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Filomat 37:19 (2023), 6453–6461
https://doi.org/10.2298/FIL2319453A
Published by Faculty of Sciences and Mathematics,
University of Niˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Remarks on the ring B1(X)
Mohammad Reza Ahmadi Zanda, Zahra Khosravia
aDepartment of Mathematical Sciences, Yazd University, Yazd, Iran
Abstract. Let Xbe a nonempty topological space, C(X)Fbe the set of all real-valued functions on Xwhich
are discontinuous at most on a finite set and B1(X) be the ring of all real-valued Baire one functions on X. We
show that any member of B1(X) is a zero divisor or a unit. We give an algebraic characterization of Xwhen
for every p∈X, there exists f∈B1(X) such that {p}=f−1(0) and we give some topological characterizations
of minimal ideals, essential ideals and socle of B1(X). Some relations between C(X)F,B1(X) and some
interesting function rings on Xare studied and investigated. We show that B1(X) is a regular ring if and
only if every countable intersection of cozero sets of continuous functions can be represented as a countable
union of zero sets of continuous functions.
1. Introduction
All our spaces are assumed to be T1unless otherwise stated and all rings will be assumed commutative
with identity and semiprime. The set of natural numbers, rational numbers and real numbers are denoted
by N,Qand R, respectively. The set of all functions from a topological space Xinto Ris denoted by F(X)
and the set of all f∈F(X) such that f−1(U) is an Fσ-set for all open set Uin Ris denoted by Fσ(X). F(X) is a
commutative ring with pointwise addition and multiplication and continuous members of F(X) which is a
subring of F(X) is denoted by C(X). Let f∈F(X). Then f−1(0) ={x∈X|f(x)=0}denoted by Z(f) is called
the zero set of fand a set is said to be a cozero set if it is the complement of a zero set of a function in F(X).
A subset Mof Xis called a Zeroσ-set or a Cozδ-set if M=Sn∈NFnor M=Tn∈NGn, respectively, where each
Fnis a zero set of a function in C(X) and each Gnis a cozero set of a function in C(X). The set of all f∈F(X)
such that f−1(U) is a Zeroσ-set for all open set Uin Ris denoted by F∗
σ(X). Clearly, F∗
σ(X)⊆Fσ(X). The set
of points at which f∈F(X) is continuous is denoted by C(f). The ring of all f∈F(X) such that X\C(f) is
a finite set is denoted by C(X)F[18]. The ring of all real-valued functions which are continuous on some
dense open subsets of X is denoted by T′(X) [1]. The characteristic function of the subset Sof X is denoted by
χS.
Recall that a commutative ring Ris called (von Neumann) regular if for each r∈R, there exists s∈R
such that r=r2s. Thus for any nonempty set Xthe commutative ring F(X) is regular. By a minimal ideal
of Rwe mean an ideal which is minimal in the poset of non-zero ideals of R. A non-zero ideal Iin Ris an
essential ideal if Iintersects every non-zero ideal of Rnon-trivially. The socle of Rdenoted by Soc(R) is the
sum of all minimal ideals of R, or the intersection of all essential ideals of R.
The set of all pointwise limit functions of sequences in C(X) is denoted by B1(X) and the element of B1(X)
2020 Mathematics Subject Classification. Primary 26A21; Secondary 54C40, 13C99.
Keywords. Baire one functions; P-space; Minimal ideal; Fixed ideal; B1(X);
Received: 30 September 2022; Revised: 09 February 2023; Accepted: 17 February 2023
Communicated by Ljubiˇ
sa D.R. Koˇ
cinac
Email addresses: mahmadi@yazd.ac.ir (Mohammad Reza Ahmadi Zand), zahra.khosravi@stu.yazd.ac.ir (Zahra Khosravi)
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6454
are called real-valued Baire one functions.B1(X) was introduced and investigated by Baire [6] and studied
extensively by many mathematiciants such as Hausdorff[20] and Lebesgue [27]. By generalizing definitions
and ideas from the book “ Rings of Continuous Functions” [19], Deb Ray and Mondal [10–12] studied B1(X)
as a subring of F(X). Our intention in the present paper is to pursue research on the ring B1(X), although
B1(X) has been studied in several aspects, see for example [7, 17, 30].
In section 2, some topological characterizations of minimal ideals, essential ideals and socle of B1(X), where
every point of Xis a zero set of a function in C(X) are given. For a topological space X, we show that any
member of B1(X) is a zero divisor or a unit. In section 3, some relations between subrings C(X), B1(X), C(X)F
and T′(X) of F(X) are studied. We give an algebraic characterization of a normal space Xwhen every subset
of Xis a Gδ-set. We give some examples of topological spaces in which the ring of Baire one functions is
regular. We give a topological characterization of B1(X) when it is a regular ring.
As usual the Stone- ˇ
Cech compactification of a T31
2-space X is denoted by βX. The reader is referred to [9],
[14] and [19] for terms and notations not described here.
1.1. Preliminaries
For our purpose we need the following results and definitions that will be used in this paper.
Theorem 1.1. Let X be an arbitrary topological space. Then B1(X)=F∗
σ(X).
Proof. See [28, Exercise 3.A.1].
Theorem 1.2. [31] Let X be a normal topological space. Then B1(X)=Fσ(X).
The collection of all zero sets of functions in B1(X) is denoted by Z(B1(X)) [10]. In the following proposition
we show that a subset Aof Xis a zero set of a function in B1(X) if and only if Acan be represented in the
form of a countable intersection of cozero sets of continuous functions.
Proposition 1.3. For a topological space X we have Z(B1(X)) ={A⊆X:A is a Cozδ-set}.
Proof. The inclusion Z(B1(X)) ⊆ {A⊆X:Ais a Cozδ-set}immediately follows from Theorem 1.1. The
inverse inclusion follows from [24, Proposition 2].
Definition 1.4. [11] A nonempty subset Fof Z(B1(X)) is said to be a ZB-filter if Fsatisfies the following conditions.
(I) ∅<F.
(II) If Z1,Z2∈ F , then Z1∩Z2∈ F .
(III) If Z ∈ F and Z′∈Z(B1(X)) which satisfies Z ⊆Z′, then Z′∈ F .
Theorem 1.5. [11] Let X be a topological space.
(I) Let I be a proper ideal in B1(X). Then ZB[I]={Z(f) : f∈I}is a ZB-filter on X.
(II) Let Fbe a ZB-filter on X. Then Z−1
B[F]={f∈B1(X) : Z(f)∈ F } is a proper ideal in B1(X).
Definition 1.6. [11] Let I be a proper ideal of B1(X). Then I is called fixed if ∩Z[I],∅, otherwise, it is called free.
2. Minimal ideals of B1(X)
In this section, we show that every member of B1(X) is a zero-divisor or a unit and we give some
topological and algebraic characterizations of minimal ideals, essential ideals and socle of B1(X).
Proposition 2.1. Let M be a subset of a topological space X. Then χM∈B1(X)if and only if M and X \M are
Zeroσ-sets.
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6455
Proof. By Theorem 1.1, f=χM∈B1(X) if and only if f∈F∗
σ(X). But f−1(U)={∅,X,M,X\M}for any open
subset Uof Rwhich completes the proof.
Clearly, every zero set of a function in C(X) is a Zeroσ-set and a Cozδ-set.
Corollary 2.2. Let K be a compact subset of a topological space X. Then for any Zeroσ-set A in X which is disjoint
from K there exists f ∈B1(X)such that f [K]={1}and f [A]={0}.
Proof. Let fn:X→[0,1] be continuous for any n∈N,A=S∞
n=1Z(fn) and K∩A=∅. Then every fnattains
its minimum bn>0 on Kand so F=T∞
n=1f−1
n([bn,1]) is a zero set of a function in C(X) such that K⊆F⊆X\A
[19]. Let f=χF. Then f[K]={1}and f[A]={0}and Proposition 2.1 implies that f∈B1(X).
Proposition 2.3. Let M be a nonempty proper subset of a topological space X which are a Zeroσ-set and a Cozδ-set.
Then B1(X)is isomorphic to the direct sum of two rings; moreover, there exists a nontrivial idempotent f in B1(X)
such that Z(f)=M.
Proof. Let N=X\M. Then ϕ:B1(X)→B1(M)⊕B1(N) defined by ϕ(1)=1|M+1|Nis a one-one
ring homomorphism. If h∈B1(M) and k∈B1(N), then 1=h∪k∈F(X) and if Uis open in R, then
1−1(U)=h−1(U)∪k−1(U) is a Zeroσ-set in X. Thus by Theorem1.1, 1∈B1(X) and ϕ(1)=h+k, and so ϕis
a ring isomorphism. By Proposition 2.1, χM∈B1(X) and so f=1−χM∈B1(X) is a nontrivial idempotent
such that M=Z(f).
Proposition 2.4. If p is a point in a topological space X, then χ{p}∈B1(X)if and only if {p}is a zero set of a function
in C(X).
Proof. Let χ{p}∈B1(X). Then by Proposition 2.1, {p}is a Zeroσ-set and so {p}is a zero set of a function in
C(X). The converse follows from Proposition 2.1.
Corollary 2.5. Let X be a completely regular space and {p}be a Gδ-set. Then χ{p}∈B1(X)
Proof. Since {p}is a Gδ-set in X,{p}is a zero set of a function in C(X). Thus by Proposition 2.4 we are
done.
Proposition 2.6. Let X be a topological space. Then any member of B1(X)is a zero-divisor or a unit.
Proof. Suppose that 0 ,f∈B1(X) is not a unit element. Then there exists p∈Z(f) by [2, Theorem 1.1] and
Z(f) is a Gδ-set by [10, Theorem 4.1]. Thus by [19, 3.11(b)], there is 1∈C(X) such that p∈Z(1)⊆Z(f). If
M=Z(1), then by Proposition 2.1, χM∈B1(X) since ∅,Mis a zero set of a function in C(X). Therefore
fχM=0 which completes the proof.
Let every singleton set of Xbe a zero set of a function in C(X). Some topological characterizations of
minimal ideals in B1(X), essential ideals in B1(X) and the socle of B1(X) are given in the following result.
Proposition 2.7. Suppose that X is a topological space in which every singleton set is a zero set of a function in C(X).
Then the following hold.
(i) An ideal I of B1(X)is minimal if and only if for some p ∈X, I is generating by χ{p}.
(ii) An ideal I of B1(X)is minimal if and only if |Z[I]|=2.
(iii) The socle of B1(X)consists of all functions which vanish everywhere except on a finite subset of X .
(iv) The socle of B1(X)is an essential ideal.
(v) The socle of B1(X)is a free ideal.
(vi) The socle of B1(X)is the intersection of all free ideals in B1(X), and of all free ideals in B∗
1(X).
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6456
Proof. If Iis an ideal in B1(X) and r=f(p) for some p∈X, then χ{p}∈B1(X) by Proposition 2.4 and
rχ{p}=fχ{p}∈I.(1)
Thus if r,0, then the ideal generated by χ{p}is contained in I.
(i) If Jis the ideal generated by χ{p}in B1(X), then by (1), J={rχ{p}:r∈R}which is a minimal ideal in B1(X).
Conversely, if Iis a minimal ideal in B1(X), then there exists f∈Iand p∈Xsuch that 0 ,r=f(p). From (1)
we infer that Iis the ideal generated by χ{p}.
(ii) Let |Z[I]|=2. Then there exists f∈Iand p∈Xsuch that 0 ,r=f(p), and so χ{p}=1
rχ{p}f∈Iby (1).
Thus from 0 ,1∈Iit follows that Z(1)=Z(χ{p})=X\ {p}and so Iis generated by χ{p}. Now, (i) implies that
Iis minimal. The converse is obvious by (i).
(iii) It follows from (i) since the socle of a commutative ring is the sum of its minimal ideals.
(iv) If 0 ,f∈B1(X), then r=f(p),0 for some p∈X. By (1), fχ{p},0 and so by (iii) fχ{p}is in the socle of
B1(X), i.e., the socle of B1(X) is essential.
(v) For any p∈X,χ{p}(p),0 and the ideal generated by χ{p}is minimal by (i). Thus by (iii), the socle of
B1(X) is free.
(vi) If Iis a free ideal in B1(X) or B∗
1(X), then for every p∈Xthere exists f∈Isuch that r=f(p),0. By (1),
χ{p}=1
rχ{p}f∈I. Thus, by (iii), the socle of B1(X) is contained in the intersection of all free ideals in B1(X)
and of all free ideals in B∗
1(X). By (v) the socle of B1(X) is free which completes the proof.
Theorem 2.8. Let X be an infinite topological space. If every singleton set of X is a cozero set of a Baire one function,
then the following statements hold.
(i) For any f ∈B1(X)which is not a unit in B1(X)there exists 1∈B1(X)such that 1,1and f =1rf, where r is
a positive real number.
(ii) Let I be a fixed ideal in B1(X)such that TZ[I]is a finite set. Then Ann(I) is a proper ideal generated by
1∈B1(X)such that 1r=1, where r is an arbitrary positive real number.
(iii) For any subset A of X there exists a subset S of B1(X)such that A =Sf∈S(X\Z(f)), in particular if A is
countable, then A is a cozero set of a function in B1(X).
Proof. (i) If 0 ,f∈B1(X) is not a unit element, then by the proof of Proposition 2.6, there exists a nonempty
proper subset Mof Z(f) such that χM∈B1(X). Hence, 1=1−χM,1 is a function with corresponding
properties.
(ii) Let S=TZ[I]={x1,· · · ,xn}be an n-element set, where n∈N. Then by our hypothesis, there exists
1i∈B1(X) such that X\Z(1i)={xi}for i=1,· · · n. Thus, 1=Pn
i=1|1
1i(xi)1i| ∈ B1(X) and 1r=1for any positive
real number rsince 1=χS. Thus for any f∈I, we have 1f=0, i.e., 1∈Ann(I) and so if the ideal generated
by 1is denoted by J, then J⊆Ann(I). If s<S, then there exists f∈Isuch that f(s),0 and so h∈Ann(I)
implies that h(s)f(s)=0, i.e., h(s)=0. Thus from X\S⊆Z(h) it follows that h=1h∈Jand so Ann(I)⊆J
which implies that Ann(I)=J.
(iii) For any subset Aof X,S={χ{x}|x∈A} ⊆ B1(X) as we have shown above. Clearly, A=Sf∈S(X\Z(f)).
By [10, Theorem 4.5] Z(B1(X)) is closed under countable intersection and so if Ais countable, then X\A∈
Z(B1(X)).
3. Some relations between B1(X) and C(X)F
Let Xbe a T1-space. Then we have the following.
C(X)⊆C(X)F⊆T′(X)⊆F(X),
C(X)⊆B1(X)⊆F(X).
Recently, some nice properties of some overrings of C(X) that are subrings of F(X) are studied and inves-
tigated, see for example [1–4, 10, 11, 18]. In this section, we will investigate some new relations between
some nice subrings of F(X). We now define an interesting subring of B1(X).
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6457
Definition 3.1. TB1(X)denotes the set of all f ∈B1(X)such that the restriction of f to a dense open subset of X is
continuous, i.e., TB1(X)=B1(X)TT′(X).
Let Xbe a T1
2-space, i.e., each singleton set of Xis either closed or open [13]. Then by [4, Remark 3.3], C(X)F
is a subring of T′(X). Thus we have the following result.
Proposition 3.2. Let X be a T 1
2-space. Then C(X)Fis a subring of B1(X)if and only if C(X)Fis a subring of TB1(X).
Recall that a space Xhas countable pseudocharacter [23] if each singleton set of Xis a Gδ-set. Let Rbe a
semiprime commutative ring. Recall that a ring Sas an overring of Ris called a ring of quotients of Rif
and only if for every 0 ,s∈Sthere is an element r∈Rsuch that 0 ,sr ∈R[16]. Now we are ready to give
a generalization of [32, Corollary 1] see also [12, Theorem 4.18].
Proposition 3.3. If X has countable pseudocharacter and X is a T31
2-space, then the following statements are
equivalent.
(i) C(X)=B1(X).
(ii) C(X)=TB1(X).
(iii) X is a discrete space.
(iv) B1(X)is a ring of quotients of C(X).
(vi) C(X)=T′(X).
(v) C(X)=C(X)F.
Proof. (iii) ⇔(vi) ⇔(v) see [18, Proposition 3.1].
(i)⇒(ii) It is obvious since C(X)⊆TB1(X)⊆B1(X).
(ii)⇒(iii) If x∈X, then by Proposition 2.4, the characteristic function f=χ{x}belongs to B1(X) and so
f∈TB1(X). Thus fmust be a continuous function since C(X)=TB1(X) by hypothesis in (ii). Hence it shows
that xis an isolated point.
(iii) ⇒(iv) It is straightforward.
(iv) ⇒(i) Let t∈X. So by Proposition 2.4, χ{t}∈B1(X). But by hypothesis, B1(X) is a ring of quotients
of C(X) and C(X) is a commutative semiprime ring, so there is f∈C(X) such that 0 ,fχ{t}∈C(X). But
f(t)χ{t}=fχ{t}is continuous and so tis an isolated point of X. Therefore Xis a discrete space which
completes the proof.
Example 3.10 below shows that the sentence “ Xhas countable pseudocharacter” cannot be dropped from
the hypotheses of Proposition 3.3. If Xis a T1
2-space, then the conditions (iii), (vi) and (v) of Proposition
3.3 are equivalent and they are not equivalent in general [4]. The one-point compactification of Nis not
discrete and so any conditions of Proposition 3.3 don’t hold for this space.
In the following example we show that the conditions (i) and (ii) of Proposition 3.3 are not equivalent
with other it’s conditions in the class of regular spaces.
Example 3.4. There is an example of a regular space X on which every continuous real-valued function is constant
[22]. Thus, R =C(X)=TB1(X)=B1(X)⊊C(X)F⊆T′(X), where R denotes the set of all constant functions on X.
An algebraic characterization of T1-spaces that singleton sets of them are zero sets of continuous functions
are given in the following theorem.
Theorem 3.5. Let X be a T1-space. Then C(X)Fis a subring of TB1(X)if and only if every singleton set of X is a zero
set of a function in C(X).
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6458
Proof. ⇒Let C(X)F⊆TB1(X) and a∈X. Then f=χ{a}∈TB1(X)⊆B1(X) since it is in C(X)F. Thus by
Proposition 2.4, {a}is a zero set of a function in C(X).
⇐Let every singleton set of Xbe a zero set of a function in C(X). Then every finite subset of Xis a zero set of
a function in C(X). Let Ube a nonempty open set of Rand f∈CF(X). Then there exists a subset Dof Xsuch
that f|D∈C(D) and X\Dis finite. Thus X\Dis a zero set of a continuous function and so it is a Zeroσ-set
and a Cozδ-set. But f−1(U)=(f|D)−1(U)SK, where Kis a subset of X\Dand so Kis a zero set of a function
in C(X). Therefore f−1(U) is a Zeroσ-set in Xsince it is a union of two Zeroσ-sets ( f|D)−1(U)=f−1(U)∩D
and Kand so f∈B1(X) by Theorem 1.1. Thus f∈TB1(X) which completes the proof.
Now we give an example of a normal space X, where C(X)Fand B1(X) cannot be compared by inclusion.
Example 3.6. Let X =βNand y ∈βN\N. Then, it is well known that {y}is not a Gδ-set [14, 19] and so {y}is not
a zero-set of a function in B1(X)by [10, Theorem 4.1]. Thus by Theorem 3.5, C(X)F⊈B1(X). On the other hand
for every n ∈N, a mapping fn:X→Rdefined by fn(x)=Pi=n
i=1iχ{i}(x)is continuous. If f :X→Ris a mapping
defined by
f(x)=(x x ∈N,
0x∈βN\N.
then for any x ∈X, f (x)=limn→∞ fn(x)and so f ∈B1(X). We note that the cardinality of X \C(f)=βN\N
is 2c, where cdenotes the cardinality of the continuum and so f <C(X)F. Thus, B1(X)⊈C(X)Fand we have the
following relations:
C(X)⊊TB1(X)=B1(X)⊊F(X)=T′(X),C(X)⊊C(X)F⊊F(X)=T′(X).
Remark 3.7. Let X be a topological space and C(X)cdenotes the set of all f ∈F(X)such that the cardinality of
X\C(f)is not greater than c. Then C(X)cis a subring of F(X)that contains C(X)F. Similar to the above example, we
note that B1(βN)⊈C(βN)cand C(βN)c⊈B1(βN).
Let X be a T31
2-space. If there is a point p in X such that {p}is not Gδ, then C(X)Fis not a subring of B1(X)by
Theorem 3.5.
3.1. When B1(X)is a regular subring of F(X)
Definition 3.8. A space X is called a B1P-space if B1(X)is a regular ring.
We recall that X is a P-space if and only if C(X) is a regular ring, see [19, 4J].
Remark 3.9. Let X be a T31
2-space. Then, X is a P-space if and only if B1(X)=C(X) [32]. Thus every P-space is a
B1P-space.
Now we give an example of a B1P-space which is normal, T′(X) is not a subring of B1(X) and C(X)Fis not a
subring of B1(X).
Example 3.10. Recall that a totally ordered set T is called an η1-set if for any countable subsets A and B such that
a<b for each a ∈A and b ∈B, there exists c ∈T such that a <c<b for each a ∈A,b∈B. Let X be an η1-set in the
order topology such that |X|=2ℵ0as it is constructed in [19, page 187]. Then X is a P-space without isolated points
by [19, Problem 13.P]. Thus B1(X)=C(X) [32], X is normal and each singleton set of X is not Gδand so
C(X)=B1(X)=TB1(X)⊊C(X)F⊆T′(X)⊊F(X).
Recall that a topological space Xis called a Q-space [5] if every subset of Xis Gδ. Clearly, every σ-discrete
space is a Q-space. An algebraic characterization of a Q-space in the class of normal spaces is given in the
following result.
Theorem 3.11. Let X be a normal topological space. Then X is a Q-space if and only if
TB1(X)=T′(X)⊆B1(X)=F(X).
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6459
Proof. ⇒Let Xbe a Q-space, Ube a nonempty open subset of Rand f∈F(X). Then f−1(U) is an Fσ-set in
Xand so by Theorem 1.2, f∈B1(X). Thus B1(X)=F(X) and so T′(X)=TB1(X)⊆F(X)=B1(X).
⇐Let TB1(X)=T′(X)⊆B1(X)=F(X) and Abe a nonempty subset of X. Then from f=χA∈F(X) it follows
that f∈B1(X) and so A=f−1(0,2) is an Fσ-set by Theorem 1.2. Thus Xis a Q-space.
If QX is the non-σ-discrete Q-space, which is perfectly normal as it is constructed in [8], then by Theorem
3.11 we have the following relations:
TB1(QX)=T′(QX)⊊B1(QX)=F(QX).
Clearly, if B1(X)=F(X), then Xis a B1P-space. Thus by Theorem 3.11, we have the following result.
Corollary 3.12. If a normal space X is a Q-space, then X is a B1P-space.
Remark 3.13. If X is a Q-space, then it is well known and easy to prove that X is a P-space if and only if X is a
discrete space. Thus by Corollary 3.12, any non-discrete normal Q-space is an example of a B1P-space which is not
a P-space. Thus a countable normal non-discrete space is a B1P-space which is not a P-space, and so we have the
following relations:
C(Q)⊊C(Q)F⊊TB1(Q)=T′(Q)⊊F(Q)=B1(Q),
and
C(N∗)⊊C(N∗)F=T′(N∗)=F(N∗)=TB1(N∗)=B1(N∗),
where N∗is the one-point compactification of N.
Theorem 3.14. X is a B1P-space if and only if every Cozδ-set is a Zeroσ-set.
Proof. ⇒Let Xbe a B1P-space and Abe a Cozδ-set. Then there exists f∈B1(X) such that A=Z(f) by
Proposition 1.3, and so there is 1∈B1(X) such that f21=f. Thus, f1−1∈B1(X) since B1(X) is a subring of
F(X). It is straightforward that Z(f1−1) =X\Z(f). By Proposition 1.3, Z(f1−1) is a Cozδ-set which implies
that A=Z(f) is a Zeroσ-set.
⇐Let every Cozδ-set be a Zeroσ-set and f∈B1(X). Then by Theorem 1.1, f−1(V) is an Zeroσ-set for any
open set Vin Rand so Z(f) is a Zeroσ-set since Z(f) is a Cozδ-set. Now consider a mapping 1∈F(X) defined
by
1(x)=
1
f(x)x<Z(f),
0x∈Z(f).
It is sufficient to prove that 1∈B1(X) since 1f2=f. Let U=(a,b) be any open interval in R. Then it is
straightforward that
1−1(U)=
f−1(1
b,1
a) 0 <(a,b),
f−1(−∞,1
a)∪f−1(1
b,∞)∪Z(f) 0 ∈(a,b).
Thus by our hypothesis, in any case 1−1(U) is an Zeroσ-set in Xand so by Theorem 1.1, 1∈B1(X) which
completes the proof.
Due to Kuratowski [26, chapter 40, VI] a topological space is called σ-space, if every Gδ-set is Fσ. Hence
by Theorem 3.14, in the class of perfectly normal spaces, B1P-spaces are exactly σ-spaces. We now give an
example of a B1P-space Xwhich is not a P-space and C(X)F=T′(X) is not a subring of B1(X).
M.R. Ahmadi Zand, Z. Khosravi /Filomat 37:19 (2023), 6453–6461 6460
Example 3.15. Let Y be an uncountable discrete space and let X =Y∪ {y}be it’s one-point compactification, where
y<Y. Let A be a Cozδ-set in X. Then by Proposition 1.3, there is f ∈B1(X)such that A =Z(f). Thus, every
continuous, and consequently every Baire-one function on X satisfies f(x)=f(y)for all but countably many points
x∈X and so Z(f)is countable or X \Z(f)is countable. In any case, A =Z(f)is a Zeroσ-set in X and so by Theorem
3.14, X is a B1P-space and we have the following relations:
C(X)⊊TB1(X)=B1(X)⊊F(X)=C(X)F=T′(X).
Thus X is not a Q-space by Theorem 3.11.
To give some conditions on a topological space Xunder which T′(X)⊈B1(X) we recall the following
definitions. A space is called Baire if the intersection of countably many open dense subsets of the space is
dense. A space is called irresolvable if it does not admit disjoint dense sets, otherwise it is called resolvable
[21]. It is well known that every locally compact space without isolated points is resolvable [21]. A perfect
subset of the space is a closed subset which in its relative topology has no isolated points.
Proposition 3.16. Let X be a topological space and ∅,A be a nowhere dense and perfect subset of X which is locally
compact. Then T′(X)is not a subring of B1(X).
Proof. D =X\Ais an open dense subset of Xsince Ais a nowhere dense and perfect subset of X. Ais
resolvable [21] and so there exists a subset Bof Asuch that Band A\Bare two disjoint dense subsets of A.
If f:X→Ris a mapping defined by
f(x)=(1x∈DSB,
0x∈A\B,
then f∈T′(X). We claim that f<B1(X) which completes the proof. On the contrary let f∈B1(X), so
f|A∈B1(A). Since Ais locally compact, Ais a Baire space and so C(f|A) must be dense in Aby [29, Theorem
48.5]. But C(f|A)=∅which is a contradiction.
The following example shows that T′(R) is not a subring of B1(R) and B1(R) is not a subring of T′(R).
Example 3.17. Let f0:R→Rbe defined as,
f0(x)=
1
qi f x =p
q,where p ∈Z,q∈Nand 1.c.d.(p,q)=1,
1if x =0,
0otherwise.
Clearly, f0<T′(R). By [15], f0∈B1(R), and so B1(R)⊈T′(R). With slight changes in [11, Example 2.7] we
observe that there is no 10∈B1(X)such that f 2
010=f0, i.e., Ris not a B1P-space. It is well known that the Cantor
set is a perfect subset of Rand it is compact and nowhere dense so by Proposition 3.16, T′(R)⊈B1(R). Thus by
Theorem 3.5, we have the following relations between some subrings of F(R).
C(R)⊊C(R)F⊊T′(R)⊊F(R)and C(R)⊊C(R)F⊊TB1(R)⊊B1(R)⊊F(R).
Acknowledgements. We record our pleasure to the anonymous referee for his or her constructive
report and many helpful suggestions on the main results of the earlier version of the manuscript which
improved the presentation of the paper.
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