Separately continuous functions on products and its dependence on $\aleph$ coordinates

0.18 MB·

by  V. V. Mykhaylyuk

SEPARATELY CONTINUOUS FUNCTIONS ON PRODUCTS
AND ITS DEPENDENCE ON ℵ COORDINATES
6
1
V.V.MYKHAYLYUK
0
2

n Abstract. Itisinvestigatednecessaryandsufficientconditionsontopological
a spaces X = Q Xs and Y = Q Yt for the dependence of every separately
J s∈S t∈T
continuousfunctionsf :X×Y →Ronatmostℵcoordinateswithrespectto
1 thefirstorthesecondvariable.
1

]
N
1. Introduction
G
. LetX = Xs betheproductofafamilyofsetsXs,Z beasetandT ⊆S. We
h
sQ∈S
t say that a mapping f : X →Z concentrated on T, if f(x′)= f(x′′) for x′,x′′ ∈ X
a
m with x′ = x′′ . Moreover, if |T| ≤ ℵ, then we say that f depends on at most ℵ
|T |T
coordinates. LetY beaset. Wesaythata mapping g :X×Y →Z concentratedon
[
T with respect to the first variable, if the mapping ϕ:X →ZY, ϕ(x)(y)=g(x,y),
1
concentrated on T. Moreover, if |T| ≤ ℵ, then we say that g depends at most
v
on ℵ coordinates with respect to the first variable. The notion of dependence of
3
4 g :X×Y →Z atmostonℵcoordinateswithrespecttothesecondvariablecanbe
3 introduced analogously. Further ℵ means an infinite cardinal. For an abridgment
2 we shalluse the term”depends onℵ coordinates”insteadthe term”depends onat
0
most ℵ coordinates”.
.
1 LetX be atopologicalspaceandℵ beaninfinite cardinal. We saythatafamily
0 α = (A : i ∈ I) of sets A ⊆ X is locally finite, if for every x ∈ X there exists
i i
6
a neighborhood U of x such that the set {i ∈ I : A ∩U 6= Ø} is finite, poinwise
1 i
finite (ℵ-pointwise), if for every x ∈ X the set {i ∈ I : x ∈ A } is finite (has the
: i
v cardinality ≤ℵ). Moreover, for arbitrary family α=(A :i∈I) the cardinality of
i
i
X I we shall call by the cardinality of the family α.
r The following properties of a topological space X will be useful for our investi-
a gation:
(I )everylocallyfinitefamilyofopennonemptysubsetsofX hasthecardinality
ℵ
≤ℵ;
(II ) every poinwise finite family of open nonempty subsets of X has the cardi-
ℵ
nality ≤ℵ;
(III )everyℵ-pointwisefamilyofopennonemptysubsetsofX hasthecardinality
ℵ
≤ℵ.
Note that for a topological space the condition (I ) coincides with pseudo-ℵ+-
ℵ
compactness, where ℵ+ is the next after ℵ cardinal. It was developed in [1] a tech-
nic of investigation of properties of topological products which based on Shanin’s
2000 Mathematics Subject Classification. Primary54B10,54C30, 54E52.
Key words and phrases. separately continuous functions, dependence functions on ℵ coordi-
nates,pseudo-ℵ-compact, Bairespace.
1
2 V.V.MYKHAYLYUK
Lemma[2,p.185]. Usingthistechnicitwasprovedin[3](Proposition1)thattopo-
logical product has (I ), (II ) or (III ) if each its finite subproduct has the same
ℵ ℵ ℵ
property. Clearly that (III ) =⇒ (II ) =⇒ (I ).
ℵ ℵ ℵ
A topological space X is called ℵ-compact if every open covering U of X with
|U|≤ℵhassomefinitesubcover. Inparticular,ℵ -compactspaceiscalledcountably
0
compact space.
Thedependenceofcontinuousfunctionsonproductsonℵcoordinateswasinves-
tigated in papers of many mathematicians 20-th century (see [2, p.187]). N. Noble
and M. Ulmer in [1] obtained a most general result in this direction. They es-
tablished a close relation between the dependence and pseudo-ℵ-compactness. In
particular, the following proposition follows from theirs results.
Theorem1.1. [Noble,Ulmer]LetX = X beaproductofafamilyofnontrivial
s
sQ∈S
(|X | > 1) completely regular spaces X besides |S| > ℵ where ℵ is an infinite
s s i i
cardinal. Then every continuous function f : X →R depends on ℵ coordinates if
i
and only if X is pseudo-ℵ -compact where ℵ is the next after ℵ cardinal.
i+1 i+1 i
It is natural to study conditions of dependence for mappings which satisfy con-
ditions weaker than continuity. In particular,it is actual for separately continuous
mappingsthat is mappingsof manyvariablescontinuouswith respectto eachvari-
able. On the other hand, it was detected in [4] that the dependence on countable
coordinates of separately continuous functions of two variables closely connected
withthe investigationofdiscontinuitypointssetofseparatelycontinuousfunctions
of two variables each of which is the product of metrizable compacts.
Conditionsofthedependenceonℵcoordinatesofseparatelycontinuousfunctions
werestudiedin[3]. Itwasobtainedin[3]thatconditions ofdependence areclosely
connected with the conditions () , () and () similarly to continuous functions.
ℵ ℵ ℵ
It was shown in [3] (Theorem 2) that if every separately continuous function f :
X ×Y → R, where X is completely regular space, Y = Y is a product of
t
tQ∈T
nontrivial completely regular spaces Y and |T| > ℵ, depends on ℵ coordinates
t
with respect to second variable, then X ×Y has (I ) and at least one of X and
ℵ
Y has (II ). Conversely (Theorems 3,4,5), every separately continuous function
ℵ
f :X×Y →R, where X and Y are the same as in Theorem 2 but not necessarily
completely regular, depends on ℵ coordinates with respect to second variable if at
least one of the following conditions holds:
(i) X has (III ) and Y has (I );
ℵ ℵ
(ii) X has (II ) and Y is a pseudo-ℵ -compact;
ℵ 0
(iii) X is countably compact space and Y has (II ).
ℵ
In the case of countable compact completely regular spaces X and Y every
s t
separatelycontinuousfunctionf :X×Y →RontheproductofspacesX = X
s
sQ∈S
andY = Y ,where|T|>ℵ,dependsonℵcoordinateswithrespecttothesecond
t
tQ∈T
variable(ordependsonℵcoordinatesasafunctionontheproduct X × Y )
s t
sQ∈S tQ∈T
if and only if X or Y has (II ) (Theorem 6).
ℵ
Among these results attention is drawn to the following fact. We impose sym-
metrical conditions on X and Y although we study the dependence with respect
to the second variable. This follows from the symmetry of the necessary condition
of the dependence (Theorem 2 from [3]) and from the symmetry of the sufficient
SEPARATELY CONTINUOUS FUNCTIONS AND ITS DEPENDENCE ON ℵ COORDINATES3
conditions(ii)and(iii)forcountablycompactspace. Inthisconnectionthefollow-
ing questions naturally arise: is it possible to interchange the conditions on X and
Y in (i); is it possible to replace the pseudo-ℵ-compactness in (ii) and countable
compactness in (iii) to (I ); is it true the inverse proposition to Theorem 2 from
ℵ
[3]?
In this paper we show that allthese questionshave the negativeanswers. More-
over,we ostendthatthe Baireproperty playanimportantrole inthe investigation
ofdependenceonℵcoordinatesofseparatelycontinuousfunctionsdefinedonprod-
ucts. In particular, we show that for a Baire space X the first question has the
positive answer.
2. Sufficient conditions of dependence
A set S ⊆ S is called a smallest set on which a mapping f : X → Z concen-
o
trated,whereX = X ,iff concentratedonS andS ⊆S foreverysetS ⊆S
s 0 0 1 1
sQ∈S
on which f concentrated. The notion of smallest set on which a mapping concen-
trated with respect to some variable can be introduce analogously. The following
proposition(see [4, Corollary])describe the smallest sets for separately continuous
mappings.
Proposition 2.1. Let X = X be the topological product of a family of topolog-
s
sQ∈S
ical spaces X , Y be a set, Z be a Hausdorff topological space and f : X ×Y →Z
s
be a mapping continuous with respect to the first variable. Then the set
S ={s∈S :(∃y ∈Y)(∃u,v ∈X)(u| =v| and f(u,y)6=f(v,y))}
0 S\{s} S\{s}
is a smallest set on which f concentrated with respect to the first variable.
The following result give us the possibility to replace a space Y with (III ) to
ℵ
the Cantor’s cube.
Theorem 2.2. Let X be a topological space, Y = Y be the product of a family
t
tQ∈T
of topological spaces Y with (III ) and f : X ×Y →R be a separately continuous
t ℵ
functionwhich does not dependon ℵ coordinates with respecttothesecondvariable.
Then there exist a set S ⊆ T |S| > ℵ and separately continuous function g :
X ×Y → R, where Y = {0,1}S, which does not depend on ℵ coordinates with
1 1
respect to the second variable.
Proof. According to Proposition 2.1, the set
T ={t∈T :(∃x∈X)(∃y,z ∈Y)(y| =z| and f(x,y)6=f(x,z))}
0 T\{t} T\{t}
is a smallest set on which f concentrated with respect to the second variable. It
follows from the conditions of Theorem that |T | > ℵ. For every t ∈ T we choose
0 0
points x ∈ X and y ,z ∈ Y, which featured in the definition of T . Using the
t t t 0
continuityoff withrespecttothesecondvariablewetakebasicopenneighborhoods
V˜ = V˜(t) and W˜ = W˜(t) of y and z in Y such that f(x ,y)6=f(x ,z) for
t s t s t t t t
sQ∈T sQ∈T
every y ∈V˜ and z ∈W˜ . Put V(t) =V˜(t) W˜(t) for every s∈T \{t}, V(t) =V˜(t)
t t s s s t t
and V = V(t). Note that V is an openTneighborhoodof y , V ⊆V˜. Moreover,
t s t t t t
sQ∈T
4 V.V.MYKHAYLYUK
for every y ∈V the point z ∈Y which defined by
t
y(s), if s∈T \{t};
z(s)=
(cid:26) zt(t), if s=t,
belongs to W˜ . Thus, f(x ,y)6=f(x ,z).
t t t
We consider the family V = (V : t ∈ T ). Since Y has (III ) and |T | > ℵ, the
t 0 ℵ 0
family V is not ℵ-pointwise. Therefore there exist a set S ⊆ T and point y∗ ∈ Y
0
such that |S| > ℵ and y∗ ∈ V for all s ∈ S. For every s ∈ S we put a = y∗(s),
s s
b =z (s) and
s s
y∗(t), if t∈T \{s};
z∗(t)=
s (cid:26) bt, if s=t.
Note that y∗| = z∗| and f(x ,y∗) 6= f(x ,z∗). Moreover, the space
T\{s} s T\{s} s s s
Y∗ = {a ,b }× {y∗(t)} is homeomorphic to the space {0,1}S, besides the
t t
tQ∈S t∈QT\S
restrictionf∗ =f isaseparatelycontinuousmapping,forwhichSisasmallest
|X×Y∗
set on which f∗ concentrated with respect to the second variable. It remains to
take the mapping ϕ:{0,1}S →Y∗, which defined by the formula
a , if t∈S and a(t)=0;
t
ϕ(a)(t)= bt, if t∈S and a(t)=1;
 y∗(t), if t∈T \S,
and to consider the function g : X ×Y → R, where Y = {0,1}S and g(x,a) =
1 1
f∗(x,ϕ(a)). (cid:3)
For a basic open set V = V in a topological product Y = Y and a set
t t
tQ∈T tQ∈T
B ⊆T the set V we denote by V| and the set {t∈T :V 6=Y } we denote by
t B t t
tQ∈B
R(V).
We shalluse the followingresultwhichisa reformulationofShanin’s Lemma [2,
p. 185] for isolated cardinals.
Proposition 2.3. Let (A :i∈I) be a family of finite sets A with |I|>ℵ. Then
i i
there exist a finite set B ⊆ A and set J ⊆I such that |J|>ℵ and A A =B
i i j
iS∈I T
for arbitrary distinct i,j ∈J.
Now we consider functions on the product Baire space and Cantor’s cube.
Theorem 2.4. Let X be a Baire space with (I ) and Y = {0,1}T. Then every
ℵ
separatelycontinuousfunctionf :X×Y →Rdepends onℵcoordinateswithrespect
to the second variable.
Proof. Suppose the contrary, that is there exists a separately continuous function
f :X×Y →Rwhichdoesnotdependsonℵcoordinateswithrespecttoy. Clearly
that |T|>ℵ. Then according to Proposition 2.1 we have |T |>ℵ where
0
T ={t∈T :(∃x∈X)(∃y,z ∈Y)(y| =z| and f(x,y)6=f(x,z))}.
0 T\{t} T\{t}
For every n∈N we put
1
T ={t∈T :(∃x∈X)(∃y,z ∈Y)(y| =z| and |f(x,y)−f(x,z)|> )}.
n T\{t} T\{t} n
SEPARATELY CONTINUOUS FUNCTIONS AND ITS DEPENDENCE ON ℵ COORDINATES5
∞
Clearly that T = T . Since ℵ is infinite cardinal, there exists integer n such
0 n 0
nS=1
that |T |>ℵ.
n0
We denote S = T , ε = 1 and for every t ∈ S choose points x˜ ∈ X and
n0 n t
y ,z ∈ Y, which featured in the definition of T . Using the continuity of f with
t t n0
respect to x for every s ∈ S we choose an open neighborhood U˜ of x˜ in X such
s s
that |f(x,y )−f(x,z )| > ε for every x ∈ U˜ . Note that the Cantor’s cube Y is a
t t s
co-Namioka space (see [5, Corollary 1.2 or Theorem 2.2]), that is for every Baire
space X′ and every separately continuous function g : X′×Y → R there exists a
dense G -set A in X′ such that g is jointly continuous at every point of set A×Y.
δ
Therefore in every open nonempty set U˜ there exists a point x such that f is
s s
jointly continuous at points (x ,y ) and (x ,z ). For every s∈S we take an open
s s s s
neighborhoodU of x in X andbasic open neighborhoodsV andW of y andz
s s s s s s
in Y such that R(V )=R(W ), V | =W | and |f(x,y)−f(x,z)|>ε for
s s s T\{s} s T\{s}
every x∈U , y ∈V and z ∈W .
s s s
We consider the family (R(V ) : s ∈ S) which consists of finite sets R(V ).
s s
According to Proposition 2.3 there exist a finite set B ⊆ S and a set S ⊆ S such
1
that |S | > ℵ, B S = Ø and R(V ) R(V ) =B for arbitrary distinct s,t ∈S .
1 1 s t 1
Sinceset{0,1}BTisfiniteandallsetsVsT|B arenonempty,wheres∈S1,thereerxists
y′ ∈{0,1}B such that the set S ={s∈S :y′ ∈V | } has the cardinality >ℵ.
0 1 s B
We show that the family (U :s∈S ) is locally finite in X. Let x ∈X. Using
s 0 0
thecompactnessofY andcontinuityofthefunctionfx0 :Y →R,fx0(y)=f(x0,y),
we choose a finite set B ⊆T such that |f(x ,y)−f(x ,z)|<ε for every y,z ∈Y
0 0 0
with y| =z| . Consider the points y ,z ∈Y which defined by
B0 B0 0 0
y′(t), if t∈B;
y0(t)= ys(t), if t∈R(Vs)\B, s∈S0\B0;
 0, otherwise,

y′(t), if t∈B;
z0(t)= zs(t), if t∈R(Vs)\B, s∈S0\B0;
 0, otherwise.
Note that accordingto the choice of sets B andS , the sets R(V )\B are pairwise
1 s
disjoint for s ∈ S . Therefore the definitions of y and z are correct. Since
1 0 0
y | =z | for every s∈S , y (t)=z (t) for every t∈B and s∈S \B .
s S\{s} s S\{s} 0 s s 0 0 0
Therefore y | = z | and |f(x ,y )−f(x ,z )| < ε. Using the continuity of f
0 B0 0 B0 0 0 0 0
with respect to x we choose a neighborhood U of x in X such that |f(x,y )−
0 0 0
f(x,z )|<ε for every x∈U .
0 0
On other hand, since S B = Ø, V | = W | . Therefore y′ ∈ W | for
0 s B s B s B
every s ∈ S0. Moreover, y0|TR(Vs)\B = ys|R(Vs)\B and z0|R(Vs)\B = zs|R(Vs)\B for
all s ∈ S \B . Taking into account that y ∈ V and z ∈ W we obtain that
0 0 s s s s
y | ∈ V | and z | ∈ W | . Therefore y ∈ V and z ∈ W ,
0 R(Vs) s R(Vs) 0 R(Vs) s R(Vs) 0 s 0 s
because R(W ) = R(V ) for every s ∈ S \B . According to the choice of U , V
s s 0 0 s s
and W , we have |f(x,y )−f(x,z )|>ε for every x∈U and s∈S \B .
s 0 0 s 0 0
Thus, U U = Ø for every s ∈S \B . Therefore the set {s ∈S : U U 6=
s 0 0 0 0 s
Ø} is finiteTand the family (Us :s∈S0) is locally finite in X. But this contTradicts
to the fact that X has (I ), because |S |>ℵ. (cid:3)
ℵ 0
Now Theorems 2.2 and 2.4 imply the main result of this section.
6 V.V.MYKHAYLYUK
Theorem 2.5. Let X be a Baire space with (I ), Y = Y be the product of a
ℵ t
tQ∈T
family of topological spaces Y which has (I ). Then every separately continuous
t ℵ
functionf :X×Y →Rdependsonℵcoordinateswithrespecttothesecondvariable.
3. Essentiality of Baireness
In this section we show that the condition of Baireness of X in Theorem 2.5 is
essential.
Proposition3.1. LetX beatopological space, U bean infinitelocally finitefamily
of open nonempty sets in X. Then there exists a family V of nonempty pairwise
disjoint open in X sets such that |V|=|U|.
Proof. Let U = (U : α< β), where β is the first ordinal with the cardinality |U|.
α
We construct the family V inductively.
Take any point x ∈ U and choose an open neighborhood V of x such that
1 1 1 1
the set A ={α<β :V U 6=Ø} is finite.
1 1 α
Assume that pairwiseTdisjoint sets Vξ for ξ < α < β are constructed such that
all sets A = {γ < β : V U 6= Ø} are finite. Note that | A | < |β|. Really,
ξ ξ γ ξ
T ξS<α
if α is a finite ordinal, then | A | < ℵ ≤ |β|. If α is an infinite ordinal then
ξ 0
ξS<α
| A |≤ℵ ·|α| =|α|<|β|. Therefore there exists γ <β such that γ 6∈ A ,
ξ 0 ξ
ξS<α ξS<α
thatis U ( V )=Ø. Takea point x ∈U and chooseanopen neighborhood
γ ξ γ γ
T ξS<α
V of x such that the set A = {ξ < β : V U 6= Ø} is finite. The family
α γ α α ξ
V =(Vα :α<β) is the required. T (cid:3)
Note that in the case ofregular space X we can constructa discrete family V of
corresponding cardinality.
Proposition 3.2. Let a topological space X has (II ), Y be a dense in X set.
ℵ
Then the subspace Y of X has (I ).
ℵ
Proof. . Let U be a locally finite family of nonempty open in Y sets. According to
Proposition3.1,there existsafamilyV =(V :i∈I)ofpairwisedisjointnonempty
i
open in Y sets V such that |I| = |U|. For every i ∈ I we choose an open set U
i i
in X such that U Y = V . Note that the family (U : i ∈ I) consists of pairwise
i i i
disjoint sets. SinceTX has (IIℵ), |I|≤ℵ, that is |U|≤ℵ. Thus, Y has (Iℵ). (cid:3)
Example 3.3. Let Y =[0,1]S, X =C (Y)bethespace of continuousfunctions on
p
Y with the topology of pointwise convergence and f :X×Y →R be the calculation
function, that is f(x,y)=x(y).
Clearly that f is a separately continuous function. Since each finite power [0,1]n
has (III ), according to Proposition 1 from [3], the space Y has (III ). Analo-
ℵ ℵ
gously the space RY has (II ). Taking into account the density X in RY and using
ℵ
Proposition 3.2 we obtain that X has (I ). Fix s ∈ S. Denote by x the func-
ℵ0 s
tion from X, which defined by x (y) = y(s). Choose points y ,z ∈ Y such that
s s s
y | = z | and y (s) 6= z (s), that is f(x ,y ) 6= f(x ,z ). Thus, accord-
s S\{s} s S\{s} s s s s s s
ing to Proposition 2.3, we have s ∈ S , where S is a smallest set on which f
0 0
concentrated with respect to y. Hence, S =S.
0
SEPARATELY CONTINUOUS FUNCTIONS AND ITS DEPENDENCE ON ℵ COORDINATES7
Taking S such that |S|>ℵ we obtain an example of separately continuous func-
tion defined on the product of spaces X and Y, where X has (I ) and Y has (III ),
ℵ ℵ
which does not depends on ℵ coordinates with respect to y. Thus, the condition of
Baireness of X in Theorem 2.4 can not be replaced by (I ) or even by (I ).
ℵ ℵ0
4. Construction some spaces
In this section we consider some spaces. They will be used in a construction of
examples which show the essentiality of corresponding properties of X and Y in
the sufficient conditions of dependence on ℵ coordinates of separately continuous
functions. Moreover,we obtain a relations between these properties.
Proposition 4.1. Every Hausdorff pseudo-ℵ -compact space is Baire.
0
Proof. Let X be a pseudo-ℵ -compact space, that is every locally finite family of
0
open nonempty sets in X is finite. We show that X is Baire.
Let U be an open nonempty set in X. Suppose that U is a meager set. Then
0 0
there exists a sequence (F )∞ of nowhere dense closed sets F ⊆ X such that
n n=1 n
∞
U ⊆ F . It easy to construct a decreasing sequence (U )∞ of open in X
0 n n n=1
nS=1
nonempty sets U such that U ⊆ U and U F = Ø for every n ∈ N. Note
n n n−1 n n
that T
∞ ∞ ∞ ∞ ∞
∩ U = ∩ U =( ∩ U )∩U =( ∩ U )∩( ∪ F )=
n n n 0 n n
n=1 n=1 n=1 n=1 n=1
∞ ∞ ∞
= ∪ (( ∩ U )∩F )⊆ ∪ (U ∩F )=Ø.
n m n n
m=1 n=1 n=1
Therefore the sequence (U )∞ is locally finite. Thus, the sequence (U )∞ is
n n=1 n n=1
locally finite too. But this contradicts to pseudo-ℵ -compactness of X. Hence, U
0 0
is of second category in X. Thus, X is a Baire space. (cid:3)
Recall that for every function f :X →R the set suppf ={x∈X :f(x)6=0} is
called the support of function f.
Let P = {x ∈ [0,1]ℵ+ : |suppx| ≤ ℵ}, that is P is the subspace of Tikhonoff
ℵ ℵ
cube with the weight ℵ+, which consists of all functions with the support of the
cardinality <ℵ+.
Proposition 4.2. a) P has (II );
ℵ ℵ0
b) P does not have (III );
ℵ ℵ
c) Pℵ has (IIIℵ′) for every infinite cardinal ℵ′ 6=ℵ;
d) P is ℵ-compact.
ℵ
Proof. Let S be a set such that |S|=ℵ+ and P ⊆[0,1]S.
ℵ
a). Let U = (U : i ∈ I) be a family of basic open nonempty sets U in P and
i i ℵ
|I| > ℵ . For every i ∈ I we denote by U˜ such basic open nonempty set in [0,1]S
0 i
such that U = U˜ P . We consider the family (B : i ∈ I), where B = R(U˜).
i i ℵ i i i
According to PropTosition 2.3, there exist a finite set B ⊆ S and an uncountable
set I ⊆ I such that B B = B for every distinct i,j ∈ I . The space [0,1]B
1 i j 1
has (IIℵ0). Therefore theTfamily (Vi : i ∈ I1), where Vi = U˜i|B, is not pointwise
finite. Thus, there exist x ∈ [0,1]B and countable set I ⊆ I such that x ∈ V
0 0 1 0 i
foe every i∈I . For every i∈I choose a point x ∈U˜ and consider the function
0 0 i i
8 V.V.MYKHAYLYUK
x:S →[0,1], which defined by:
x (s), if s∈B;
0
x(s)= xi(s), if s∈Bi\B, i∈I0;
 0, if s∈S\( Bi).
i∈SI0
SinceB B =B,(B \B)(B \B)=Øforarbitrarydistincti,j ∈I . Therefore
i j i j 0
the functTion x is defined cTorrectly. Moreover, suppx ⊆ Bi and | Bi| ≤
i∈SI0 i∈SI0
|I |·ℵ ≤ℵ. Thus, |suppx|≤ℵ. Hence, x∈P . Note that x| = x ∈V =U˜ |
0 0 ℵ B 0 i i B
and x| = x | ∈ U˜| . Therefore, x| ∈ U˜| , that is x ∈ U˜ for every
Bi\B i Bi\B i Bi\B Bi i Bi i
i∈I . Thus, x∈U foreveryi∈I andthefamily U isnotpointwisefinite. Thus,
0 i 0
P has (II ).
ℵ ℵ0
b). For every s ∈ S put U = {x ∈ P : x(s) > 0}. Clearly that the family
s ℵ
(U :s∈S) is ℵ-pointwise in P , besides |S|>ℵ. Thus, P does not have (III ).
s ℵ ℵ ℵ
c). Let ℵ′ be an infinite cardinal which not equals to ℵ and U = (U : i ∈I) be
i
a family of nonempty open basic sets in P , besides |I| = ℵ′+. For every i ∈ I we
ℵ
choose a basic open set U˜ in [0,1]S such that U =U˜ P and put B =R(U˜).
i i i ℵ i i
Firstly we consider the case of ℵ′ < ℵ. Put B = TBi. The space [0,1]B has
iS∈I
(IIIℵ′), therefore the family (Vi : i∈I), where Vi =U˜i|B, is not ℵ′-pointwise, that
is there exist x∈[0,1]B and I ⊆I such that |I |>ℵ′ and x∈V for every i∈I .
1 1 i 1
Define a point y ∈[0,1]S by:
x(s), if s∈B;
y(s)=
(cid:26) 0, if s∈S\B.
Since suppy ⊆B and |B|≤|I|·ℵ ≤ℵ′+·ℵ ≤ℵ, y ∈P . Now it easy to see that
0 0 ℵ
y ∈Ui fpr every i∈I1. Thus, U is not ℵ′-pointwise. Hence, Pℵ has (IIIℵ′).
Nowletℵ′ >ℵ+. WeconsiderthesystemAofallfinitesubsetsofsetS. Clearly
that |A|=|S|=ℵ+. For every A∈A we put I ={i∈I :B =A}.
A i
Suppose that |I | ≤ ℵ′ for every A ∈ A. Then |I| = | I | ≤ |A|·ℵ′ =
A A
AS∈A
ℵ+ ·ℵ′ ≤ ℵ′. But this contradicts to |I| = ℵ′+. Thus, there exist A ∈ A and
0
I ⊆I such that|I |>ℵ′ and B =A for every i∈I .
0 0 i 0 0
Takingintoaccountthat[0,1]A0 has(IIIℵ′)weobtainthatthereexistx∈[0,1]A0
andI ⊆I suchthat|I |>ℵ′andx∈U˜ . Thenapointy ∈[0,1]S whichdefined
1 0 1 i|A0
by:
x(s), if s∈A ;
y(s)= 0
(cid:26) 0, if s∈S\A0,
belongs to P , besides y ∈U for every i∈I .
ℵ i 1
Thus, U is not ℵ′-pointwise and Pℵ has (IIIℵ′).
d). This assertion follows from the next fact: the closure in P of any set with
ℵ
the cardinality ≤ℵ is compact. (cid:3)
This proposition implies that all properties (III ) are not comparable,although
ℵ
it easy to see that (Iℵ) =⇒ (Iℵ′) and (IIℵ) =⇒ (IIℵ′) for ℵ < ℵ′. Moreover, the
discrete space with the cardinality ℵ shows that the implication (IIIℵ) =⇒ (Iℵ′)
is not true for ℵ > ℵ′. The Aleksandroff compactification (see [2, p. 261]) of
discrete space with the cardinality ℵ+ show that the compactness does not imply
any properties (II ) and (III ).
ℵ ℵ
SEPARATELY CONTINUOUS FUNCTIONS AND ITS DEPENDENCE ON ℵ COORDINATES9
Thus, the properties which featured in the investigation of dependence on ℵ
coordinates of separately continuous functions can be represented as the following
table. NotethattheimplicationsfromProposition4.1aretrueforHausdorffspaces
and all non-marked implications are not true.
countably ⇒ pseudo−ℵ0− ⇒ (I ) ⇒ (I ) ⇒ ... ⇒ (I ) ⇒ ...
compact. compact. ℵ0 ℵ1 ℵ
⇑ ⇓ ⇑ ⇑ ⇑
ℵ− compact. Baire (II ) ⇒ (II ) ⇒ ... ⇒ (II ) ⇒ ...
ℵ0 ℵ1 ℵ
⇑ ⇑ ⇑ ⇑
compact. (III ) (III ) ... (III ) ...
ℵ0 ℵ1 ℵ
We will use the following auxiliary result in investigation of properties of the
second multiplier.
Proposition 4.3. Let X be a topological space with (II ), U = (U : i ∈ I) be a
ℵ i
family of open nonempty sets in X with |I| > ℵ. Then there exists x ∈ X such
0
that for every neighborhood U of x in X the set {i ∈ I : U U 6= Ø} has the
0 i
cardinality >ℵ. T
Proof. Suppose the contrary. That is for every x ∈ X there exists an open neigh-
borhood V of x such that the cardinality of the set {i ∈ I : U V 6= Ø} is
x i x
≤ℵ. T
Using transfinite induction we construct a family (G : α < β), where β is the
α
firstordinalofthecardinality|I|,ofopennonemptypairwisedisjointsetsG . This
α
contradicts to (II ) of X.
ℵ
We take i ∈ I and x ∈ U and put G = V U . Clearly that the set
1 1 i1 1 x1 i1
I1 ={i∈I :Ui G1 6=Ø} has the cardinality is ≤ℵ.T
Assume thatTthe sets Gα for α < γ < β are constructed, moreover all sets
I = {i ∈ I : U G 6= Ø} have the cardinality is ≤ ℵ. Note that | I | ≤
α i α α
α<γ
T S
|γ|·ℵ ≤ ℵ2 = ℵ < |I|. Therefore there exists i ∈ I such that U G = Ø for
γ iγ α
every α < γ. Take a point xγ ∈ Uiγ and put Gγ = Uiγ Vxγ. ClTearly that the
cardinality of Iγ is ≤ℵ. T (cid:3)
Denote by D the discrete space of the cardinality ℵ+. Let Q = D {∞},
ℵ ℵ ℵ
besides neighborhoods of ∞ in Qℵ are all sets A {∞}, where A⊆Dℵ andS|Dℵ\
A|≤ℵ. S
Proposition 4.4. a) Q has (I );
ℵ ℵ
b) every subfamily of ({d} : d ∈ D ) with the cardinality ℵ is locally finite in
ℵ
Q ;
ℵ
c) P ×Q has (I ).
ℵ ℵ ℵ
Proof. a). LetU =(U :i∈I)bealocallyfinitefamilyofnonemptyopeninQ sets
i ℵ
U . We choose a neighborhood U of ∞ such that the set J ={i∈I :U U 6=Ø}
i i
is finite. The set B = Dℵ\U has the cardinality ≤ ℵ. Moreover, for evTery b ∈ B
the set I ={i∈I :b∈U } is finite. Since I =J ( I ), |I|≤ℵ +ℵ·ℵ ≤ℵ.
b i b 0 0
S bS∈B
b). It follows immediately from the definition of D .
ℵ
c). Let W = (W : i ∈ I) be a locally finite family of nonempty basic open sets
i
W = U ×V in R = P ×Q . We may assume that V = {d }, where d ∈ D .
i i i ℵ ℵ i i i ℵ
For every d ∈ D we put I = {i ∈ I : V = {d}}. Clearly that the sets I are
ℵ d i d
pairwise disjoint. Since the family W is locally finite in R, for every d ∈ D the
ℵ
10 V.V.MYKHAYLYUK
family (U : i ∈ I ) is locally finite in P . Since P has (II ), |I | ≤ ℵ for every
i d ℵ ℵ ℵ0 d 0
d∈D .
ℵ
Now we show that the set B ={d∈D :I 6=Ø} has the cardinality ≤ℵ.
ℵ d
Suppose that |B| > ℵ. For every b ∈ B we choose i ∈ I and put U = U ,
b b b ib
V = V . We consider the family (U : b ∈ B). Since P has (II ), according
b ib b ℵ ℵ
to Proposition 4.3 there exists p ∈ P such that for every neighborhood U of
ℵ
p the set B = {b ∈ B : U U 6= Ø} has the cardinality > ℵ. For every
U b
neighborhood V of ∞ in Qℵ wTe have |Dℵ \V| ≤ ℵ. Therefore, |BU \V| ≤ ℵ,
thus, the set B = B V has the cardinality > ℵ. For every b ∈ B we have
0 U 0
Ub U 6=Ø and Vb ={bT}⊆V. Hence, B0 ⊆{b∈B :(U ×V)(Ub×Vb)6=Ø} and
{ibT: b ∈ B0} ⊆ {i ∈ I : (U ×V) Wi 6= Ø} = I′. Since all indexes ib are distinct
for b∈B0, because Vib =Vb ={bT},the set I′ is infinite. Thus, the family W is not
locally finite in the point (p,∞), a contradiction.
Thus, |B| ≤ ℵ. Then, taking into account that I ⊆ I and |I | ≤ ℵ we
b b 0
bS∈B
obtain that |I|≤ℵ ·|B|≤ℵ ·ℵ=ℵ.
0 0
Thus, R has (I ). (cid:3)
ℵ
5. Essentiality of some sufficient conditions
Firstly we generalize a construction of separately continuous functions from [3].
Families U = (U : i ∈ I) and V = (V : i ∈ I) of sets U and V in topological
i i i i
spacesX and Y respectively are calledconcerted, if for everyx∈X and y ∈Y the
families V = (V : i ∈ Ix) and U = (U : i ∈ I ), where Ix = {i ∈ I : x ∈ U }
x i y i y i
and I = {i ∈ I : y ∈ V }, are locally finite in Y and X respectively. Clearly that
y i
pointwise finite families U and V are concerted.
Theorem 5.1. Let X, Y be topological spaces, (ϕ : i ∈ I), (ψ : i ∈ I) be
i i
families of continuous functions ϕ :X →R and ψ :Y →R such that the families
i i
U = (suppϕ : i ∈ I) and V = (suppψ : i ∈ I) are concerted, (f : i ∈ I) be
i i i
a family of separately continuous functions f : X ×Y → R. Then the function
i
f :X ×Y →R, which defined by formula
f(x,y)= ϕ (x)ψ (y)f (x,y),
i i i
Xi∈I
is separately continuous.
Proof. Fix x ∈ X and put I = {i ∈ I : x ∈ suppϕ }. Since the fami-
0 0 0 i
lies U and V are concerted, the family (suppψ : i ∈ I ) is locally finite on
i 0
Y. Therefore the locally finite sum ϕ (x )ψ (y)f (x ,y) of continuous on Y
i 0 i i 0
iP∈I0
functions ϕ (x )ψ (y)f (x ,y) is continuous on Y. On the other hand, f(x ,y) =
i 0 i i 0 0
ϕi(x0)ψi(y)fi(x0,y). Thus, the function fx0 is continuous on Y.
iP∈IThe continuity of the function f with respect y can be verify analogously. (cid:3)
Theorem 5.2. Let Z =[0,1]S, where |S|=ℵ+, X =P , Y =Q ×Z, X =Q ,
1 ℵ 1 ℵ 2 ℵ
Y =P ×Z. Then there exists a function f :P ×Q ×Z →R which separately
2 ℵ ℵ ℵ
continuous on X ×Y and on X ×Y and such that S is a smallest set on which
1 1 2 2
f concentrated with respect to z.
Proof. Let P = {x ∈ [0,1]S : |suppx| ≤ ℵ} and Q = S {∞}. We consider
ℵ ℵ
the families (ϕs : s ∈ S), (ψs : s ∈ S) and (fs : s ∈ S) ofScontinuous functions