CC BY
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24
YAK 515.12
Вестник Самарского университета. Естественнонаучная серия. Том 24 № 2 2018
DOI: 10.18287/2541-7525-2018-24-2-24-27
T.F. Zhuraev, Q.R. Zhuvonov, Zh.Kh. Ruziev1
SHAPE PROPERTIES OF THE SPACE OF PROBABILITY MEASURES
AND ITS SUBSPACES
In this article we consider covariant functors acting in the categorie of compacts, preserving the shapes of infinite compacts, ANR-systems, moving compacts, shape equivalence, homotopy equivalence and A(N)SR properties of compacts. As well as shape properties of a compact space X consisting of connectedness components 0 of this compact X under the action of covariant functors, are considered. And we study the shapes equality ShX = ShY of infinite compacts for the space P(X) of probability measures and its subspaces.
Key words: Covariant functors, A(N)R-compacts, ANR-systems, probability measures, moving compacts, retracts, measures of finite support, shape equivalence, homotopy equivalence.
Citation. Zhuraev T.F., Zhuvonov Q.R., Ruziev Zh.Kh. Shape properties of the space of probability measures and its subspaces. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia [Vestnik of Samara University. Natural Science Series], 2018, no. 24, no. 2, pp. 24-27. DOI: http://doi.org/10.18287/2541-7525-2018-24-2-24-27 [in Russian].
MSC: 54B15, 54B30, 54B35, 54C05, 54C15, 54C60, 54D30.
For a compact X by P(X)denote the space of probability measures. It is known that for an infinite compact X, this space P(X) is homeomorphic to the Hilbert cube Q. For a natural number n e N by Pn (X) denote the set of all probability measures with no more than n support, i.e. Pn (X) = = [p e P (X) : |suppp| ^ n}. The compact Pn (X) is a convex linear combination of Dirac measures in the form
p = m\5Xl + m2SX2 + ... + mriSXn,J2n=i mi = l,mi > 0,xi e X, SXi- the Dirac measure at a point xi. By S (X) denote the set of all Dirac measures. Recall that the space Pf (X) C P (X) consists of all probability measures in the form p = m1SXl + m2SX2 + ... + mkSXk of finite support, for each of which mi > k+i for some i. For a positive integer n put Pf,n = Pf nPn. For a compact X we have Pf,n (X) = [p e Pf (X) : |suppM| < n}; PC = Pf n PC, Pfn = Pf n Pn n PC.PC = PC n P. For the compact X by PC (X) denote the set of all measures p e P (X) the support of each of which lies in one of the components of the compact X [12].
1. Introduction
For a space X by OX denote the expansion (partition) of the space X consisting of all the connected components. If f : X ^ Y is a continuous mapping, then the continuous mapping Of : OX ^ OY is uniquely determined by condition °f = Of ■ , where : Y ^ OY and : X ^ OX, i.e. we have the following diagram
X A Y
nx I I nY (1.1)
OX a OY
Lemma 1. If X is a compact ANR-space, then the map PC (nx) isahomotopy equivalence.
!© Zhuraev T.F., Zhuvonov Q.R., Ruziev Zh.Kh., 2018
Zhuraev Tursunboy Faizievich ([email protected]), Department of General Mathematics, Tashkent State Pedagogical University named after Nizami, 27, Bunyodkor Street, Tashkent, 100070, Republik of Uzbekistan.
Zhuvonov Qamariddin Rizokulovich ([email protected]), Department of Higher Mathematics, Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, 39, Kari Niyazov Street, Tashkent, 100000, Republic of Uzbekistan.
Ruziev Zhamshid Khudaikulovich ([email protected]), Department of General Mathematics, Tashkent State Pedagogical University named after Nizami, 27, Bunyodkor Street, Tashkent, 100070, Republik of Uzbekistan.
Proof. Let be an ANR-compact, then the space PC (X) is a finite set or is a finite union of Hilbert cubes and points. The space PC(DX) consists of finitely many points, because the space X is an ANA-compact. For any n e PC(DX) the transformation (PC(f ))-1(^) is the Hilbert cube, or one point, i.e. Sh((PC(f ))-1(^)) is trivial, then by Theorem 7 [5] the map PC (f) is a shape equivalence, and thus, is a homotopy equivalence. The proof is complete.
Theorem 1. Let X be a compact and let nx : X ^ DX be a quotient map. Then the mapping PC (nx) induces a shape equivalence, i.e. Sh(PC (X)) = Sh (DX).
Proof. Suppose X is compact, DX is also compact, then by V.I.Ponamareva theorem [6] dim DX = 0. Hence, dim PC (X) = 0 and PC (DX) =0. By Theorem 2 [5] the mapping PC (nx) is a shape equivalence. This means that Sh (PC (X)) = ShPC (DX) and |DPc (DX)| = |DX|. This proves the theorem.
Definition [10]. A normal subfunctor F of the functor Pn is called locally convex if the set F(n) is locally convex.
We say that a functor F1 is a subfunctor (respectively nadfunktorom?) of a functor F2 if there exists a natural transformation h : F1 ^ F2 that the map h (X) : F1 (X) ^ F2 (X) is a monomorphism (epimorphism) for each object X. By exp denote the hyperspace functor of closed subsets. For example, the identity functor Id is a subfunctor of expn, where expn X = {F e expX : |F| < n}, and the nth degree functor n is a nadfunktorom of functors expn and SPg. A normal subfunctor F of the functor Pn is uniquely determined by its value F(n) at an n-point space. Note that Pn(n) is the (n — 1)-dimensional simplex. Any subset of the (n — 1)-dimensional simplex an-1 defines a normal subfunctor of the functor Pn if it is invariant under simplicial mappings.
An example of not normal subfunctor of the functor Pn is the functor of probability measures PnC whose supports lie in one of components. One of the examples of locally convex subfunctors of Pn, is a functor SPn = SPn .
S n
Corollary 1. If for compacts X and Y the equality |DX| = |DY| = H0 holds, then Sh (PC (X)) = = Sh (PC (Y)) and ShP (X) = ShP (Y), where Z| is the cardinality of a set Z.
Proof. Suppose the sets |DX| and |DY| are countable. In this case, by Arkhangelskii's result [8], the spaces |DX| and |DY| are compact and metrizable. Note that |DX| and |DY| have a dense set of isolated points. Then the compacts P(X) and P(Y) are homeomorphic to the Hilbert cube Q. On the other hand, PC [X] = DX and PC [DY] = DY. Consequently, Sh (PC (DX)) = Sh (PC (DY)). The corollary is proved.
By Ma we denote the class of all compacts X such that DX is metrizable. From corollary it follows that if X,Y e Ma, then DX and DY have a countable dense set of isolated points [9].
Corollary 2. If X,Y e Ma, then either Sh (PC (X)) > Sh (PC (Y)) or Sh (PC (X)) < Sh (PC (Y)). Therefore, if DX and DY are infinite, then Sh (PC (X)) = Sh (PC (Y)), i.e. Sh (PC (X)) > Sh (PC (Y)) and Sh (PC (X)) < Sh (PC (Y)) .
Proof. Suppose that X and Y are elements of the family Ma. Then DX and DY are the zero-dimensional compacta. In particular, if DX and DY are finite sets, then by Theorem 1 we obtain the desired.
If |DX| > H0, then DX contains Cantor's discontinuum. In this case, DY can be embedded into DX, then the compact DY is a retract for DX [10]. Sh (DX) > Sh (DY) and Sh(PC (DX)) > Sh(PC (DY)).
Consequently, by Theorem 1 we have ShPC [DX] > ShPC [DY]. If DX < H0 and DY < H0, then compacts DX and DY are homeomorphic to Mazurkiewicz-Sierpinski ordinal compact [11]. Last, suppose DX and DY are infinite sets, then Sh (DX) ^ Sh (DY) if and only if DX and DY are homeomorphic [3]. If |DX| > |DY| or |DX | < |DY |, then either DY or DX is retract for DX or DY, respectively. By Theorem 1 we have Sh (PC (X)) > Sh (PC (Y)). Corollary 2 is proved.
Remark. In [11] it is shown that the Borsuk's definition of shapes of compacts is equivalent to the shapes of ANR-systems.
Lemma 2. For any compact X we have |DP^ (X)| = |DX|.
Proof. Let X b an arbitrary compact, DX its set of connected components, i.e. DX = {xi e X : nX (xi) — is connected component of the point xi}. It is obvious that DX is compact and DX c X. Hence, Sh (DX) ^ ShX. On the other hand, the commutativeness of the diagram
nX : X ^ DX
t t (1.2) Pf (nx): Pf (X) ^ S (DX)
implies D1hPf (X)| = |DX|. From (1.2) we get |DPf (X)| = |DX|. Lemma 2 is proved.
26
T.F. Zhuraev, Q.R. Zhuvonov, Zh.Kh. Ruziev
Let us note that for all x e X and y € X between sets (^r - 1 j (x) and (r- 1 j (y) there is a one-one
correspondence, i.e. to an arbitrary point px e (p—^J (X) we assign py € (Pf^J , where
Px = moSx0 + miSx1 + ... + mkSxk, Py = m0Sy0 + ... + mkSxk. In the case of the infinite compacts X and Y the spaces P(X) and P(Y)are homeomorphic to the Hilbert cube Q. If A and B are Z-sets lying in the compacts P(X) and P(Y), then by Chapman's theorem [2], ShA = ShB if and only if P (X)\A is homeomorphic to P (Y) \B. In [10,12] it is shown that the subspaces F(X) and F(Y)are Z-sets in the compacts P(X) and P(Y), where F = Pf (X) ,Pf,n (X) ,PCn (X) ,PC (X). Moreover, it was noted that this space X is a strong deformation retract for F(X). So the following is valid.
Theorem 2. For infinite compacts X and Y the following conditions are equivalent:
1. ShX = ShY;
2. P (X) \P f (X) ~ P (Y) \P f (Y);
3. P (X) \S (X) ~ P (Y) \S (Y);
4. P (X) \F (X) ~ P (Y) \F (Y), where F = PCnn-,PC.
Theorem 3. Suppose that X and Y are elements of Mq, X e M^and Y e Mq. Then the following conditions are equivalent:
1. Sh (OX) = Sh (OY);
2. P (X) \PC (X) ~ P (Y) \PC (Y).
Theorem 4. Suppose that X and Y are elements of Mq. Then Sh (OX) = Sh (OY) if and only if ShX = Sh (OX).
It is known that from the inequality ShX ^ ShY it follows Sh (OX) ^ Sh (OY). In particular, the equality ShX = ShY implies Sh (OX) = Sh (OY).
Now let Sh (OX) = Sh (OY). From the fact that the compacts OX and OY are zero-dimensional and metrizable, and by Mardeschicha Segal theorem [3], OX and OY are homeomorphic. If for any y e OX the set n-1 (y) has the trivial shape, then by Theorem 7 [5] we have ShY = Sh (OX); By virtue of the zero-dimensionality and equality ShY = Sh (OX) it follows Y ^ OX ^ OY.
Note that in this case ShX = ShY and X ~ Y, i.e. ShX = Sh (OX) is equivalent to ShX = ShY.
Corollary 3. a) The space PC (X) is an ASR if and only if X is connected; b) PC (X) is an ANSR if and only if X has finitely many connected components.
Theorem 5. For any infinite zero-dimensional compacts X and Y the followings are true:
a) If ShX = ShY, then Pn (X) ~ Pn (Y);
b) if ShX = ShY, then P (X) \Pn (X) ~ P (Y) \Pn (Y);
c) ShPn (X) = ShPn (Y) if and only if P (X) \Pn (X) ~ P (Y) \Pn (Y);
d) ShF (X) = ShF (Y) if and only if P (X) \F (X) ~ P (Y) \F (Y), where Fare locally convex subfunctors of the functor Pn;
e) ShX = ShY if and only if P (X) \S (X) ~ P (Y) \S (Y).
Theorem 6. For any infinite zero-dimensional compacts X and Y the following conditions are equivalent:
1. ShX = ShY;
2. ShF (X) = ShF (Y), where F = Pf ,n, P£n, Pf, Pf;
3. X ~ Y;
4. P (X) \F (X) ~ P (Y) \F (Y);
Theorem 7. For any infinite compacts X and Y we have: a) if ShX = ShY, then P (X) \Pn (X) ~ P (Y) \Pn (Yfor any n e N;
b) if ShX = ShY, then P (X) \F (X) ~ P (Y) \F (Y), where F are locally convex subfunctors of the functors Pn.
Theorem 8. For any infinite compacts X e Mq and Y e Mq we have:
a) ShX = ShY if and only if P (X) \Pn (X) ~ P (Y) \Pn (Y);
b) ShX = ShY if and only if P (X) \f (X) ~ P (Y) \F (Y).
References
[1] Oledski J. On summetric products. Fund. Math., 131, 1988, pp 185-190 [in English].
[2] Borsuk K. Shape theory. Mir, 1976, p. 187.
[3] Mardesic S., Segal J. Shapes of compacta and ANR-systems. Fund. Math., LXXII, 1971, pp. 41-59 [in English].
[4] Basmanov V.N. Covariant functors, retracts and dimension. DAN USSR, 1983, Vol. 271, no. 5, pp. 1033-1036 [in Russian].
[5] Kodama Y., Spiez S., Watanabe T. On shapes of hyperspaces. Fund. Math., 1978, Vol. 11, pp. 59-67 [in English].
[6] Ponomarev V.I. On continuous decompositions of bicompacta. Uspekhi Mat. Nauk, 12, 1957, pp. 335-340 [in English].
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[9] Pelczynski A. A remark on spaces for zerodimensional X. Bull. Acad. Polon. Scr. Sci. Math. Astronom. Phys., 19, 1965, pp 85-89 [in English].
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[11] Mardesic S., Segal J. Equivalence of the Borsuk and the ANR-system approach to shapes. Fund. Math., LXXII, 1971, pp. 61-66 [in English].
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Т.Ф. Жураев, К.Р. Жувонов, Ж.Х. Рузиев2
СВОЙСТВА ФОРМЫ ВЕРОЯТНОСТНОГО ПРОСТРАНСТВА
И ЕГО ПОДПРОСТРАНСТВ
В этой заметке мы рассмотрим ковариантные функторы, действующие в категории компактов, сохраняющие формы бесконечных компактов, ANR-систем, движущиеся компакты, эквивалентность формы, гомотопическую эквивалентность и A(N)SR свойства компактов. Рассмотрены свойства формы компактного пространства X, состоящего из компонент связности 0 этого компактного X под действием ковариантных функторов. И мы изучаем равенство форм ShX = ShY бесконечных компактов для пространства вероятностных мер P(X) и его подпространств.
Ключевые слова: Ковариантный функтор, шейп компакта, компонента, связности и гомотопическая эквивалентность.
Цитирование. Zhuraev T.F., Zhuvonov Q.R., Ruziev Zh.Kh. Shape properties of the space of probability measures and its subspaces // Вестник Самарского университета. Естественнонаучная се-рия. 2018. Т. 24. № 2. С. 24-27. DOI: http://doi.org/10.18287/2541-7525-2018-24-2-24-27.
MSC: 54B15, 54B30, 54B35, 54C05, 54C15, 54C60, 54D30.
Статья поступила в редакцию 23/V/2018.
The article received 23/V/2018.
This work is licensed under a Creative Commons Attribution 4.0 International License.
2Жураев Турсунбой Файзиевич ([email protected]), кафедра общей математики, Ташкентский государственный педагогический университет имени Низами, 100070, Республика Узбекистан, г. Ташкент, ул. Бунедкор, 27.
Жувонов Камариддин Ризокулович ([email protected]), кафедра высшей математики, Ташкентский институт инженеров ирригации и механизации сельского хозяйства, 100000, Республика Узбекистан, г. Ташкент, ул. Кары-Ниязи, 39.
Рузиев Жамшид Худайкулович ([email protected]), кафедра общей математики, Ташкентский государственный педагогический университет имени Низами, 100070, Республика Узбекистан, г. Ташкент, ул. Бунедкор, 27.
