Baire classification of separately continuous functions and Namioka property

0.17 MB·

by  V. V. Mykhaylyuk

BAIRE CLASSIFICATION OF SEPARATELY CONTINUOUS
FUNCTIONS AND NAMIOKA PROPERTY
6
1
V.V.MYKHAYLYUK
0
2

n Abstract. Weprovethefollowingtworesults.
a 1. IfX isacompletelyregularspacesuchthatforeverytopological space
J Y each separately continuous function f : X ×Y → R is of the first Baire
0 class,theneveryLindel¨ofsubspaceofX bijectivelycontinuouslymapsontoa
2 separablemetrizablespace.
2. If X is a Baire space, Y is a compact space and f : X ×Y → R is
] a separately continuous function which is a Baire measurable function, then
N there exists adense inX Gδ-setA such that f is jointlycontinuous at every
pointofA×Y (thisgivesapositiveanswertoaquestionofG.Vera).
G
.
h
t
a
m 1. Introduction
[ A function f : X → R, which defined on a topological space X, is called a

1 function of the first Baire class, if there exists a sequence (fn)∞n=1 of continuous
v functions f : X → R such that f(x) = lim f (x) for every x ∈ X. For every at
n n
7 most countable ordinal α a function f :Xn→→∞R is called a function of the α Baire
9
class, if there exists a sequence (f )∞ of functions f :X →R of Baire class <α
1 n n=1 n
5 suchthatf(x)= lim fn(x)foreveryx∈X. Afunctionf :X →RiscalledBaire
n→∞
0 measurability if f is a function of α Baire class for some at most countable ordinal
.
1 α.
0 A function f : X → R is called a function of the first Lebesgue class, if f−1(G)
6 is a F -set in X for every open set G⊆R.
1 σ
InvestigationsofBaireandLebesgueclassificationofseparatelycontinuousfunc-
:
v tions(thatisfunctionsofmanyvariableswhicharecontinuouswithrespecttoevery
i
X variable) were started by A.Lebesgue in the classical paper [1]. These investiga-
tions were continued in papers of many mathematicians (see, for example, [2] the
r
a literature given there).
In particular, W. Moran and H.Rosental [3,4] proved that for a compact X the
following conditions are equivalent:
(i) foreverycompactspaceY everyseparatelycontinuousfunctionf :X×Y →
R is a first Baire class function;
(ii) X hasthecountablechainconditionthatiseverysystemofpairwisedisjoint
open in X nonempty set is at most countable.
A topological space X is called a Moran (weakly Moran) space, if for every
compactspaceY eachseparatelycontinuousfunctionf :X×Y →RisafirstBaire
2000 Mathematics Subject Classification. Primary54C08,54C30, 54C05.
Key words and phrases. separately continuous functions, first Baire class function, Namioka
space, dependence functions on ℵ coordinates, Baire space, quarter-stratifiable space,Lindelo¨f
space.
1
2 V.V.MYKHAYLYUK
classfunction (a Baire measurablefunction). These notionswere introducedin [5],
where the following result was obtained.
Theorem 1.1. Let X be a completely regular space with the countable chain con-
dition, Y be a compact, f : X ×Y → R be a separately continuous function and
ϕ:Y →C (X), ϕ(y)(x)=f(x,y). Then the following conditions are equivalent:
p
(i) f is a first Baire class function;
(ii) the space ϕ(Y) is metrizable.
For a topologicalspace X by C (X) we denote the space of all continuous func-
p
tions z :X →R with the topology of pointwise convergence.
Moreover, relations between Moran spaces and Namioka property was investi-
gated in [5]. A mapping f : X ×Y → R which is defined on the product X ×Y
oftopologicalspacesX andY has Namioka property, if there exists aneverywhere
dense in X G -set A ⊆ X such that f is jointly continuous at every point of set
δ
A×Y. AtopologicalspaceX iscalleda Namioka space,ifforeverycompactspace
Y each separately continuous functions f :X×Y →R has the Namioka property.
The following question was formulated in [5].
Question 1.2. Is every Baire Moran space a Namioka space?
AtopologicalspaceX iscalleda space with B-property (L-property),ifforevery
topological space Y each separately continuous function f : X ×Y → R is a first
Baire (Lebesgue) function.
A standard reasoning (see [6, p.394]) shows that every first Baire class function
is a first Lebesgue class function. Therefore every space with B-property has L-
property. Note that spaces with B-property were called by Rudin spaces in [7].
But this term is not very successful because W. Rudin in [8] used partitions of the
unit for proof that every separately continuous mappings which is defined on the
productofatopologicalspaceandametrizablespaceandvaluedinalocallyconvex
space is a Baire first mapping. W. Rudin is not considered a mapping with valued
in R separately.
A development of Rudin’s result leads to the following notions.
A topological space X is called a PP-space, if there exist a sequence
((h :i∈I ))∞
n,i n n=1
of locally finite partitions of the unit (h : i ∈ I ) on X and sequence (α )∞
n,i n n n=1
of families α = (x : i ∈ I ) of points x ∈ X such that for every x ∈ X and
n n,i n n,i
neighborhood U of x in X there exists an integer n ∈ N such that x ∈ U, if
0 n,i
n≥n andx∈supph ,wherebysupphwedenotethesupport{x∈X :h(x)6=0}
0 n,i
of h.
These notion was introdused in [9]. It was obtained in [10, Theorem 1] that
every PP-space has the B-property.
A topologicalspace X with the topology T is called quarter-stratifiable, if there
exists a function g :N×X →T such that
(i) X = g(n,x) for every n∈N;
S
x∈X
(ii) if x∈g(n,x ) for every n∈N, then x →x.
n n
If there exists a weaker metrizable topology T on a qarter-stratifiable space X
such that all covers U = {g(n,x): x ∈ X} can be chosen T-open, then the space
n
X is called metrically quarter-stratifiable.
BAIRE CLASSIFICATION AND NAMIOKA PROPERTY 3
These notions were introduced in [7]. Moreover, it was shown in [7] (Theorem
6.2(3)) that every metrically quarter-stratifiable space has the B-property.
In the connections with an investigation of properties of spaces with the B-
property the following questions were formulated in [7, question 6.3].
Question1.3. IsthereaspacewiththeB-propertywhichisnotquarter-stratifiable?
Has a (compact) space X the B-property if the set {(x,y) : y(x) = 0} is a G -set
δ
in X ×C (X)? Is a compact space with B-property metrizable?
p
Clearly that analogical questions can be formulated for spaces with L-property.
Inparticular,sinceitfollowsfrom [11,Proposition2.1]thateveryquarter-stratifiable
space has L-property,the following question arises naturally.
Question1.4. IsthereaspacewiththeL-propertywhichisnotquarter-stratifiable?
Note that it was shown in [7]([12]) that every topological space X has the B-
property (L-property) if and only if the calculation function c : X ×C (X) →
X p
R, c (x,y) = y(x), is a first Baire (Lebesgue) function. Since the space C (X)
X p
the countable chain condition, it follows from Theorem 1.1 that every compact
subspaceofacompletelyregularspacewiththeB-propertyismetrizable(thisgives
a positive answer to the third part of Question 1.3. In other words, every compact
subspace of completely regularspace with the B-property is submetrizable, that is
it can be continuously bijectively mapped on a metrizable space. Moreover, every
Lindelo¨f space with the B-property is separable [7, Theorem 6.2.(6)]. Generally,
everyLindelo¨fsubspaceM ofcompletelyregularspaceX withL-propertycontained
in the closure of some countable set A ⊆ X [13, Proposition 4.7]. Therefore, the
following question arises naturally.
Question 1.5. Let Z be completely regular space with the B-property and X ⊆Z
be a Lindel¨of subspace of Z. Does there exist a continuous bijection which defined
on X and valued in a separable metrizable space?
In this paper using dependence onsome coordinatesoffunction we givepositive
answers to Questions 1.2 and 1.5.
2. Some properties of B-spaces and L-spaces
A set G⊆X in a topologicalspace X is called functionally open, if there exists
a continuous function ϕ : X → [0,1] such that G = ϕ−1((0,1]), and a set E ⊆ X
is a functionally G -set, if E = G , where (G )∞ is a sequence of fuctionally
δ T n n n=1
n∈N
open in X sets. The sets X \G and X \E are called functionally closed and a
functionally F set respectively.
σ
The following propositions refine characteristic properties of spaces with B-
property and L-property[7, Theorem6.2.(1)] and [12, Proposition2.3], which give
the possibility to consider the calculation function only.
Proposition 2.1. Let X be a topological space. Then the following conditions are
equivalent:
(i) X has the B-property;
(ii) the set E ={(x,y):y(x)=0} is a functionally G -set in X ×C (X).
δ p
Proof. (i) ⇒ (ii). Since X has the B-property, there exists a sequence (f )∞
n n=1
of continuous functions f : X ×Y → R, where Y = C (X), which pointwise on
n p
4 V.V.MYKHAYLYUK
X converges to the separately continuous calculation function f = c . For every
X
m,n ∈ N we put G = f−1((−1, 1)) and E = G . Since the sets E
mn n m m mn S mk mn
k≥n
are functionally open and E = E , the condition (ii) is true.
T mn
m,n∈N
(ii)⇒(i). According to [7, Theorem 6.2.(1)] it enough to prove that the calcu-
lation function f =c is a first Baire class function on X×C (X).
X p
Let a,b∈ R, a< b and g : R →R be a continuous function such that g−1(0)=
(−∞,a]∪[b,+∞). Clearly that the mappings ϕ : C (X) → C (X), ϕ(y)(x) =
p p
g(y(x)), and h : X ×C (X) → X ×C (X), h(x,y) = (x,ϕ(y)), are continuous.
p p
Since f−1((−∞,a]∪[b,+∞)) = h−1(E), the set f−1((a,b)) is a functionally F -
σ
set. Therefore according to [11, Theorem 3.8] the function f is a first Baire class
function. (cid:3)
Proposition 2.2. Let X be a topological space. Then the following conditions are
equivalent:
(i) X has L-property;
(ii) the set E ={(x,y):y(x)=0} is a G -set in X ×C (X).
δ p
Proof. The implication (i) ⇒ (ii) is obvious. The implication (ii) ⇒ (i) can be
proved analogously as in the previous proposition using [12, Proposition 2.3]. (cid:3)
Remark 2.3. It follows from Proposition 2.1 and 2.2 that thesecond part of Ques-
tion 1.3 is a variant of the problem on the coincidence of Baire’s and Lebesgue’s
classifications of separately continuousfunctions of two variables. Accordingto [13,
Theorem 4.12] the product of a family (X : s ∈S) of nontrivial separable linearly
s
ordered spaces X has L-property if and only if |S| ≤ 2ℵ0. Therefore the compact
s
space [0,1][0,1] is an example of a compact space with L-property which does not
have B-property, because all compact spaces with B-property are metrizable. This
gives the negative answer to the second part of Question 1.3.
Remark 2.4. The space X = [0,1][0,1] is an example of a space with L-property
which is not a quarter-stratifiable space. This gives the negative answer to Ques-
tion 1.4. Really, assuming that X is a quarter-stratifiable space according to [7,
Theorem 2.3], we obtain that X is a metrically quarter-stratifiable space, hence has
B-property.
The following proposition refines a structure of spaces with B-property.
Proposition2.5. LetX beacompletelyregularspacewithB-property. Thenevery
onepoint set is a G -set in X.
δ
Proof. Let (f )∞ be a sequence of continuous functions f : X ×Y →R, where
n n=1 n
Y = C (X), which converges to the separately continuous calculation function
p
f = c pointwise on X and x ∈ X. Taking into account the continuity of f
X 0 n
and the countable chin conditions of Y, for every n ∈ N we construct sequences
(U )∞ of open neighborhood of x ∈ X and (V )∞ of open sets in Y such
nm m=1 0 nm m=1
∞
that the set V is dense in Y and |f (x′,y′) − f (x′′,y′′)| < 1 for every
S nm n n n
m=1
m ∈ N, x′,x′′ ∈ U and y′,y′′ ∈ V . Then |f (x,y)−f (x ,y)| ≤ 1 for every
nm nm n n 0 n
BAIRE CLASSIFICATION AND NAMIOKA PROPERTY 5
∞ ∞ ∞
x ∈ U and y ∈ Y. Thus, f(x,y) = f(x ,y) for every x ∈ U and
T nm 0 T T nm
m=1 n=1m=1
∞ ∞
y ∈Y. Since X is a completely regular space, {x }= U . (cid:3)
0 T T nm
n=1m=1
3. Quarter-stratifiable and PP-space
In this section we show that metrically quarter-stratifiable spaces coincide with
Hausdorff PP-spaces.
The notion of quarter-stratifiable space has the following equivalent reformula-
tion (see [7, Theorem 1.4]) which is approximate to the definition of PP-space.
A topological space X is a quarter-stratifiable space if and only if there exist a
sequence (U )∞ of open covers U = (U : i ∈ I ) of X and sequence (α )∞
n n=1 n n,i n n n=1
of families α = (x : i ∈ I ) of points x ∈ X such that for every x ∈ X and
n n,i n n,i
neighborhood U of x in X there exists n ∈ N such that x ∈ U if n ≥ n and
0 n,i 0
x∈U .
n,i
Remark3.1. Formetricallyquarter-stratifiablespaceX usingtheparacompactness
of (X,T) we cam choose T-locally finite T-open covers V = (V : j ∈ J )
n n,j n
of X, which inscribed in the cover U . Now putting y = x , where i ∈ I
n n,j n,i n
is an index such that V ⊆ U , and taking into account that all sets V are
n,j n,i n,j
functionally open in X, we obtain that every metrically quarter stratifiable space is
a PP-space. Moreover, according to [14, Lemma 5.1.8], analogical reasoning show
that the condition of local finiteness of partitions of the unit in the definition of
PP-space is not essential.
The following proposition show that every Hausdorff PP-space is metrically
quarter-stratifiable.
Proposition 3.2. Let X be a topological space and (h : i ∈ I) be a partition of
i
the unit on X. Then the function p : X2 → R, p(x,y) = |h (x)−h (y)|, is a
P i i
i∈I
continuous pseudo-metric on X.
Proof. Sinceh (x)≥0foreveryi∈I andx∈X and h (x)=1foreveryx∈X,
i P i
i∈I
the function p is correctly defined, moreover, p(x,y)≤2 for every x,y ∈X.
It easy to see that p satisfies the axioms od pseudo-metric. It remains to verify
that p is continuous. It enough to show that for every x ∈ X and ε > 0 the set
0
B (x )= {x∈ X : p(x ,x) <ε} is a neighborhood of x in X. We choose a finite
ε 0 0 0
set I ⊆ I such that h (x ) > 1− ε. Using the continuity of h we found a
0 P i 0 4 i
i∈I0
neighborhood U of x in X such that |h (x )−h (x)| < ε for every x ∈ U.
0 P i 0 i 4
i∈I0
Then h (x)>1− ε and
P i 2
i∈I0
ε ε ε
p(x0,x)≤ X|hi(x0)−hi(x)|+ X hi(x0)+ X hi(x)< + + =ε
4 4 2
i∈I0 i∈I\I0 i∈I\I0
for every x∈U, that is U ⊆B (x ). (cid:3)
ε 0
Proposition 3.3. Let X be a topological space and ((h : i ∈ I ))∞ be a se-
n,i n n=1
quence of partition of the unit on X. Then there exists a continuous pseudo-metric
p on X such that all function h are continuous with respect to p.
n,i
6 V.V.MYKHAYLYUK
Proof. For every x,y ∈X we put
1
p(x,y)= X 2n X|hn,i(x)−hn,i(y)|.
n∈N i∈In
It follows from Proposition 3.2 that p is a continuous pseudo-metric on X. More-
over,|h (x)−h (y)|≤2np(x,y)foreveryn∈N, i∈I andx,y ∈X. Therefore
n,i n,i n
all function h are p-continuous. (cid:3)
n,i
Corollary 3.4. Every Hausdorff PP-space is a metrically quarter-stratifiable.
4. Lindel¨of space with B-property
InthissectionwestudypropertiesofLindelo¨fsubsetsofspaceswithB-property.
Thenotionofdependenceonsomecoordinatesoffunctionsisamaintechnicaltool
in this investigation.
Let X ⊆ RS, Y be a set and f : X ×Y → R. We say that f concentrated
on a set T ⊆ S with respect to the first variable, if f(x′,y) = f(x′′,y) for every
x′,x′′ ∈ X with x′| = x′′| and y ∈ Y; and f depends on ℵ coordinates with
T T
respect to the first variable, where ℵ is an infinite cardinal, if there exists a set
T ⊆ S such that f concentrated on T and |T| ≤ ℵ. The notions of dependence
with respect to the second variable or dependence of mapping f : X → R can be
introduced analogously.
Clearly that for a Lindelo¨f space X ⊆RS every continuous function f :X →R
depends on at most countable quantity of coordinates.
The next result take an important place in the study of Lindelo¨f subsets od
spaces with B-property.
Theorem 4.1. Let X ⊆ RS be a Lindel¨of space, Z be a completely regular space,
Y =C (Z), B ⊆Y be a dense in Y set and f :X×Y →R be a function which is
p
continuous with respect to the second variable and jointly continuous at every point
of set X×B. Then f depends on countable quantity of coordinates with respect to
the first variable.
Proof. Suppose the contrary, that is for every at most countable set T ⊆ S there
exist x′,x′′ ∈ X and y ∈ Y such that x′| = x′′| and f(x′,y) 6= (x′′,y). Note
T T
that since f is continuous with respect to the second variable and B is dense in Y,
without loss of generality we can propose that y ∈B.
For every y ∈ B and ε > 0 using the joint continuity of f at every point of set
X ×{y} and the Lindelo¨f property of X we find a cover (U(y,ε,n) : n ∈ N) of X
by open basic sets U(y,ε,n) and a sequence (V(y,ε,n) : n ∈ N) of neighborhoods
of y in Y such that |f(x′,y′)−f(x′′,y′′)| < ε for every n ∈ N (x′,y′),(x′′,y′′) ∈
U(y,ε,n)×V(y,ε,n). Using the structure of basic sets in the space X ⊆ RS we
choose a countable set T(y,ε) ⊆ S such that for every n ∈ N and x′,x′′ ∈ X
the conditions x′| = x′′| and x′ ∈ U(y,ε,n) imply x′′ ∈ U(y,ε,n). Put
T(y,ε) T(y,ε)
∞
T(y) = T(y, 1). Clearly that T(y) is countable, in particular, the function
S m
m=1
f :X →R, f (x)=f(x,y), concentrated on T(y).
y y
Let ω is the first uncountable ordinal. Using the transfinite induction we con-
1
struct an increasing sequence (T : α < ω ) of countable sets T ⊆ S, a sequence
α 1 α
(ε : α < ω ) of reals ε > 0, sequences (x′ : α < ω ), (x′′ : α < ω ) and
α 1 α α 1 α 1
BAIRE CLASSIFICATION AND NAMIOKA PROPERTY 7
(y : α < ω ) of x′ ,x′′ ∈ X and y ∈ Y and a sequence (V : α < ω ) of neigh-
α 1 α α α α 1
borhoods V of y in Y such that for every α < ω the following conditions are
α α 1
satisfied:
(a) x′ | =x′′| ;
α Tα α Tα
(b) T(y )⊆T ;
α α+1
(c) |f(x′ ,y)−f(x′′,y)|>ε for every y ∈V .
α α α α
Let T ⊆ S is a countable set. Using the assumption we choose x′,x′′ ∈X and
1 1 1
y ∈ B such that f(x′,y ) 6= f(x′′,y ). Put ε = 1|f(x′,y )−f(x′′,y )|. Using
1 1 1 1 1 1 2 1 1 1 1
the continuity of f with respect to the second variable we choose a neighborhood
V of y in Y such that the condition (c) holds for α=1.
1 1
Suppose that the sequences (T : α < β), (ε : α < β), (x′ : α < β), (x′′ :
α α α α
α < β), (y : α < β) and (V : α < β), where β < ω , satisfy the conditions (a),
α α 1
(b) and (c). If β is a limited ordinal, then we put T = T . If β = γ +1 for
β S α
α<β
someat mostcountable ordinalγ, then we put T =T ∪T(y ). Further using the
β γ γ
assumption analogously as for α = 1 we find x′ ,x′′ ∈ X, y ∈ B, ε > 0 and a
β β β β
neighborhood V of y in Y such that the conditions (a) and (c) hold.
β β
For every α < ω we choose a finite set C ⊆ Z and δ > 0 such that V˜ =
1 α α α
{y ∈ Y : |y(z)−y (z)| < δ for every z ∈ C } ⊆ V . Since ℵ = |ω | is a regular
α α α α 1 1
cardinal, using Shanin’s Lemma [14, p.185] we obtain that there exist ε > 0,
0
δ > 0, a finite set C ⊆ Z and a set Γ ⊆ [1,ω ) such that |Γ| = ℵ , ε ≥ ε and
0 1 1 γ 0
δγ ≥ δ0 for every γ ∈ Γ and Cγ′ ∩Cγ′′ = C for every distinct γ′,γ′′ ∈ Γ. We
consider the continuous mapping ϕ : {y : γ ∈ Γ} → RC, ϕ(y ) = (y (z)) .
γ γ γ z∈C
Since RC is a separable metrizable space, there exists γ ∈ Γ such that |Γ | = ℵ ,
0 0 1
where Γ ={γ ∈Γ:|y (z)−y (z)|<δ for every z ∈C}.
0 γ γ0 0
Show that for every neighborhood V of y = y the set Γ(V) = {γ ∈ Γ :
0 γ0 0
V ∩V˜ = Ø} is finite. Let C′ ⊆ Z is a finite set, δ′ > 0 and V = {y ∈ Y :
γ
|y(z)−y (z)| < δ′ for every z ∈ C′}. Since Z is a completely regular space, the
0
condition V ∩V˜ = Ø for some γ ∈ Γ implies the existence of z ∈ C′∩C such
γ 0 γ γ
that |y (z)−y (z)|≥δ +δ′ >δ . It follows from the definition of the set Γ that
γ 0 γ 0 0
zγ 6∈C. Taking into account that Cγ′ ∩Cγ′′ = C for every distinct γ′,γ′′ ∈ Γ0 we
obtainthatzγ′ 6=zγ′′ foreverydistinctγ′,γ′′ ∈Γ(V). Thus,|Γ(V)|≤|C′\C|<ℵ0.
Choose m ∈ N such that 1 < ε . Since the set Γ′ = Γ(V(y , 1 ,n)) is
0 m0 0 nS∈N 0 m0
at most countable, |Γ \Γ′| = ℵ . Therefore there exist β ∈ Γ such that β > γ
0 0 0 0
and V(y , 1 ,n)∩V˜ 6= Ø for every n ∈ N. Recall that (U(y , 1 ) : n ∈ N) is
0 m0 β 0 m0
a cover of X. We take n ∈ N such that x′ ∈ U(y , 1 n )). Since (b) implies
0 β 0 m0 0
that T(y , 1 ) ⊆ T(y ) = T(y ) ⊆ T and x′ | = x′′| according to (a),
0 m0 0 γ0 β β Tβ β Tβ
x′′ ∈U(y , 1 ,n ). Now take y ∈V˜ ∩V(y , 1 ,n ). According to (c) we obtain
β 0 m0 0 β 0 m0 0
|f(x′ ,y)−f(x′′,y)|>ε ≥ε .
β β β 0
Onotherhand,accordingtothe choiceofthe setsU(y , 1 ,n )andV(y , 1 ,n ),
0 m0 0 0 m0 0
we have
1
|f(x′ ,y)−f(x′′,y)|< <ε .
β β m 0
0
Hence, we obtain a contradiction and the Theorem is proved. (cid:3)
The following result gives the positive answer to the Question 1.5.
8 V.V.MYKHAYLYUK
Theorem 4.2. Let Z be a completely regular space with B-property and X ⊆ Z
be a Lindel¨of subset of Z. Then there exist a separable metrizable space H and a
continuous bijection ϕ:X →H.
Proof. WithoutloosofgeneralitywecanassumethatZ ⊆RS. PutY =C (Z)and
p
weconsidertheseparatelycontinuousfunctiong :Z×Y →R,g(z,y)=y(z). Since
Z has B-property, there exists a sequence of continuous function g : Z ×Y → R
n
such that g(z,y) = lim g (z,y) for every (z,y) ∈ Z × Y. Put f = g | .
n n n X×Y
n→∞
According to Theorem 4.1, for every n ∈ N there exists an at most countable set
T ⊆ S such that f concentrated on T with respect to the first variable. Then
n n n
∞
the function f concentrated on the set T = T as the poinwise limit of the
S n
n=1
sequence (f )∞ .
n n=1
It remains to consider the continuous mapping ϕ: X →RT, ϕ(x)=X| . Note
T
that ϕ is a bijection with valued in the separable metrizable space H = ϕ(X).
Really if ϕ(x ) = ϕ(x ), then y(x ) = y(x ) for every y ∈ Y. Therefore x = x ,
1 2 1 2 1 2
because Z is completely regular. (cid:3)
Remark 4.3. Every countable space X has B-property, because the space C (X)
p
is metrizable. Therefore every nonmetrizable countable space is an example which
showsthatTheorem4.2cannotbestrengthenedtometrizabilityofLindel¨ofsubspace
of spaces with B-property (analogously as for compact spaces).
5. Namioka spaces and Maran spaces
We willuse the followingtwoauxiliaryresults from[15],whichgivea possibility
to use the technic of the dependence on some coordinates for the obtaining the
Namioka property.
Proposition5.1. Let Y ⊆RT beacompact space, (Z,|·−·|) beametricspace, f :
Y →Z beacontinuousmappingε≥0andsetS ⊆T suchthat|f(y′)−f(y′′)| ≤ε
Z
for every y′,y′′ ∈ Y with y′| = y′′| . Then for every ε′ > ε there exists a finite
S S
set S ⊆ S and δ > 0 such that |f(y′)−f(y′′)| ≤ ε′ for every y′,y′′ ∈ Y with
0 Z
|y′(s)−y′′(s)|<δ for every s∈S .
0
Proposition5.2. LetX beaBairespase,Y ⊆RT beacompactspace,f :X×Y →
R bea separately continuous function. Then the following statement areequivalent:
(i) f has the Namioka property;
(ii) for every open in Xnonempty set U and a real ε>0 there exist an open in
X nonempty set U ⊆U and an at most countable set S ⊆T such that |f(x,y′)−
0 0
f(x,y′′)|≤ε for every x∈U and y′,y′′ ∈Y with y′| =y′′| .
0 S0 S0
LetX beatopologicalspace.DefinetheShoquetgameonX inwhichtwoplayers
α and β participate. A nonempty open in X set U is the first move of β and a
0
nonempty open in X set V ⊆ U is the first move of α. Further β chooses a
1 0
nonemptyopeninX setU ⊆V andαchoosesanonemptyopeninX setV ⊆U
1 1 2 1
∞
and so on. The player α wins if V 6=Ø. Otherwise β wins.
T n
n=1
A topological space X is called α-favorable if α has a winning strategy in this
game. AtopologicalspaceX iscalledβ-unfavorableifβ hasnowinningstrategyin
thisgame. Clearly,anyα-favorabletopologicalspaceX isaβ-unfavorablespace. It
wasshownin[6]thatatopologicalgameX isBaireifandonlyifX isβ-unfavorable.
BAIRE CLASSIFICATION AND NAMIOKA PROPERTY 9
It is well-known (see [17]) that β-unfavorability of X is equivalent to the fact
that X is a Baire space.
The following result occupies a central place in this section.
Theorem 5.3. Let X be a Baire space, Y ⊆ RT be a compact space and f :
X ×Y → R be a separately continuous Baire measurability function. Then f has
the Namioka property.
Proof. Sincef isaBairemeasurability,thereexistsancountablefamilyF ofcontin-
uousfunctionsonX×Y suchthatthefunctionf canbeobtainedasatmostcount-
able times pointwise limit of sequence of functions from F. Let F =(f :n∈N).
n
Suppose that f has not the Namioka property. Then |T|>ℵ and according to
0
Proposition 5.2, there exist an open in X nonempty set U and ε > 0 such that
0 0
for every open in X nonempty set U ⊆U and at most countable set S ⊆T there
0
exist x∈U and y′,y′′ ∈Y such that y′| =y′′| |f(x,y′)−f(x,y′′)|>ε .
S S 0
We construct a strategy τ for the player β in the Shoquet game on the space
X. The set U is the first move of β. Let V ⊆ U is a nonempty in X set
0 1 0
which is the first move of the player α. Since the continuous function f has the
1
Namiokaproperty,accordingtoProposition5.2thereexistanopeninX nonempty
set U˜ ⊆V and countable set S ⊆T such that
1 1 1
|f (x,y′)−f (x,y′′)|<1
1 1
for every x ∈ U˜ and y′,y′′ ∈ Y with y′| = y′′| . Using the assumption we find
1 S1 S1
points x ∈U˜ and y′,y′′ ∈Y such that y′| =y′′| and
1 1 1 1 1 S1 1 S1
|f(x ,y′)−f(x ,y′′)|>ε .
1 1 1 1 0
Since f is continuous with respect to the second variable, there exists an open in
X neighborhood U ⊆U˜ of x such that
1 1 1
|f(x,y′)−f(x,y′′)|>ε
1 1 0
for every x∈U . Now we put U =τ(U ,V ), that is U is the second move of β.
1 1 0 1 1
Further, let V ⊆U be a nonempty open in X set which is the second move of
2 1
α. Choose an open in X nonempty set U˜ ⊆ V and countable set S ⊆ T which
2 2 2
contains the set S such that
1
1
|f (x,y′)−f (x,y′′)|<
2 2
2
for every x∈U˜ and y′,y′′ ∈Y with y′| =y′′| . Using the assumption and the
2 S2 S2
continuityoff with respectto the firstvariablewe chooseanopeninX nonempty
set U ⊆U˜ and points y′,y′′ ∈Y such that y′| =y′′| and
2 2 2 2 2 S2 2 S2
|f(x,y′)−f(x,y′′)|>ε
2 2 0
for every x∈U , and put U =τ(U ,V ,U ,V ).
2 2 0 1 1 2
Continuing this process to infinity we obtain sequences (U )∞ and (V )∞ of
n n=1 n n=1
openinX nonemptysetsU andV ,(y′)∞ and(y′′)∞ ofpointsy′,y′′ ∈Y and
n n n n=1 n n=1 n n
anincreasingsequence (S )∞ ofcountable sets S ⊆T suchthatfor everyn∈N
n n=1 n
the following conditions:
(a) U ⊆V ⊆U ;
n n n−1
(b) y′| =y′′| ;
n Sn n Sn
(c) |f (x,y′)−f (x,y′′)| < 1 for every x ∈ U and y′,y′′ ∈ Y with y′| =
n n n n Sn
y′′| ;
Sn
10 V.V.MYKHAYLYUK
(d) |f(x,y′)−f(x,y′′)|>ε for every x∈U ;
n n 0 n
are true.
Since the spaceX is Baire,the strategyτ isnota winning strategyforβ. Thus,
thereexistagameinwhichthe playerlosesplayingaccordingtoτ. Inotherwords,
there exist corresponding sequences with the properties (a), (b),(c) and (d) such
∞
that U 6=Ø.
T n
n=1
∞ ∞
Take a point x ∈ U and put S = S . Let a function g : X ×Y → R
0 T n 0 S n
n=1 n=1
is the pointwise limit of a subsequence (f )∞ of sequence (f )∞ . Taking into
nk k=1 n n=1
account that S ⊆ S and x ∈ U for every k ∈ N, and going to the limit
nk 0 0 nk
in the condition (c) by k → ∞, we obtain that g(x ,y′) = g(x ,y′′) for every
0 0
y′,y′′ ∈ Y with y′| = y′′| . Thus, for every pointwise limit g of a sequence
S0 S0
(f )∞ its vertical section gx0 : Y → R, gx0(y) = g(x ,y), concentrated on S .
nk k=1 0 0
Since pointwiselimitofasequence offunctions g :Y →R,whichconcentratedon
n
setS ,concentratedonS too,theverticalsectionfx0 :Y →R, fx0(y)=f(x ,y),
0 0 0
of the function f concentrated on S . According to Proposition 5.1 there exists
0
a set T ⊆ S such that |f(x ,y′) − f(x ,y′′)| ≤ ε for every y′,y′′ ∈ Y with
0 0 0 0 0
y′| = y′′| . Since the sequence (S )∞ increase, there exists an integer n ∈N
T0 T0 n n=1 0
such that T ⊆ S . Then it follows from the condition (b) and the choice T
0 n0 0
that |f(x ,y′ ) − f(x ,y′′ )| ≤ ε . On other hand, according to (d), we have
0 n0 0 n0 0
|f(x ,y′ )−f(x ,y′′ )|>ε , a contradiction. (cid:3)
0 n0 0 n0 0
The following corollary gives the positive answer to Question 1.2.
Corollary 5.4. Every Baire weakly Moran space is a Namioka space.
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