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3ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
ON THE PRESERVING OF CENTERED OPEN SET SYSTEMS UNDER THE ACTION OF A FUNCTOR O( X)
Mamadaliev N. K., Manasypova R. Z.
National University of Uzbekistan named after Mirzo Ulugbek DOI: 10.31618/ESSA.2782-1994.2022.1.84.305
Introduction
Nowadays, most scientific and practical research at the world level is reduced to the study of problems in the theory of semi-additive functionals and covariant functors. Comparison of cardinal invariants of the space of semi-additive functionals with compact supports is the object of research in such areas as functional analysis, geometry, and topology. The study of cardinal and topological properties of the space of semi-additive functionals is one of the most important problems of the theory of cardinal and topological properties of various spaces, such as general topology, spaces of weak functionals, algebraic topology, the theory of cardinal invariants, and the theory of covariant functors.
In the works of E. V. Shchepin [1], a very advanced and very meaningful general theory of covariant functors is constructed. He identified a number of natural and non-restrictive properties of a functor and defined a normal functor. T.Radul [2] began to investigate the functor O of weakly additive order-preserving normed functionals in the category of compacts and their continuous maps. T.Radul [2] showed that the functor O : Comp ^ Comp is not normal and, therefore, the study of this functor is much more complicated than that of normal functors. The appearance of this functor is due to the recent development of the theory of nonlinear functionals. In addition, P is the probability measure functors, exp
is the exponential functor, and X is the superextension functor are subfunctors of the functor O.
D. E. Davletov [3] investigated weakly additive, order-preserving, normed, positive-homogeneous, and
semi-additive functionals defined on the space C ( X )
of all continuous real functions of a compact X set , and studied the algebraic dimension of the space of semi-additive functionals.
In recent years, such scientists as R.B. Beshimov, A.A. Zaitov, N.K. Mamadaliev and F.G. Mukhammadiev have been working closely on the development of the theory of covariant functors in Uzbekistan.
In particular, it was proved in [4], that an arbitrary product of r -bounded spaces is r -bounded and vice versa, that the r -boundedness property is preserved by continuous maps. In particular, continuous maps preserve r the-boundedness of topological spaces. In addition, the authors of [4] investigated the behavior of minimal tightness and functional tightness of
topological spaces under the influence of an exponential functor of finite degree. It was shown that this functor preserves functional tightness and minimal tightness of compacts.
In [5], the action of the functors exp, exp , and exp was investigated on finally compact, pseudocompact, extremely disconnected spaces and N -spaces. The authors studied some topological properties of a uniform space and its hyperspace. It was proved that if a uniform space (X, U) is uniformly
paracompact, then (exp, exp , exp) is uniformly paracompact. It is also shown that if a uniform space (X, U) is uniformly R -paracompact, then a uniform
space (exp X, exp U) is uniformly R -paracompact.
In [6], the notion of a functor OSa of semiadditive a -smooth functionals was introduced into the category of Tychonoff spaces Tych , which continues the functor OS: Comp ^ Comp of semiadditive functionals. It is proved that the functor OSa : Tych ^ Tych transforms Z -embeddings
into embeddings and that the space OSa ( X) is closed in the space of weakly additive a -smooth functionals, in particular, OSa (X) is complete according to Hewitt for any Tychonoff space X e Tych .
In this paper, we investigate the acting of the functor O on centered and linked systems of a space X , and also prove that the weight of the space X is conserved under the action of a functor P . Moreover,
it is proved that the functor of idempotent probability measures of finite support preserves the density of compact spaces.
Preliminary information
Let ^ = {G,M } be a family of elements of two
varieties. Elements of Gare called objects, and elements of M are called morphisms. For each morphism f a unique ordered pair X,Yof objects is defined, and f is called a morphism from X to Y. In this case X is denoted by dom f, and Y : rng f .
The family of all morphisms from X and Y is denoted by [X,Y] .
I IB
A family g = ( 6, M } is called a category [7] if the following conditions hold:
a) for each pair of morphisms f and g with
rng f = dom g , a unique morphism h with dom h = dom f and rng h = rng g is defined. It is called the composition of morphisms f and g and denoted by g ° f :
b) (hogJo f = ho(go f) for every triple of morphisms with rng f = dom g and rngg = domh.
Let g = (6, M ) and g = (6, M) be two
categories. A map F : g ^ g ,that transfers objects into objects and morphisms into morphisms is called a covariant functor [7] from category g to category g if:
1) for every morphism f : X ^ Y in the category g , the morphism F( f) acts from F(X ) in F(Y).
2) F(idx) = idP(x) for any X e 6.
3) F(fog) = F(f)oF(g).
Note that in this paper, by "functor" we mean a covariant functor.
A functor F : Comp ^ Comp A is called weight-preserving if coF(X) = a>(X) for any
infinite compact X .
A functor F : Comp ^ Comp is called
monomorphic if for any embedding i of a compact X into a compact Y, the map F(i) : F(X) ^ F(Y) is also an embedding.
A functor F : Comp ^ Comp is called epimorphic if preserves the surjective maps of compacts.
A functor F : Comp ^ Comp is called intersection-preserving if for any family (Xa : a e Q } of closed subsets of an arbitrary compact, we have
n(F(Xa ): a eQ} = F( ^(Xa :aeQ})
A functor F : Comp ^ Comp is called preimage-preserving if for any continuous map f of a compact Y and for every closed subset B c Y is
F(f 'B) = F(f)~'F(B).
A covariant functor F : Comp ^ Comp is called normal if it is continuous, preserves weight, intersections and preimages, is monomorphic and epimorphic, and transfers singletons into singletons, and an empty set into an empty set.
A covariant functor F : Comp ^ Comp is called weakly normal if it satisfies all the normality conditions except for preserving preimages.
Let X be a compact, F : Comp ^ Comp be a functor and x e F(X) . The degree of a point x (denoted by deg( x)) is the smallest natural number n such that x e F(f)(F(K)) for a map f : K ^ X of a n -point space K . The degree of a
functor F (denoted by deg F ) is a finite or infinite number such that
deg F = sup{deg(x): x e F(X), and X -compact}. The conditions of a preserving intersections and preimages ensure that the functor F is connected to the exponent exp .
A closed subset suppF(x) c X such that the relations A 3 suppF (x) and x e F(A) are equivalent for any closed set A c X is called the support of the point x e F(X) .
The condition for preserving preimages for an intersection-preserving functor is equivalent to the condition for preserving carriers under maps, i.e., the condition that for any map f and any x the relation holds
f (supp(x)) = supp F( f )(x) .
Thus for a preimage-preserving and intersection-preserving functor F : Comp ^ Comp, the support
suppF (x) performs a natural transformation of this functor F : Comp ^ Comp into an exponent exp, i.e.
suppF(x): F(X) ^ exp X
V. N. Basmanov proposed a modification of the degree of the functor. Here is this modification.[8]
For a functor F : Comp ^ Comp through Fn
we denote a functor that corresponds to a space X the set of all those elements a e F(X) which supports consist of at most n points. A discrete space consisting of k points is denoted by k . Consider such a mapping nF,x,k C(k, X) x F(k) ^ F(X) defined by the equality nFX k k(%, a) = F(^)a at
C(k, X), a e F(k) . This map KF X k is continuous if the functor F is continuous. The subfunctor [7] Fk of a functor F is defined as follows:
Let F : Comp ^ Comp be an arbitrary normal functor and X e Ob(Tych). Assume that
F((X) = {x e F((X): supp(x) c X}. If f : X ^ Y is a morphism of a category Tych, then we admit F ( (f) = F( (f)\F ((X) where (f : (X ^ (Y is the Stone-Chekhov continuation of the map f . Using the normality of the functor F , we can show that the definition F ( (f ) is correct, that is F((f)F ( (X) c F ( (Y) . It is also easy to see
that F(is a covariant functor of a category Tych in itself.
Let X and Y be topological spaces. For sets A c X , B c Y assume that
M(A,B) = {f e YX :f(A) c B} . Denote by F the family of all finite subsets of the set X , and let G be the topology on Y. The family ^ of all sets of the form n{M(A,BJ:i = \,2,...,k} , where Ai e F and Ui eG at i = l,2,...,k generates a
topology on Y X called the pointwise convergence topology. A family ^ is a base of a space YX with the pointwise convergence topology. By R X we denote the space of all real functions on the space X and by Cp(X) we denote the space of continuous functions with a pointwise convergence topology. The family
Let X be compact. By C(X) denote the set of all continuous functions (p : X □ with the usual algebraic operations. There is the sup-norm on this set is defined as follows:
\\f\\ = sup f (x)|
where x e X.
A constant function is denoted by cX and defined by the equality
cx (x) = c
for all x of X and for each real number c . Let p, y be elements from e C(X) . We will write p<y iff p(x) < y(x) for all xof X .
Definition 5. [2]. A functional /u : C(X) —> □ is called:
1) weakly additive if ju(p + cX ) = j(p) + c for any (p e C{X) and ceD ;
2) order-preserving if the inequality <j)<V implies < for any pair of functions <ye C(X) ;
3) normed if j(1x) = l.
0(<p;x1.....xn;e) = {g (= C/XJ :\ <p(x,)-g(x,) ^^P-^y^ honpgcncous if v(A<f>) = Av(0)
for all (j) e C(X) ,2e0 where □ + = [0,+oo) ;
forms the base of the topology on the space C ( X ) . It is easy to see that C ( X ) everything is
dense in space R X .
Let X e Comp. The set of all continuous functions (p : X □ with the usual sup - norm:
|| p ||= sup{| p(x) |: x e X}
called a
denotes by C(X) . The mapping v : C(X) —> □ is functional.
Definition 2 [9]. A family F = {Fs }seS of sets is
called centered if <F * 0 and F fl... Pi * 0 for
1 sk
each finite system s1,..., sk e S .
Definition 3 [7]. A system £ of closed subsets of a space X is called linked if any two elements of £ have nonempty intersection.
Definition 4 [9]. The weight of a topological space
X is a cardinal such that w(X) = min{|B|} , where B is the base of the space X .
5) semi-additive, if v( f + g) <v( f) + v(g) for all f, g e C(X) ;
We denote the set of all weakly additive, order-preserving, normed functionals of a compact X by O(X) . For the sake of brevity, we call the elements of a set O(X) "weakly additive functionals". Note [6] that every functional v e O(X) is a continuous map from C( X) to □ . Hence, the set 0(X) is a subset of the space C (C(X)) of all continuous functions on C(X) provided by the pointwise convergence topology. Fitting O(X) with this topology, it can be considered as a subspace of a space C (C(X)) . The
neighborhood base of a weakly additive functional v e O (X) is formed by sets of the form:
(v;pi,...,(k;e) = {v' e O(X): |v(p)-v()| <e,i = 1,2,...,k},
where p e C(X), i = 1,2,...,k , and £> 0.
Let X, Y e Comp and f : X ^ Y be a continuous map. Define the mapping O(f) : O(X) ^ O(Y) using the formula
(Q<jyJu)№ = K<P°f)i
where ju e O(X) and p e C (7) . In the paper [2], T.Radul proved that the covariant functor O : Comp ^ Comp of weakly additive normalized and order-preserving functionals on the category of compacts satisfies all the normality conditions except for preserving preimages.
For each compact X the space of probability measures P(X) , (i.e., the space of all linear, nonnegative, normalized functionals) is a subspace of the space O(X) .
Main results
Let X be a compact. We define operations ©
and D on (.' (X) by the rules
(p®\{f = max{(p,ys} and (pU y/ = (p + i/7,
where C(X) . A functional
JU '. C(JQ —> D [10] is called an idempotent
probability measure on X if it has the following properties:
1) JU{A,X) = A, for all X G □ (normalization).
2) JU(A □ (p)=AU ju((p) for all
A G □ and (p G C(X) (unifonnity);
3) ju(p ©y) = ju(p) © u(^) for all p,Y e C(X) (additivity).
For a compact set X , we denote by I ( X ) the set of all idempotent probability measures on X , and by Iw (X) the set of all idempotent probability measures which cardinality of supports does not exceed
n.
The set D max = D U {—cc j is assigned a metric p
P(x,y) =1 e
als0 G max = (D max
Here is an example of an idempotent measure. Let xl,...,xn el and G max satisfy the
property } = 0. We define
JU\C{X)^U as follows:
j(p) = max{p(x) + A | i = 1,...,n}.
Each idempotent probability measure j with a finite support is represented as
LI = 1U 8 ©10 8
~ I Z X-, 77 xn
defined by
„y
e | (agreement e
y.
the only way (up to permutation) where
{x1 ,x2,...,xn} is the support of j. Here the
coefficients A. satisfy the following conditions
A > 0 = -ro . i 0 „ ,
i , I = 1,2,...,n and
... ©A= 1=o
For an open subset U of a compact Hausdorff space X , numbers a and £ > 0, we define the following sets
(U,a;£) = e O(X): ^{a%u) > £}, if a > o,
{u, a;-£) = {m eo(x): h(a%u) < -£} if a < 0
In [11], A. A. Zaitov proved the following Proposition 1. For every open subset U of a compact Hausdorff space X and an arbitrary £ > 0 the sets (U,1;£ and (U, —1; — £ are open in
O(X) with respect to the pointwise convergence topology.
Using this sentence, we prove the following theorem:
Theorem 1. For any two intersecting open subsets U1 and U2 of space X , the intersection of the sets
(lU1,1;£ and (U2,1;£ is not empty, where
0<£<1.
Proof. Let us take a function (p: X I ,constructed as follows:
(( x) =
|Xif x eU 10, if x e X \ U
a formula = 0). Let
Then the following conditions will be met for the
Dirac measure 8 :
x0
8Xo(<p) = <p(x0) = l
and
^ (Zu) = sup{^ (<P): 0A- < ^ < ^ } > {<p) = 1 > e ^
where x is a fixed point in U .
Let two open subsets U1 and U 2 of X have a nonempty intersection. Then there exists a functional ju e O(X) , such that j(xXJ ) > £ , because for any
Fi = X \ U we can construct a continuous function
(~pi : X —> / suchtliat:
q>(x) =
[1, if x^UlC\U2 0 ,if xeFt
and
öAv,) = V,(x0) = \>e
Ob(X)
are
homeomorphic,
where
The space X is embedded in a space p (X) , since the map 5: X ^ P(X) that transfers a point x to a measure 5x is continuous and, therefore, is a homeomorphism. Based on this, we have:
w(Pß(X)) > w(X)
(2)
where i = 1.2 and x0 is a certain point from
u,nu2.
Based on the above, it can be argued that if
^flU2 then(Up1;s)n(U2,1;s)*0.
The following corollary easily follows from Theorem 1:
Corollary 1. Let be a linked system £ = {Ua : a e A} of open subsets of an infinite
compact X . Then X = {(Ua,1,s} :a e A} is a
linked system of open sets in the space O(X) .
Corollary 2. For any centered family £ = {Ua : a e A} of open sets in X , the family
X = {(Ua,1, s) :a e A} is a centered system of open sets in O (X) .
We mean by p (x) the set of all probability measures of the space p((X), where (X is the Stone-
Czech compactification, which support lies in X .
In 2000, R. B. Beshimov proved the following result.
Theorem [12]. Let X be a Tychonoff space, (X be its Stone-Czech compact extension, and bX
be some compact extension. Then O ( X ) and
Ob(X) = {je O(bX): supp c X} .
Using the above property, we can prove the following
Theorem 3. For any infinite Tychonoff space X we have
w( X) = w(P(( X))
Proof. Let t be the weight of the space X . Then the weight of the compact extension bX is equal to r and greater than or equal to the weight of the space
P((X), ie.
t = w( X) = w(bX) > w(Pp (X)) Connecting the beginning and end, we get:
w(P((X)) < w(X)_ (1)
It follows from statements (1) and (2) that w(X) = w(P((X)) .
Theorem 4. For any infinite compact X we have
d (X) = d (In (X)).
Proof. First, we prove the inequality d(In (X)) < d(X) . Let X be an infinite
compact and d(X) = T> N. It is clear that
d (Xn) = r for anyone n e N. By virtue of V. Basmanov's theorem [8], we have that the space I (X) is represented as a continuous image of the
space Xn X Xn 1, where , X. = R
f ^ Zn-1 x ' max
¿=1
for each i = 1,2,____ The mapping
Tt: Xn X Xn 1 ^ In (X) is determined by the formula
n
i=1 ,
where > 0 = —Xi, i = 1,2,...,n and ... ®\= 1 = 0, 5 are Dirac measures at points x , respectively. Since
d ( Xn xE"—1 )<
T and the density of a topological space is preserved under a continuous map, we obtain d (In (X)) < T .
Now let us prove the inequality d(X) < d (In (X)) . Let
d(In (X)) = T> N . Then there exists a dense subset Q = {j : a e A} of In (X) such that
|A| = T . We define a subset M of the space X follows:
M = U{ SUPP(ja): | supp
aeA
(Ja)|<", JaeO)
as
Clearly, |M| = T.
We show that the set M is everywhere dense in X . Assume the opposite, i.e. there exists a point x
0
that
and its neighborhood Oxo such M n Ox0 = 0. By regularity X, there exists a neighborhood O\ of a point X0 such that [Ox ] ^ Oxo . Since X is normal, there is a function pe C(X) such that p([O,X0]) = 0 and p(X \ O\ ) = 1.
Consider the neighborhood O(S ; p; 0,5) of point Ô from In ( X) . For every measure VeQ we have SUpp(v) n Oxo =0. Let
supp(v) = {Xi,X2,...,Xk}, k < n. p(X ) = 1, i = 1,2,..., k. Therefore,
Then
— 8x («| = ©A, I 8^ —8x = ©©A <fr(x,) — <Kx0) = = |max{A +^(x2),...,A )} — ^(x0 )| = |max{A +1} = 1
This contradicts density of the set Q in I (X).
Hence, the set M is dense in X, i.e d (X) < T . Theorem 4 is proved.
References
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[9]. Engelking R. General topology, Moscow: Mir., 1986, 576 p.
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