Abstract
If 
is a topology of open sets on a set X, a real-valued function on X is of Baire class one over 
, 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 
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 
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 
, that is 
extended with inductive definitions with rules of countably many premises.
Introduction
If 
is a topology of open sets on a set X, a function 
is of Baire class one over
, 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 
, and if 
is the set of all Baire class one-functions, one defines for every ordinal 
, where 
is the first uncountable ordinal 
, the set
 |
where, if 
is the set of real-valued functions on X, 
, and 
denotes that f is the pointwise limit of 
, 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 class1. 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 
. 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 [18–27] 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 
to 
. A function 
is called (Bishop) continuous, if it is uniformly continuous on every bounded subset B of 
i.e., if for every bounded subset2
B of 
and for every 
there exists 
such that
 |
where the function 
, is called a modulus of continuity for 
on B. Their set is denoted by 
, and two functions 
are equal, if 
, for every 
. The restriction of this notion of continuity to a compact interval [a, b] of 
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 
, intuitionistic mathematics 
, and intuitionistic computable mathematics 
.
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 
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 
is summarized by Theorem 1, according to which the set 
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 
, that is 
extended with inductive definitions with rules of countably many premises. A formal system for 
is Myhill’s system 
, developed in [17], or 
with dependence choice3 (see [6], p. 12), and some very weak form of Aczel’s regular extension axiom (see [1]).
Fundamentals of Bishop Spaces
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 
, let 
and 
. Hence, 
. If 
, let 
, where for all definitions related to 
see [4], chapter 2. If 
and 
, the pointwise convergence
and the uniform convergence
on 
are defined, respectively, by
 |
 |
 |
A set X is inhabited, if it has an element. We denote by 
, or simply by a, the constant function on X with value 
, and by 
their set.
Definition 1
A Bishop space is a pair 
, where X is an inhabited set and F is an extensional subset of 
i.e., 
, such that the following conditions hold:
.
If 
, then 
.
If 
and 
, then 
.
If 
and 
such that 
on X, then 
.
We call F a Bishop topology on X. If 
is a Bishop space, a Bishop morphism from 
to 
is a function 
such that 
. We denote by 
the set of Bishop morphisms from 
to 
. If 
, we say that h is open, if 
.
A Bishop morphism 
is a “continuous” function from 
to 
. If 
is a bijection, then 
i.e., h is a Bishop isomorphism, if and only if h is open. Let 
be the Bishop space of reals
. It is easy to show that if F is a topology on X, then 
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 
and 
are defined pointwise, and 
. If 
denotes the bounded elements of 
, then 
is a Bishop topology on X. If 
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
 |
is tight i.e., 
. 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 
, then 
. In Proposition 5.1.2. of [18] we show that 
.
Definition 2
Turning the definitional clauses 
into inductive rules, the least topology 
generated by a set 
, called a subbase of 
, is defined by the following inductive rules:
 |
 |
where the last rule is reduced to the following rule with countably many premisses
 |
The above rules induce the corresponding induction principle 
on 
. If 
, the relative topology 
on A has the set 
as a subbase. Unless otherwise stated, from now on, X, Y are inhabited sets, and F, G are Bishop topologies on X and Y, respectively.
The Bishop Topology of Baire Class One-Functions
Definition 3
A function 
is called of Baire class one over F, or simply of Baire class one when F is clear from the context, if there is a sequence 
such that 
on X. We denote their set by 
.
Lemma 1
Let 
and 
with 
. If 
, there is 
, such that 
Proof
Let 
such that if 
, then 
, hence 
. If 
then 
, for every 
, and 
.
Lemma 2
If 
and 
, then 
.
Proof
Let 
such that 
. If 
and 
are fixed, there is 
such that for every 
we have that 
. Since 
, for every 
, and 
, by the uniform continuity of 
we have that
 |
Hence, for every 
we have that 
. Since 
is arbitrary, we get 
. Since 
is arbitrary, we get 
.
Note that 
is not a distributive lattice, since not even 
is one. For the properties of 
and 
used in the next proof see [7], p. 52.
Lemma 3
If 
and 
, the following hold.
(i) If 
, then 
and 
.
(ii) 
.
Proof
(i) Since 
, we get 
. Since also 
, we get 
. Since 
, and since also 
, we get 
.
(ii) By Corollary 2.17 in [4], p. 26, 
or 
. If 
, then 
and, since 
we also get 
, hence 
. If 
, and since also 
, we get 
, hence 
. Moreover, 
and the required equality holds.
Lemma 4
If 
is bounded by some 
, there is a sequence 
such that 
and 
is bounded by M, for every 
.
Proof
If 
such that 
, let 
, for every 
. We show that 
. Let 
, 
and 
, such that for every 
we have that 
, or equivalently 
By Lemma 3(i), and since 
, we get
 |
Hence
 |
i.e., 
. Since 
, by Lemma 3(i) we get 
. Since 
, we get 
hence by Lemma 3(ii) we get
 |
i.e., 
, which implies 
. Since we have already shown that 
, by the definition of 
we conclude that 
, for every 
. Of course, 
, for every 
.
The proofs of the following two lemmas for 
(see [8]) are constructive.
Lemma 5
Let 
and 
with 
, for every 
, and 
. If 
, for every 
, then 
Proof
Since 
is bounded by 
, for every 
, by Lemma 4 there is 
with 
and 
. If 
, let 
Let 
. Since 
, there is 
with 
. If 
, there is 
, such that for every 
we have that 
, for every 
. If 
, then
 |
Lemma 6
If 
and 
with 
, then 
.
Proof
Using dependent choice there is a subsequence 
of 
with 
, for every 
. Let 
If 
, then
 |
By Lemma 5 we have that 
. Since
 |
we get 
, as 
is trivially closed under addition.
Theorem 1
is a Bishop topology on X that includes F.
Proof
is an extensional subset of 
, since if 
and 
such that 
on X, then if 
, we also get 
on X. Clearly, 
, and hence 
. Moreover, 
is closed under addition. By Lemma 2
is closed under composition with elements of 
, and by Lemma 6
is closed under uniform limits.
By Theorem 1, if 
, then 
, and 
are in 
. These facts also follow trivially by the definition of 
. 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 
that, consequently, depend only on the Bishop space-structure of 
.
Corollary 1
(i) 
is a Bishop topology on X.
(ii) If 
such that 
, for some 
with 
, then 
.
(iii) If 
such that 
, then 
.
(iv) The collection 
of zero sets of 
, where 
, is closed under countable intersections.
(v) [Urysohn lemma for 
-zero sets] If 
, then there is 
with 
and 
if and only if there are 
, and 
. such that 
, 
, and 
.
(vi) [Urysohn extension theorem for 
Let 
such that 
, for every 
. If for every 
, whenever A, B are separated by some function in 
, then A, B are separated by some function in 
, then every 
is the restriction of some 
.
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 
. Notice that in [20] we avoid quantification over the powerset of Y in the formulation of the Urysohn extension theorem, formulating it predicatively.
Proposition 1
Let 
.
(i) If 
with 
, there is 
such that 
.
(ii) 
.
(iii) The apartness 
is tight if and only if the apartness 
is tight.
(iv) 
separates the points of X if and only if F separates them.
(v) 
separates the points of X if and only if 
separates them.
Proof
(i) Since 
, let the well-defined function
 |
is in 
, 
and 
. If 
with 
, then
 |
If 
, then 
with 
and 
, hence 
.
(ii) If 
, there is 
such that 
. By (i) we get 
. Conversely, if 
, there is 
with 
. Since f is also in 
, we get 
.
(iii) Let 
be tight. If 
, then by (ii) we get 
, hence 
. 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 
separates them (see Proposition 5.7.2. in [18]).
Proposition 2
Let 
and 
.
(i) If 
, then 
.
(ii) Let 
be a surjection with 
a modulus of surjectivity4 for h i.e., 
. If 
is open, then 
is open.
Proof
(i) We need to show that 
. If we fix 
, let 
such that 
. Then, we get 
. Since 
, we have that 
, for every 
, and hence 
.
(ii) By case (i) 
. By Definition 1, if 
, we prove 
. Let 
and 
with 
on X. By the principle of countable choice (see [6], p. 12) there is 
such that 
, for every 
. Let 
, defined by 
. First we show that 
. If 
, we show that
 |
Since 
we get
 |
i.e., 
for every 
. Since 
and 
, we get 
. Since 
is an extensional subset of 
and 
, we conclude that 
too. To prove 
, we show that 
. If 
, then
 |
Since 
, we conclude that 
.
Examples of Functions of Baire Class One over F
First we find an unbounded Baire class one-function over some Bishop topology. If 
, let 
defined by
 |
Clearly, 
, for every 
. If 
, then 
and 
, hence 
. If 
, then 
and 
, hence 
. Hence,
 |
If 
, we consider the Bishop topology 
on 
. Let the function 
, defined by
 |
Clearly, g is unbounded on its domain. We show that 
, hence 
. If 
, then 
. Let 
We fix some 
, and we find 
such that 
. Hence, if 
, then 
too. Since then 
and 
, 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 
to F i.e., if every 
is bounded, then every 
is bounded (see Proposition 3.4.4 in [18], p. 46), the topology 
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 
and 
. Next we find a Baire class one-function over some F that is not in F. Let 
be equipped with the relative Bishop topology 
, where 
is the Bishop topology of uniformly continuous functions on [0, 1], and 
, with 
being defined similarly to 
. Let 
, where 
, for every 
. By the definition of relative Bishop topology (see Definition 2) we have that 
, for every 
, and 
, where 
is given by
 |
Since Y is dense in [0, 1], g is not in 
; if it was, by Proposition 4.7.15. in [18] we get 
with 
, which is impossible.
A similar example is the following. Let 
be equipped with the relative topology 
. If 
, let 
and 
. If 
,
 |
Let 
be the restriction of 
to Z, for every 
. Clearly, 
, where
 |
Since Z is dense in 
(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 
are Baire class one-functions over 
. We reformulate the definition in [4], p. 44, as follows.
Definition 4
Let 
, 
(uniformly) continuous on [a, b], and 
. We say that f is differentiable on [a, b] with derivative 
and modulus of differentiability 
, in symbols 
, if
 |
If 
, we say that 
is differentiable with derivative 
, in symbols 
, if for every 
we have that 
.
Proposition 3
If 
such that 
, then 
.
Proof
If 
, let 
. We show that 
. Let 
and 
. Let 
with 
. Since 
and 
, there is 
such that for every 
we have that 
and 
, hence 
and by Definition 4 we have that
 |
Concluding Comments
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 
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 
.
For constructive topology, the fact that 
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 
. Since by definition 
, the elements of 
are Bishop morphisms from the resulting least Bishop space 
to the Bishop space 
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 
and the Heaviside step function H “continuous”. We consider a certain set 
of linear maps on the test functions on 
, where 
, and the Bishop topology 
is used to define the set of distributions on 
. The second way, is to start from a Bishop topology F and find elements of 
i.e., Bishop morphisms from 
to 
, 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 
is dense in 
, 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 
over a Bishop topology F (see [22]) is a Baire class-one function over the relative topology 
? Can we show constructively other classical characterisations of 
, like for example through 
-sets? What is the exact relation between 
and 
, or between 
and the product Bishop topology (see [18], Sect. 4.1 for its definition) 
? 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 
is the uniform limit of a sequence in B. If B is a base of F, it follows easily that
 |
hence for the uniform closure 
of B in 
we get
 |
i.e., 
is a base of 
. If 
is a subbase of F i.e., 
, we have that 
, hence 
. When does the inverse inclusion also hold?
We hope to address some of these questions in a future work.
Footnotes
For that see the Lebesgue-Hausdorff theorem in [15], p. 393, and Lorch’s comment in [16], p. 751, on the “coextension” of the two theories.
It suffices to say that 
is uniformly continuous on every interval 
, and the quantification over the powerset of 
is replaced by quantification over 
.
Here we use the principle of dependent choice in the proof of Lemma 6.
We use 
in order to avoid the general axiom of choice in the proof.
Contributor Information
Marcella Anselmo, Email: manselmo@unisa.it.
Gianluca Della Vedova, Email: gianluca.dellavedova@unimib.it.
Florin Manea, Email: flmanea@gmail.com.
Arno Pauly, Email: arno.m.pauly@gmail.com.
Iosif Petrakis, Email: petrakis@math.lmu.de.
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