Insertion of A Baire-one Function
MAJID MIRMIRAN
Department of Mathematics, University of Isfahan, Isfahan , 81746-73441, Iran ∗Corresponding Author: [email protected]
Copyright c2013 Horizon Research Publishing All rights reserved.
Abstract
A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a Baire-one function between two comparable real-valued functions on the topological spaces that Λ−sets areGδ−sets.
Keywords
Insertion, Strong binary relation, Baire-one function, Λ−sets, Lower cut set.1
Introduction
A generalized class of closed sets was considered by Maki in 1986 [5]. He investigated the sets that can be represented as union of closed sets and called them
V−sets. Complements ofV−sets, i.e., sets that are in-tersection of open sets are called Λ−sets [5].
Results of Katˇetov [2] , [3] concerning binary relations and the concept of an indefinite lower cut set for a real-valued function, which is due to Brooks [1], are used in order to give a necessary and sufficient condition for the insertion of a Baire-one function between two compara-ble real-valued functions on the topological spaces that Λ−sets areGδ−sets.
A real-valued functionfdefined on a topological space
X is calledBaire-oneif the preimage of every open sub-set ofR is aFσ−set inX.
Ifgandf are real-valued functions defined on a space
X, we writeg ≤ f (resp. g < f) in case g(x) ≤ f(x) (resp. g(x)< f(x)) for all xin X.
The following definitions are modifications of condi-tions considered in [4].
A property P defined relative to a real-valued func-tion on a topological space is a B1−property provided that any constant function has propertyP and provided that the sum of a function with property P and any Baire-one function also has propertyP. IfP1andP2are
B1−properties, the following terminology is used: (i) A spaceXhas theweakB1−insertion property for(P1, P2) iff for any functionsgandf onX such thatg≤f, ghas propertyP1 andf has propertyP2, then there exists a Baire-one functionhsuch that g≤h≤f. (ii) A space
X has theB1−insertion property for(P1, P2) iff for any functionsg andf onX such thatg < f, g has property
P1andf has property P2, then there exists a Baire-one functionhsuch thatg < h < f.
In this paper, for a topological space that Λ−sets are
Gδ−sets, is given a sufficient condition for the weak
B1−insertion property. Also for a space with the weak
B1−insertion property, we give a necessary and sufficient condition for the space to have the B1−insertion prop-erty. Several insertion theorems are obtained as corol-laries of these results.
2
The main results
Before giving a sufficient condition for insertability of a Baire-one function, the necessary definitions and terminology are stated.
Definition. Let A be a subset of a topological space (X, τ). We define the subsetsAΛ andAV as follows:
AΛ=∩{O :O ⊇A, O ∈(X, τ)} andAV =∪{F :F ⊆
A, Fc∈(X, τ)}.
AΛ is calledkernelofA.
The following first two definitions are modifications of conditions considered in [2], [3].
Definition. If ρ is a binary relation in a set S
then ¯ρ is defined as follows: xρ y¯ if and only ify ρ ν
impliesx ρ νandu ρ ximpliesu ρ yfor anyuandvinS. Definition. A binary relationρin the power setP(X) of a topological space X is called a strong binary re-lation in P(X) in case ρ satisfies each of the following conditions:
1) If Ai ρ Bj for anyi∈ {1, . . . , m} and for any j ∈
{1, . . . , n}, then there exists a setC in P(X) such that
Ai ρ C and C ρ Bj for any i ∈ {1, . . . , m} and any
j ∈ {1, . . . , n}.
2) If A⊆B, thenAρ B¯ .
3) If A ρ B, thenAΛ⊆B andA⊆BV.
The concept of a lower indefinite cut set for a real-valued function was defined by Brooks [1] as follows:
Definition. If f is a real-valued function defined on a spaceX and if{x∈X:f(x)< `} ⊆A(f, `)⊆ {x∈X :
f(x) ≤`} for a real number `, then A(f, `) is a lower indefinite cut set in the domain off at the level`.
We now give the following main results:
withg≤f. If there exists a strong binary relationρon the power set ofX and if there exist lower indefinite cut setsA(f, t) andA(g, t) in the domain of f and gat the level t for each rational number t such that if t1 < t2 then A(f, t1) ρ A(g, t2), then there exists a Baire-one functionhdefined onX such thatg≤h≤f.
Proof. Letgandf be real-valued functions defined on the X such that g ≤ f. By hypothesis there exists a strong binary relationρon the power set ofX and there exist lower indefinite cut sets A(f, t) and A(g, t) in the domain off andgat the leveltfor each rational number
tsuch that ift1< t2 thenA(f, t1)ρ A(g, t2).
Define functionsF andGmapping the rational num-bers Q into the power set of X byF(t) = A(f, t) and
G(t) = A(g, t). If t1 and t2 are any elements of Q with t1 < t2, then F(t1) ¯ρ F(t2), G(t1) ¯ρ G(t2), and
F(t1)ρ G(t2). By Lemmas 1 and 2 of [3] it follows that there exists a functionH mappingQinto the power set of X such that if t1 and t2 are any rational numbers with t1 < t2, then F(t1) ρ H(t2), H(t1) ρ H(t2) and
H(t1)ρ G(t2).
For anyxin X, leth(x) = inf{t∈Q:x∈H(t)}. We first verify that g ≤h ≤f: Ifx is in H(t) then
xis in G(t0) for any t0 > t; since xin G(t0) = A(g, t0) implies that g(x)≤t0, it follows that g(x)≤t. Hence
g ≤ h. If x is not in H(t), then x is not in F(t0) for anyt0 < t; sincexis not inF(t0) =A(f, t0) implies that
f(x)> t0, it follows thatf(x)≥t. Hence h≤f. Also, for any rational numbers t1 and t2 with
t1 < t2, we haveh−1(t1, t2) = H(t2)V \H(t1)Λ. Hence
h−1(t1, t2) is a Fσ−set in X, i.e., h is a Baire-one
function onX.
The above proof used the technique of theorem 1 of[2].
Theorem 2.2. LetP1 andP2 beB1−property andX be a space that satisfies the weakB1−insertion property for (P1, P2). Also assume thatgand f are functions on
X such that g < f, g has property P1 and f has prop-ertyP2. The spaceXhas theB1−insertion property for (P1, P2) iff there exist lower cut setsA(f−g,3−n+1) and there exists a decreasing sequence{Dn}of subsets ofX
with empty intersection and such that for eachn, X\Dn
andA(f−g,3−n+1) are completely separated by Baire-one functions.
Proof. Assume that X has the weak B1−insertion property for (P1, P2). Let g and f be functions such that g < f, g has property P1 and f has property P2. By hypothesis there exist lower cut setsA(f−g,3−n+1) and there exists a sequence (Dn) such thatT∞n=1Dn=∅
and such that for eachn, X\DnandA(f−g,3−n+1) are
completely separated by Baire-one functions. Letkn be
a Baire-one function such thatkn= 0 onA(f−g,3−n+1)
andkn= 1 onX\Dn. Let a functionkonXbe defined
by
k(x) = 1/2
∞ X
n=1
3−nkn(x).
By the Cauchy condition and the properties of Baire-one, the function k is a Baire-one function. Since
T∞
n=1Dn =∅and sincekn = 1 onX\Dn, it follows that
0< k. Also 2k < f−g: In order to see this, observe first that ifxis inA(f−g,3−n+1), thenk(x)≤1/4(3−n). If
xis any point inX, thenx /∈A(f−g,1) or for somen,
x∈A(f−g,3−n+1)−A(f−g,3−n);
in the former case 2k(x)<1, and in the latter 2k(x)≤ 1/2(3−n) < f(x)−g(x). Thus if f
1 = f −k and if
g1 = g+k, then g < g1 < f1 < f. Since P1 and P2 areB1−properties, theng1 has propertyP1 andf1 has propertyP2. SinceX has the weakB1−insertion prop-erty for (P1, P2), then there exists a Baire-one function
h such that g1 ≤ h≤ f1. Thus g < h < f, it follows that X satisfies theB1−insertion property for (P1, P2). (The technique of this proof is by Katˇetov[3]).
Conversely, let g and f be functions on X such that
g has property P1, f has property P2 and g < f. By hypothesis, there exists a Baire-one functionhsuch that
g < h < f.We follow an idea contained in Lane [4]. Since the constant function 0 has propertyP1, sincef−hhas propertyP2, and sinceXhas theB1−insertion property for (P1, P2), then there exists a Baire-one functionksuch that 0 < k < f−h. Let A(f−g,3−n+1) be any lower cut set forf−g and letDn ={x∈X :k(x)<3−n+2}.
Sincek >0 it follows thatT∞n=1Dn=∅. Since
A(f −g,3−n+1)⊆ {x∈X : (f−g)(x)≤3−n+1} ⊆
{x∈X:k(x)≤3−n+1}
and since {x ∈ X : k(x) ≤ 3−n+1} and {x∈X :k(x)≥3−n+2}=X\D
n are completely
sepa-rated by sup{3−n+1,inf{k,3−n+2}}that is a Baire-one function, it follows that for eachn, A(f−g,3−n+1) and
X\Dn are completely separated by Baire-one functions.
2.1
ApplicationsDefinition. A real-valued functionf defined on a space
X is called upper semi-Baire-one (resp. lower semi-Baire-one) iff−1(−∞, t) (resp. f−1(t,+∞)) is aFσ−set
for any real numbert.
The abbreviations usc, lsc, usB1 and lsB1 are used for upper semicontinuous, lower semicontinuous, upper semi-Baire-one, and lower semi-Baire-one, respectively. Remark 1. [2], [3]. A space X has the weak
c−insertion property for (usc, lsc) iffX is normal. Before stating the consequences of theorems 2.1, 2.2, we suppose that X is a topological space that Λ−sets areGδ−sets.
Corollary 3.1. For each pair of disjoint Gδ−sets
G1, G2, there are two Fσ−sets F1 and F2 such that
G1 ⊆F1,G2 ⊆F2 and F1∩F2 =∅ iff X has the weak
B1−insertion property for (usB1, lsB1).
Proof. Let g and f be real-valued functions defined on the X, such that f is lsB1, g is usB1, and g ≤ f.If a binary relationρis defined by A ρ Bin caseAΛ⊆BV,
then by hypothesis ρ is a strong binary relation in the power set ofX. Ift1andt2 are any elements ofQwith
t1< t2, then
{x∈X :g(x)< t2} ⊆A(g, t2);
since{x∈ X :f(x)≤t1} is a Gδ−set and since{x∈
X :g(x)< t2} is a Fσ−set, it follows thatA(f, t1)Λ ⊆
A(g, t2)V. Hencet
1< t2implies thatA(f, t1)ρ A(g, t2). The proof follows from Theorem 2. 1.
On the other hand, let G1 and G2 are disjoint
Gδ−sets. Setf =χGc
1 andg=χG2, thenf islsB1, g is usB1, andg≤f. Thus there exists Baire-one function
hsuch that g ≤h≤f. SetF1 ={x∈X : h(x)< 12} and F2 = {x ∈ X : h(x) > 12}, then F1 and F2 are disjointFσ−sets such thatG1⊆F1 andG2⊆F2.
Before stating the consequences of theorem 2.2, we state and prove the necessary lemmas.
Lemma 3.2. The following conditions on the spaceX
are equivalent:
(i) Every two disjointGδ−sets ofX can be separated
byFσ−sets ofX.
(ii) If G is a Gδ−set of X which is contained in a
Fσ−setF, then there exists aFσ−setH such thatG⊆
H ⊆HΛ⊆F.
Proof. (i)⇒(ii) Suppose thatG⊆F, whereGandF
areGδ−set andFσ−set ofX, respectively. Hence,Fc is
aGδ−set and G∩Fc=∅.
By (i) there exists two disjoint Fσ−sets F1, F2 s.t.
G⊆F1 andFc⊆F2. But
Fc⊆F2⇒F2c⊆F,
and
F1∩F2=∅ ⇒F1⊆F2c hence
G⊆F1⊆F2c ⊆F
and sinceFc
2 is aGδ−set containingF1we conclude that
FΛ
1 ⊆F2c, i.e.,
G⊆F1⊆F1Λ⊆F.
By settingH=F1, condition (ii) holds.
(ii) ⇒ (i) Suppose that G1, G2 are two disjoint
Gδ−sets ofX.
This implies thatG1⊆Gc2andGc2is aFσ−set. Hence
by (ii) there exists aFσ−setHs.t.,G1⊆H ⊆HΛ⊆Gc2. But
H ⊆HΛ⇒H∩(HΛ)c=∅ and
HΛ⊆Gc2⇒G2⊆(HΛ)c.
Furthermore, (HΛ)c is a Fσ−set of X. Hence
G1⊆H, G2 ⊆(HΛ)c and H∩(HΛ)c =∅. This means that condition (i) holds.
Lemma 3.3. Suppose that X is the topological space s.t. we can separate every two disjoint Gδ−sets by
Fσ−sets. If G1 andG2 are two disjoint Gδ−sets ofX,
then there exists a Baire-one function h : X → R s.t.
h(G1) ={0}andh(G2) ={1}.
Proof. SupposeG1 andG2are two disjointGδ−sets of
X. SinceG1∩G2 =∅, henceG1 ⊆Gc2. In particular,
sinceGc
2is aFσ−set ofXcontainingG1, by Lemma 3.2,
there exists aFσ−setH1/2 s.t.,
G1⊆H1/2⊆H1Λ/2⊆G
c
2.
Note that H1/2 is a Fσ−set and contains G1, and Gc2 is a Fσ−set and contains H1Λ/2. Hence, by Lemma 3.2,
there existsFσ−setsH1/4 andH3/4 s.t.,
G1⊆H1/4⊆H1Λ/4⊆H1/2⊆H1Λ/2⊆H3/4⊆H3Λ/4⊆G
c
2.
By continuing this method for every t ∈ D, where
D ⊆[0,1] is the set of rational numbers that their de-nominators are exponents of 2, we obtain Fσ−sets Ht
with the property that if t1, t2 ∈ D and t1 < t2, then
Ht1 ⊆Ht2. We define the real-valued function hon X
by h(x) = inf{t:x∈Ht}for x6∈G2 and h(x) = 1 for
x∈G2.
Note that for every x ∈ X,0 ≤h(x) ≤ 1. Also, we note that for any t∈ D, G1 ⊆Ht; hence h(G1) ={0}. Furthermore, by definition, h(G2) = {1}. It re-mains only to prove that h is a Baire-one function on X. For every α ∈ R, we have if α ≤ 0 then {x ∈ X : h(x) < α} = ∅ and if 0 < α then {x ∈ X : h(x) < α} = ∪{Ht : t < α}, hence,
they are Fσ−sets of X. Similarly, if α < 0 then
{x ∈ X : h(x) > α} = X and if 0 ≤ α then {x∈ X :h(x) > α} =∪{(HtΛ)c : t > α} hence, every
of them is a Fσ−set. Consequently h is a Baire-one
function.
Lemma 3.4. Suppose thatX is the topological space such that every two disjoint Gδ−sets can be separated
byFσ−sets. The following conditions are equivalent:
(i) Every countable convering of Fσ−sets of X has a
refinement consisting of Fσ−sets s.t., for every x∈X,
there exists aFσ−set containingxsuch that it intersects
only finitely many members of the refinement.
(ii) Corresponding to every decreasing sequence{Gn}
of Gδ−sets with empty intersection there exists a
de-creasing sequence {Fn} of Fσ−sets s.t., T∞n=1Fn = ∅
and for every n∈N, Gn ⊆Fn.
Proof. (i) ⇒ (ii). suppose that {Gn} be a
decreas-ing sequence ofGδ−sets with empty intersection. Then
{Gcn : n∈ N} is a countable covering of Fσ−sets. By
hypothesis (i) and Lemma 3.2, this covering has a re-finement {Vn : n ∈ N} s.t. everyVn is a Fσ−set and
VΛ
n ⊆Gcn. By settingFn= (VnΛ)c, we obtain a
decreas-ing sequence ofFσ−sets with the required properties.
(ii) ⇒ (i). Now if {Hn : n∈N} is a countable
cov-ering of Fσ−sets, we set for n ∈ N, Gn = (Sni=1Hi)c.
Then {Gn} is a decreasing sequence of Gδ−sets with
empty intersection. By (ii) there exists a decreasing se-quence{Fn}consisting ofFσ−sets s.t.,T∞n=1Fn=∅and
for every n∈ N, Gn ⊆Fn. Now we define the subsets
Wn ofX in the following manner:
W1 is aFσ−set ofX s.t. F1c⊆W1andW1Λ∩G1=∅.
W2 is aFσ−set ofX s.t. W1Λ∪F2c ⊆W2 andW2Λ∩
G2=∅, and so on. (By Lemma 3.2,Wn exists).
Then since {Fnc :n ∈N} is a covering for X, hence
{Wn:n∈N}is a covering forX consisting ofFσ−sets.
Moreover, we have (i)WΛ
n ⊆Wn+1 (ii)Fc
(iii)Wn⊆Sni=1Hi.
Now suppose thatS1=W1 and forn≥2, we setSn =
Wn+1\WnΛ−1. Then since WΛ
n−1 ⊆ Wn and Sn ⊇ Wn+1\Wn, it
follows that{Sn :n∈N}consists ofFσ−sets and covers
X. Furthermore, Si∩Sj 6= ∅ iff |i−j| ≤ 1. Finally,
consider the following sets:
S1∩H1, S1∩H2
S2∩H1, S2∩H2, S2∩H3
S3∩H1, S3∩H2, S3∩H3, S3∩H4
and continue ad infinitum. These sets areFσ−sets, cover
X and refine {Hn :n ∈ N}. In addition, Si∩Hj can
intersect at most the sets in its row, immediately above, or immediately below row.
Hence if x ∈ X and x ∈ Sn∩Hm, then Sn ∩Hm
is a Fσ−set containing x that intersects at most
finitely many of sets Si ∩ Hj. Consequently,
{Si∩Hj:i∈N, j= 1, . . . , i+ 1} refines{Hn :n∈N}
s.t. its elements are Fσ−sets, and for every point
in X we can find a Fσ−set containing the point that
intersects only finitely many elements of that refinement.
Remark 2. [2] , [3]. A space X has the c−insertion property for (usc, lsc) iff X is normal and countably paracompact.
Corollary 3.5. X has the B1−insertion property for (usB1, lsB1) iff every two disjointGδ−sets ofX can be
separated byFσ−sets, and in addition, every countable
covering of Fσ−sets has a refinement that consists of
Fσ−sets s.t., for every point ofX we can find aFσ−set
containing that point s.t., it intersects only a finite num-ber of refining memnum-bers.
Proof. Suppose that G1 and G2 are disjoint Gδ−sets
. Since G1∩G2 =∅, it follows thatG2 ⊆Gc1. We set
f(x) = 2 forx∈Gc
1,f(x) =12 forx6∈G
c
1, andg=χG2.
SinceG2 is a Gδ−set, and Gc1 is aFσ−set, therefore
g is usB1,f is lsB1 and furthermore g < f. Hence by hypothesis there exists a Baire-one functionhs.t., g < h < f. Now by setting F1 = {x ∈ X : h(x) < 1} and F2 = {x ∈ X : h(x) > 1}. We can say that F1 and F2 are disjoint Fσ−sets that contain G1 and G2, respectively. Now suppose that {Gn} is a decreasing
sequence ofGδ−sets with empty intersection. SetG0=
Xand define for everyx∈Gn\Gn+1,f(x) = n+11 . Since
T∞
n=0Gn =∅ and for every x∈X, there existsn∈N,
s.t.,x∈Gn\Gn+1, f is well defined. Furthermore, for every r∈ R, ifr ≤0 then{x∈X :f(x) > r}=X is aFσ−set and ifr >0 then by Archimedean property of
R, we can findi∈N s.t. i+11 ≤r. Now suppose thatk
is the least natural number s.t. k+11 ≤r. Hence k1 > r
and consequently, {x∈ X : f(x) > r} = X \Gk is a
Fσ−set. Therefore,f islsB1. By settingg= 0, we have
g isusB1 and g < f. Hence by hypothesis there exists a Baire-one functionhonX s.t.,g < h < f.
By settingFn ={x∈ X : h(x)< n+11 }, we have Fn
is aFσ−set. But for everyx∈Gn, we havef(x)≤n1+1
and since g < h < f therefore 0 < h(x) < n+11 , i.e.,
x ∈ Fn therefore Gn ⊆ Fn and since h > 0 it follows
thatT∞n=1Fn=∅. Hence by Lemma 3.4, the conditions
holds.
On the other hand, since every two disjointGδ−sets
can be separated byFσ−sets, therefore by Corollary 3.1,
X has the weakB1−insertion property for (usB1, lsB1). Now suppose that f and g are real-valued functions on
X with g < f, s.t., g isusB1 and f is lsB1. For every
n∈N, set
A(f−g,3−n+1) ={x∈X: (f−g)(x)≤3−n+1}.
Since g is usB1, and f is lsB1, therefore f −g is
lsB1. Hence A(f −g,3−n+1) is a Gδ−set of X.
Con-sequently, {A(f −g,3−n+1)} is a decreasing sequence of Gδ−sets and furthermore since 0< f−g, it follows
that T∞n=1A(f −g,3−n+1) = ∅. Now by Lemma 3.4, there exists a decreasing sequence {Dn} of Fσ−sets
s.t. A(f −g,3−n+1) ⊆D
n and
T∞
n=1Dn = ∅. But by
Lemma 3.3, A(f −g,3−n+1) and X \D
n of Gδ−sets
can be completely separated by Baire-one functions. Hence by Theorem 2.2, there exists a Baire-one function
h defined on X s.t., g < h < f, i.e., X has the
B1−insertion property for (usB1, lsB1).
Remark 3. [6]. A space X has the weakc−insertion property for (lsc, usc) iffX is extremally disconnected. Corollary 3.6. For every F of Fσ−set, FΛ is a
Fσ−set iff X has the weak B1−insertion property for
(lsB1, usB1).
Before giving the proof of this corollary, the necessary lemma is stated.
Lemma 3.7. The following conditions on the space
X are equivalent:
(i) For everyF ofFσ−set we haveFΛ is aFσ−set.
(ii) For each pair of disjointFσ−sets asF1andF2we have FΛ
1 ∩F2Λ=∅.
The proof of lemma 3.7 is a direct consequence of the definition Λ−sets.
We now give the proof of corollary 3.6.
Proof. Let g and f be real-valued functions defined on the X, such thatf islsB1, gisusB1, andf ≤g.If a binary relationρis defined byA ρ B in caseAΛ⊆F ⊆
FΛ⊆BV for someF
σ−setF inX, then by hypothesis
and Lemma 3.7ρis a strong binary relation in the power set ofX. Ift1andt2are any elements ofQwitht1< t2, then
A(g, t1) ={x∈X :g(x)< t1} ⊆ {x∈X:f(x)≤t2}; =A(f, t2);
since {x∈ X : g(x) < t1} is a Fσ−set and since {x∈
X :f(x)≤t2}is aGδ−set, by hypothesis it follows that
A(g, t1)ρ A(f, t2). The proof follows from Theorem 2.1. On the other hand, Let F1 and F2 are disjoint
Fσ−sets. Set f =χF2 andg=χF1c, thenf is lsB1, g is usB1, andf ≤g.
Thus there exists Baire-one function h such that
f ≤ h ≤ g. Set G1 = {x ∈ X : h(x) ≤ 13} and
G2={x∈X :h(x)≥2/3}thenG1 andG2are disjoint
Gδ−sets such that F1 ⊆ G1 and F2 ⊆ G2. Hence
FΛ
1 ∩F2Λ=∅.
and if for any decreasing sequence{Gn}of open subsets
of X with empty intersection there exists a decreasing sequence {Fn} of closed subsets of X with empty
intersection such thatGn⊆Fn for eachn.
Corollary 3.8. For everyF ofFσ−set, FΛ is aFσ−set
and in addition for every decreasing sequence {Fn} of
Fσ−sets with empty intersection, there exists a
decreas-ing sequence{Gn} of Gδ−sets with empty intersection
s.t. for everyn∈N, Fn⊆GniffXhas theB1−insertion property for (lsB1, usB1).
Proof. Since for every F of Fσ−set, FΛ is a Fσ−set,
therefore by corollary 3.6,X has the weakB1−insertion property for (lsB1, usB1). Now suppose that f and g are real-valued functions defined on X with g < f, g is
lsB1, and f is usB1. Set A(f −g,3−n+1) ={x∈X : (f−g)(x)<3−n+1}. Then sincef −g is usB1, hence {A(f−g,3−n+1)} is a decreasing sequence of Fσ−sets
with empty intersection. By hypothesis, there exists a decreasing sequence{Dn}ofGδ−sets with empty
inter-section s.t., for every n ∈ N, A(f −g,3−n+1) ⊆ Dn.
Hence X \ Dn and A(f −g,3−n+1) are two disjoint
Fσ−sets and therefore by Lemma 3.7, we have
A(f −g,3−n+1)Λ∩(X\Dn)Λ=∅
and therefore by Lemma 3.3,X\DnandA(f−g,3−n+1)
are completely separable by Baire-one functions. There-fore by Theorem 2.2, there exists a Baire-one function
h on X s.t., g < h < f, i.e., X has theB1−insertion property for (lsB1, usB1).
On the other hand, suppose that F1 and F2 be two disjointFσ−sets. SinceF1∩F2=∅. We have F2⊆F1c. We set f(x) = 2 for x∈Fc
1, f(x) = 1
2 for x6∈F
c
1 and
g=χF2.
Then since F2 is a Fσ−set and F1c is a Gδ−set, we
conclude that g is lsB1 and f is usB1 and further-more g < f. By hypothesis, there exists a Baire-one function h on X s.t., g < h < f. Now we set
G1={x∈X:h(x)≤34}andG2={x∈X :h(x)≥1}. Then G1 and G2 are two disjoint Gδ−sets contain F1 andF2, respectively. HenceF1Λ⊆G1andF2Λ⊆G2and consequentlyF1Λ∩F2Λ=∅. By Lemma 3.7, for everyF ofFσ−set, the setFΛis a Fσ−set.
Now suppose that {Fn} is a decreasing sequence of
Fσ−sets with empty intersection.
We set F0 =X and f(x) = n+11 for x∈ Fn\Fn+1. Since T∞n=0Fn = ∅ and for every n ∈ N there exists
x∈Fn\Fn+1, f is well-defined. Furthermore, for every
r∈R, ifr≤0 then{x∈X :f(x)< r}=∅ is aFσ−set
and if r >0 then by Archimedean property ofR, there exists i∈ N s.t. i+11 ≤r. Suppose that k is the least natural number with this property. Hence 1
k > r. Now
if k+11 < rthen {x∈X :f(x)< r} =Fk is a Fσ−set
and if k+11 =r then {x ∈ X : f(x)< r} =Fk+1 is a
Fσ−set. Hence f is a usB1 on X. By settingg = 0, we have conclude that g is lsB1 onX and in addition
g < f. By hypothesis there exists a Baire-one function
honX s.t.,g < h < f.
Set Gn = {x ∈ X : h(x) ≤ n1+1}. This set is a
Gδ−set. But for every x ∈ Fn, we have f(x) ≤ n+11 and since g < h < f thush(x)< n+11 , this means that
x∈Gn and consequentlyFn ⊆Gn.
By definition of Gn,{Gn} is a decreasing sequence
of Gδ−sets and since h > 0,
T∞
n=1Gn = ∅. Thus the
conditions holds.
Acknowledgements
This research was partially supported by Centre of Excellence for Mathematics(University of Isfahan).
References
[1] . Brooks, Indefinite cut sets for real functions, Amer. Math. Monthly, 78(1971), 1007-1010. [2] . Katˇetov, On real-valued functions in topological
spaces, Fund. Math., 38(1951), 85-91.
[3] . Katˇetov, Correction to, ”On real-valued functions in topological spaces”, Fund. Math., 40(1953), 203-205.
[4] . Lane, Insertion of a continuous function, Pacific J. Math., 66(1976), 181-190.
[5] . Maki, Generalized Λ−sets and the associated clo-sure operator, The special Issue in commemoration of Prof. Kazuada IKEDA’s Retirement, (1986), 139-146.
