysohn extension theorem, formulating it predicatively. Proof. (i) Since |g(x) − g(y)| > 0, let the well-defined function

The converse implication is shown similarly. (iv) It follows from (iii) and the result mentioned in Sect. 2 that a Bishop topology separates the points if and only if its induced apartness is tight.

(v) It follows from the general fact that F separates the points if and only if F * separates them (see Proposition 5.7.2. in [18] ).

First we find an unbounded Baire class one-function over some Bishop topology. If n ≥ 1, let f n : {0} ∪ (0, 1] → R defined by

Clearly, f n ≤ n, for every n ≥ 1. If 0 < x < 1 n , then 0 < n 2 x < n and n < 1

If Clearly, g is unbounded on its domain. We show that f n

We fix some ε > 0, and we find n 0 ≥ 1 such that 1 n0 < x. Hence, if n ≥ n 0 , then 1 n < x too. Since then n < n 2 x and 1

x < n, we have that

A pseudo-compact Bishop topology is a topology all the elements of which are bounded functions. Since boundedness is a "liftable" property from F 0 to F i.e., if every f 0 ∈ F 0 is bounded, then every f ∈ F 0 is bounded (see Proposition 3.4.4 in [18] , p. 46), the topology F 0 of the previous example is pseudo-compact, and hence the above construction is also an example of an unbounded Baire-class one function over a pseudo-compact Bishop topology! It is immediate to show that B 1 (F(X)) = F(X) and B 1 (Const(X)) = Const(X). Next we find a Baire class one-function over some F that is not in 

In this paper we introduced the notion of a function of Baire class one over a Bishop topology F , translating a fundamental notion of classical real analysis and topology into the constructive topology of Bishop spaces. Our central result, that the set B 1 (F ) of Baire class one-functions over F is a Bishop topology that includes F , is used to apply concepts and results from the general theory of Bishop spaces to the theory of functions of Baire class one over a Bishop topology. These first applications suggest that the structure of Bishop space, treated classically, would also be useful to the classical study of function spaces like B 1 (X). For constructive topology, the fact that B 1 (F ) is a Bishop topology provides a second way, within the theory of Bishop spaces, to treat classically discontinuous, real-valued functions as "continuous" i.e., as Bishop morphisms. The first way is to consider such discontinuous functions as elements of a subbase F 0 . Since by definition F 0 ⊆ F 0 , the elements of F 0 are Bishop morphisms from the resulting least Bishop space F to the Bishop space R of reals. In [27] , and based on a notion of convergence of test functions introduced by Ishihara, we follow this way to make the Dirac delta function δ and the Heaviside step function H "continuous". We consider a certain set D 0 (R) of linear maps on the test functions on R, where δ, H ∈ D 0 (R), and the Bishop topology D 0 (R) is used to define the set of distributions on R. The second way, is to start from a Bishop topology F and find elements of B 1 (F ) i.e., Bishop morphisms from F 1 to R, that are pointwise discontinuous, as the functions g and h in the last two example before Definition 4. This second way is sort of a constructive analogue to the classical result that the points of pointwise continuity of some f ∈ B 1 (R) is dense in R, hence f is almost everywhere continuous.

There are numerous interesting questions stemming from this introductory work. Can we prove constructively that the characteristic function of a (complemented) Baire set B = B 1 , B 0 over a Bishop topology F (see [22] ) is a Baire class-one function over the relative topology F |B 1 ∪B 0 ? Can we show constructively other classical characterisations of B 1 (X), like for example through F σ -sets? What is the exact relation between B 1 (F ) |A and B 1 (F |A ), or between B 1 (F × G) and the product Bishop topology (see [18] , Sect. 4.1 for its definition) B 1 (F ) × B 1 (G)? How far can we go constructively with the study of Baire class two-functions?

A base of a Bishop topology F is a subset B of F such that every f ∈ F is the uniform limit of a sequence in B. If B is a base of F , it follows easily that i.e., Lim p (B) is a base of B 1 (F ). If F 0 ⊆ F is a subbase of F i.e., F = F 0 , we have that Lim p (F 0 ) ⊆ Lim p ( F 0 ) = B 1 (F ), hence Lim p (F 0 ) ⊆ B 1 (F ). When does the inverse inclusion also hold?

We hope to address some of these questions in a future work.

Constructive Set Theory, Book Draft

Sur les fonctions des variable réelles

Foundations of Constructive Analysis

Constructive Analysis

Reflections on function spaces

Varieties of Constructive Mathematics

Techniques of Constructive Analysis

On rings of Baire one functions

Certain subsclasses of Baire-1 functions with Banach space applications

Baire one functions

Relating Bishop's function spaces to neighborhood spaces

Space of Baire functions. I, Annales de l'institut Fourier

A classification of Baire class 1 functions

Classical Descriptive Set Theory

Continuity and Baire functions

Constructive set theory

Constructive Topology of Bishop Spaces

Completely regular Bishop spaces

The Urysohn extension theorem for Bishop spaces

A constructive function-theoretic approach to topological compactness

Borel and Baire sets in Bishop spaces

Constructive uniformities of pseudometrics and Bishop topologies

Direct spectra of Bishop spaces and their limits

Embeddings of Bishop spaces

A constructive theory of C * (X)

Towards a constructive approach to the theory of distributions

Since h ∈ Mor(F, G), we have that g n •h ∈ F , for every n ≥ 1, and hence g•h ∈ B 1 (F ).By the principle of countable choice (see [6] , p. 12) there is (g n ) ∞ n=1 ⊆ G such that f n = g n • h, for every n ≥ 1. Let g * : Y → R, defined bySince f n (σ(y)) n −→ f * (σ(y)) := g * (y), we conclude that g n (y) n −→ g * (y).Since Y is dense in [0, 1], g is not in Bic([0, 1]) |Y ; if it was, by Proposition 4.7.15. in [18] we get g = h |Y with h ∈ C u ([0, 1]), which is impossible.A similar example is the following. Let Z := (−∞, 1) ∪ {1} ∪ (1, +∞) be equipped with the relative topology Bic(R) |Z . If n ≥ 1, let φ n = nid R + (1 − n) ∈ Bic(R) and θ n := −nid R + (1 + n) ∈ Bic(R). If ψ n := (φ n ∨ 0) ∧ (θ n ∨ 0) ∈ Bic(R),Let ψ * n be the restriction of ψ n to Z, for every n ≥ 1. Clearly, ψ *Since Z is dense in R (see Lemma 2.2.8. of [18] ), and arguing as in the previous example, h cannot be in the specified Bishop topology on Z.As in the classical case, all derivatives of differentiable functions in F(R) are Baire class one-functions over Bic(R). We reformulate the definition in [4] , p. 44, as follows. on [a, b] , and δ f, [a,b] : R + → R + . We say that f is differentiable on [a, b] with derivative f and modulus of differentiability δ f, [a,b] , in symbols Dif(f, f , δ f, [a,b] If φ, φ ∈ Bic(R), we say that φ is differentiable with derivative φ , in symbols Dif(φ, φ ), if for every n ≥ 1 we have that Dif(φ |[−n,n] , φ |[−n,n] , δ φ |[−n,n] ,[−n,n] ).