Tikhonov spaces and continuous mappings into themselves and dimension dim.

1. 19(2): S.I (1), 9-18, 2024 www.thebioscan.com  − mappings into themselves and dimension dim. Gulchekhra Mukhamedova, −functors in the category of Tych Tikhonov spaces and continuous .. . pic Candidate of pedagogical sciences, docent of the department of "General Mathematics" of Tashkent State Pedagogical University named after Nizami. e-mail: muxamedovagulchehra74@gmail.com, https://orcid.org/ 0000-0002-1252-5790. Feruza Saydalieva, Candidate of pedagogical sciences, docent of the department of "General Mathematics" of Tashkent State Pedagogical University named after Nizami. e-mail: fsaidaliyeva@gmail.com Dilnoza Ibragimova, assistant of the department of "General Mathematics" of Tashkent State Pedagogical University named after Nizami. e-mail: ibragimovadilnoza1983@gmail.com Kamariddin Zhuvonov, Assistant of the department of "Higher mathematics" Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, National Research University. e-mail: qamariddin.j@mail.ru, Tursunboy Zhuraev, Doctor of Physical and Mathematical Sciences, Professor of the "Department of General Mathematics" at Tashkent State Pedagogical University named after Nizami. e-mail: tursunzhuraev@mail.ru, https://orcid.org/0009-0005-5379-3862 DOI: https://doi.org/10.63001/tbs.2024.v19.i02.S.I(1).pp09-18 ABSTRACT KEYWORDS Normal functor, locally convex subfunctor, weakly countable dimension, finitely open functor, projectively inductively closed functor, functor, finite functor. Receivedon:  − pic ..  − This note studies the geometric, topological and dimensional properties of the the category of Tych−Tychonov spaces and continuous self-maps. It is shown that weakly countable-dimensional spaces, –paracompact,  − paracompact, p− paracompact, metrizable and stratifiable spaces. Considering a number of subfunctors F of the functor P of probability measures that are their various topological and dimensional properties are studied, properties in the category of Top − topological spaces and continuous mappings into themselves. -functor and its subfunctors in .. pic . Functors preserve  − pic .. .– functors, 25-07-2024 Accepted on: 09-11-2024 MSC.54В15, 54В30, 54В35, 54С05, 54С60, 54О30. 9

2. INTRODUCTION For metric compact sets X ( ) P X simplices of an arbitrarily large number of dimensions, is infinite- dimensional [4]. By Cayley's theorem, the convex compact set ( ) ( ) P X R  is affinely Consequently, by Keller's theorem, the compact set ( ) there is a simple ( ) P X is also a compact space [3]. Further, it, containing topological classification of the spaces probability measures. In the case of a finite n− point space ={ } X n are convex linear combinations of Dirac measures: = (0) m m   + Therefore, they are naturally identified with the points of the ( 1) n− dimensional simplex ( ) i  are formed by the vertices of the simplex, and the masses i m placed at points i are the barycentric coordinates of the measure  . Thus, the compact set  In the case of an infinite compact space X , the space of all C X the points  of the space embedded in P X . ( ) = P n P n ( ) 2 n , as 1  (1) ... + + 1  − − an infinite-dimensional convex compact set lying in 2, is m ( n 1) 0 n  Q = I 0 homeomorphic to the Hilbert cube [2]. On the other −  n 1 . In this case, Dirac ( ) P X hand, the space compact set X is called the set of all regular Borel probability measures on X , equipped with the weakest of the topologies : ( ) the measure  to ( ) U  of all probability measures on a → uf C X (U is an open set in X ) [5]. R P n ( ) for which each functional , which takes is affinely − n 1 homeomorphic to the simplex [1-2].  1  -degree of a non-one- the smallest of the closed sets F ( ) = ( ) F X   sup ( ) = { : p  ( ) ={ the set of all measures   X for which It is also known that any point compact set K , the space of all probability measures ( ) P K is homeomorphic to the Tikhonov cube   , I − segment [0,1] . Note, in particular, that all these spaces are topologically homogeneous. A for spaces ( ) P K for different [3]. For an arbitrary compact set X and a measure ( ) P X  , its support supp  = ( ) m x m   + i.e.    A A  = , ( ): P X A ( )} P A supp I 1 1 ; ,    − nP X n } P K ( ) = I 1 1 at most n supported, − the set of all probability    > 1 ( ) = P X  measures  with finite supports[4-5 ]. Recall that the space ( ) ( ) P X  consists of all probability measures [1] P X ( ) the situation is 1 n n =1 fP X ( ) is defined, which is  ( ) ... x  + + n + ( ) x  n N fP X ( ) ,( ) f n P X m nP X ( ) , the sets P X , , are 1 1 2 2 n n closed in ( ) .  m with finite supports, for each of which for some i  n 1 P X ( )  ( ) P X Consequently, the subspace ( ) P X  ( ) P X . ( ) P X ( ) P X  , the support of each of which lies in one of the connected components of the space (compact space) X [2]. ; i is  − compact, everywhere dense in and     ,( ) ={ f n P X ( ): P X supp ( ) n }. c f Let denote the set of all measures fP X Hence, the compact subspace is also the union of ,( ) f n P X compacts . those. c P X ( )  The closedness of the set ( ) P X is ( )( ( )) P f P X the functor P , like any normal functor, preserves supports, i.e. in the compact ( ) = P X P P ( ) X . It is obvious that f f n , =1 ( ) n P X n set quite P Y obvious. The inclusion   ,( ) f n X fP X ( ) P X ( )  and .  c c ( ) follows from the fact that Obviously, for a metric compact set X and any 10

3.   supp P f ( ( ))( ) = f supp P f C ( ) . . 1  c of the functor P is I of a Cantor perfect set P does not clearly c P  − is called the pseudo- = \ S Q BdQ− the cube Q , BdQ = W Thus, the subfunctor i → : i =1 defined. An epimorphism boundary of the cube Q , and pseudo-interior of the cube Q  In the theory of infinite-dimensional manifolds, the following objects play an important role: the Hilbert cube Q , c onto an interval shows that the functor . It is known that P preserve epimorphisms in the same way that ( ) = = P n n P P P   , P X , while 1  ( 1,1)i − S = . c P n ( ) i =1 as . Therefore, P we P get 1  c f c c f n c P P , , f , f P n  1 2n c c c c − P P P ( )  P = .  the separable Hilbert space 2, the linear hull of the n n For an infinite compact set X Comp Comp → and a    Q = [0, ] standard brick in the Hilbert space F Comp : n =1 normal or seminormal functor of or  denotes the linear subspace of the Hilbert infinite degree [1]. Let us accept the following notation[6]: ( ) = ( )\ F X F X  ( ) F X F   ; 2) ( ) ={ ( ): a F X  ( ) supp a denotes the support of the point a ; ( ) = n F X F X  4) ( ) ( ) ={ F S A a F X  A and } A X  ; 5) ( ) = ( )\ nk n k F X F X F X n ( ) = F X  ( ) = F X F X   ( ) = n F X F X  Recall that the topological space Y is called an absolute (neighborhood) retract in ( ) ( ), Y A N R K  homeomorphism h, mapping Y onto a closed subset h(Y) of the space X from the class K, the set h(Y) is a retract (neighborhood) of the space X [7]. Recall that a topological space X manifold modeled on the space Y , or a Y − manifold [7], if every point in the space X has a neighborhood homeomorphic to an open subset of the space X . A Q -manifold is a separable metric space locally f 2, via  ( ) X 2 n =1 1) ; For F space 2, consisting of all points only a finite number of Q Hilbert cube Q , consisting of only a finite number of points the number of coordinates of which is different from zero. By the Anderson-Kadets theorem [9], the Hilbert space ( ) X we identify ; f 1 coordinates of which is nonzero, and is the subspace of the  n F X supp a ( ) n a } .  ( ) F X 2 is homeomorphic to C [9]. From Bessaga-Pelczynski's results it follows that  f f Q  ; ; { = ( ) | < <1 n n x x Q x t  Further, it is obvious that rintQ true: BdQ . It is known that the spaces Q , and ( )\ F X ( ) 3) . is homeomorphic torintQ and n  : supp a ( ) A [9]. Here rintQ denotes the set , 2  n N } for BdQ all .  , which means it is k n ( ), > ,  2 ; F X ( ) F X  F X 6) ; n n =1 2 are ( )\ ( )\  ( ) ( ) 7) ; ,  and f strongly infinite-dimensional, and the spaces Q are weakly infinite-dimensional and all these spaces are 2 8) ; n f the class K K (written homogeneous.  if Y and for any Definition [10]. A closed setA of spaceXis called Z -set in X if the identity mapping be approximated arbitrarily : \ f X X A → . A countable union of Z -sets in X is called a Z  − −set in X .  − Qis called a boundary set in Q (denoted by ( ) \ . Q B  Z  − − set whose complement is a manifold. From the above it follows that the pseudo-boundary of the space X can closely by the id X mappings is called a −set B of a Hilbert cube Z Following [9], a B Q ) if cube Q homeomorphic  ={( i W to the Hilbert [7], where More generally, a boundary set in a Q - 2 − Q = [ 1,1] ;  2− Hilbert cube [8], manifold is a i | i =1 g  j − th face of the Hilbert = 1  ) Q g j i 11

4. BdQ of the Hilbert cubeQ is its boundary set. Let X be a topological space. A set A X  if there is a homotopy ( , ): h x t that ( ,0) = X h x id An A set A  if \ X A is homotopy dense in X . : e Y (resp. homotopy negligible) if ( ) (resp. homotopy negligible) in X [5]. Definition 1. : F Comp Comp → conditions are met: continuous; monomorphic; epimorphic; saves: weight; intersections; prototypes; point and empty set; Definition 2. : F Comp Comp → is called [11] regular if the following are satisfied: monomorphic; epimorphic, continuous, preserving intersections and preimages. Definition [2]. A normal subfunctor F of a functor nP is said to be locally convex if the subset ( ) Let F : T G → F : , X Y T G T → F supp a is the intersection of all closed sets of the space X such that ( ) = { F supp a A X A  ( )} a F X  ( ) support of a point  a F A ( ) i.e. is called homotopy dense in X [10] [0,1] X  and ( (0,1]) h X  X : →  is closed and X such A . → F Tych ( ) F K Comp For , the functor F is compact, but : Tych is homotopy negligible in X Definition 3. A functor called compact ( − compact) if compact) for any space X Comment : F Comp → is is compact ( − → X  4. An embedding is homotopy dense . e Y is a homotopy dense set any functor Comp A covariant functor not every compact functor in Tych is of type F. As is called normal [3] if the following R P and P of Radon examples, we can consider the functors and  -additive probability measures, respectively (see [11]). The functor → F Comp : Comp Returning to the functors, we obviously have ( a F supp a  If the functor F preserves preimages. then F preserves supports [11], i.e. ( ( )= ( ( )( )). f supp a supp F f a The (1.4) property can be converted. Proposition 5 [1]. Any functor in Comp is support- preserving if and only if it is preimage-preserving. From the definition of the functor F and the property (4) it follows that ( ( ) = F X f supp a supp  for any preimage-preserving : F Comp Comp → : f X → ( )). F n of the ( ) (3) nP n is locally convex. and G simplex be functors. A natural transformation between us is the system of : G → F : f X Y → (4) , where X is a mappings space such that for any mapping X ( ) = X G f T A functor F is called a subfunctor of a functor G between the Y ( ) f spaces and T the following holds: F . Y T G→F such that : if there is a natural transformation T Main part embedding for any X . F ( )( ) f  a X (5) ( ) F ( ) Y  → f X : Y functor For a continuous mapping spaces X X Y   → conditions ( )( ( F f F   between , Tikhonov spaces X and Y , Y . and a  and Y the  Tikhonov : f , a mapping F X ( )  continuous mapping . is defined, which satisfies the For the category C we denote by O(C) and M(C) the family of all objects C and the family of all morphisms C respectively. Let G and F be functors in some category C Top  Xi ( ) X O C  , such that ( ) = ( ) X Y F f i i G f ( ) f M C  . Xi transformation i . A functor G is called a subfunctor of a → F X  . A natural transformation : i G F X )) F Y ( )  is a (1) → : ( ) G X ( ) family of mappings ,  f supp a where ( ) ={ F X  X  ( ( )) ( ( )( )) supp F f a (2)    a F ( X ): supp a ( ) X } (6) , is the Stone Chekhov extension of the space X . For a functor F and an element for any  a ( ) F X The mappings are called components of the , the 12

5. → functor F if there is a natural transformation : i G F and G its regular subfunctor. Then such ( ) = G A ( ) G X F A ( ) Xi that all its components transformation of i is called the embedding of G into F. If : i G F → is an embedding, then (6) implies ( ) ( ) =Y G f i F f Comment 7. Usually subfunctors arise in a fairly natural way. For example, in the category Comp the functor expcof subcontinuum hyperspaces is built into the functor exp of hyperspaces of closed subsets from the very beginning. Let's assume this is a standard situation. This means that the functor G is a subfunctor of F if the natural transformation (embedding) : i G F → consists of identical embeddings, i.e. G(X) is a subspace in F(X) for any space X. In this case, the condition (7) is equivalent to the condition ( )( ( )) ( ). F f G X G Y  k F -subfunctor of the functor F in are embeddings. This natural for any compact set X and its closed subset A. From this definition of a carrier we get Proposition 13[14]. Let F be a functor in Comp, and G its subfunctor. Then supp a G X  Proposition 14[14]. Let F be a regular functor in Comp, G its regular subfunctor. Then supp a G X  From propositions 9 and 14 it follows Proposition 15[15]. Let F be a regular functor of k  in Comp, and let G be its regular subfunctor. Then degG k  . From the equality (7) we immediately obtain Proposition 16[15]. Let F be a continuous functor in  ( ) a supp ( ) a − 1 i F X ( ) ( ) G X ( ) (7) X for any compact set X and . ( ) = a ( ) supp ( ) a F X ( ) G X ( ) for any compact set X and . degree (8)  G F Comp, and G its continuous subfunctor. Then any natural number k . Proposition 17[15]. Let F be a regular functor in the category Comp, and let G be a regular subfunctor in F. Then ( ) = G X  for any Tikhonov space X. Proposition 18[11]. Let F - be a functor in the category Comp, and G be its subfunctor. Then G is a subfunctor of F. About closed subfunctors and their sums Definition 19[16]. Let F be a functor acting in the category ! Top  , and let G be a subfunctor of F. A functor G is called a closed subfunctor of a functor F if for any ( ) X O C  the space ( ) G X ( ) F X . The next statement is trivial. Proposition 20[16]. In the category Comp, each subfunctor is closed. From Proposition 8 and Proposition 20 we get Corollary 21[6]. For every continuous functor : F Comp Comp → k F is a closed subfunctor of F. Returning to the category Tych, for a Tikhonov space X, : F Comp positive integer k, we set ( ) = ( ( ), k F k F X C k X   We also put ( ) = ( ). F   for k k Definition 8. Let ( ) =  ( )  . For k F ( ) f X ( ) F f ( , ) C k Y F Comp, defined as follows. A-priory. k F X a non-empty compactum X,  is the image of the Y → , ( )  G ( X ) F X ( )  : k F f ( ) mapping . For a mapping F X k , , k F X the : ( , f C k X takes  onto the composition f f id   is restriction of to . Let → ) denote the mapping that  . It's easy to see that  = F f ( ) (9) Y k , F ({ }) k X k ,  ( )( F f ( )) F X kF is a subfunctor of F with ( ) F X for an arbitrary F Y ( ) is a closed subspace of Therefore, . k k k F is a functor. Clear. k F X compact set X. A functor F is called a degree functor k (they write degF ( ) = ( ) F X 1( ) ( ) X F X −  Proposition 9[11]. For any continuous functor F and compact set X we have ( ) ={ a F X  The media definition and properties (3) imply Proposition 10[13]. For a functor F, a compact space X and a closed subset A in X we have ( )={ F A a F X  Proposition 11[13]. Let F be a functor in Tych, A be a closed ( ) a F X  and ( )( )= F f a a for any mapping | = A f A id Proposition 12[14]. Let F be a regular functor in Comp, Therefore  ( ) unit embedding for any compact set X , and k F X = k) if and a natural number k , the k F for some X[3]. functor → Comp  k F X ( ): supp a ( ) k }. a continuous functor and a  ) ( ). F k (10) , ,  ( ): supp a ( ) A }.  → supp a : f X ( ) A X subset of X, . Then k F (11) such  Let us now denote the restriction of to that .  F , X k , 13

6.  → F X k  → C k continuous map, then ( ) F k ( ) g X : Y k F Comp : Comp this extends the functor to the by . If is a , , category Tych. The following statement follows from Proposition 9. F Comp k F Tych subfunctor of the functor F, and ( ) = ( ) k k F X F X F  g F X   f F ( )( ( )) F Y = ( ), g  → : Comp Tych k k Proposition 22. [17]. If taking into account the equality (9) to display . → : is a continuous functor, then is a Therefore the setting  → ( ) = F g ( ): F g F ( g )| F Y ( ), F X ( ) k k F X ( ) we obtain the mapping . k k k  ( X ). (12) Thus, we have defined a covariant functor → k F Tych : Tych , The equality (12) gives Corollary 23[17]. For every continuous functor F Comp Comp → and a natural number k, the functor ( ) particular, the functor expk, is a p.i.c.-functor. : F Tych , cl F F n    = ), n F F any Tikhonov space X[17]. Definition : F Tych Tych →  , where each : → Tych Let be a functor and k F Comment 24[17]. Proposition 9 states that the kF [1] and Basmanov [3] coincide for any continuous functor F in Comp . However, we will use both  is a closed subfunctor of F. n n F ( . We say that F is a union of i.e.   n n ( ) = F X F ( ) X if for definitions of the functor =0 n =0 32[17]. is called  The functor , k F definitions depending on the situation and denote them F respectively. As for the equality (10), we can assume ( )k b F be written in the form ( ) = ( ) = ( k b k b k b F F F   Proposition 25[17]. If G is a regular subfunctor of a regular functor F in Comp, then G is a closed subfunctor of F. Definition 26. [11]. A functor F is said to be finitely ( 1) is open in natural number k. The dual of this definition states that ( 1)\ ( 1) k F k F k + + Comment 27[17]. As an example of a finitely open functor, we can take any finitary functor (or finite functor) F, i.e. ( ) F k is finite for any positive number k. In particular, the hyperspace functor exp[3] is finite and therefore finitely open. Proposition 28[11,17]. If F is a regular finitely open functor, and G is its regular subfunctor, then G is finitely open. -p.i.c. functor if n n F F = ( F ) and is a p.i.c-functor of , k b  n =0 finite degree. that it defines the functor . Thus, the equality (12) can , Example 33[17]. Taking into account Remark 27 and exp  functor  -p.i.c. The next statement is obvious.  = expn Theorem 30, the functor is the ) .  n =1 (13) , , ,  n F = F Proposition 34[17]. If , where n =0 is a functor  - p.i.c., then F is a functor  -p.i.c. : → functor of Borel regular probability measures. This functor is normal [1,2,4]. According to Proposition 22 and Remark 24, we will denote ( )  by n F each nP Comp Comp Recall that denotes the F k + k F k + ( 1) open if the set for any kP kP . F k + ( 1) is closed in . kP is a  -p.i.c.-functor Theorem 35. The functor for any natural number k. Comment 36. In fact, we have proven more: the functor kP is the union of its normal finitely open subfunctors ( . We will say that the  -p.i.c.-functor F has degree k if = F a functor F such that n kP )  → f X : Y Recall that an epimorphism is said to  n F in the representation (from Definition be inductively closed if there is a closed subset A in X such that ( )= f A Y and |A f Definition 29[16-17]. : F Comp Comp → is said to be projectively n =0 is a closed mapping.  n F k 32[17]) each functor has degree . A continuous functor G and : F Comp G Proposition 37[17]. Let be normal Comp 1 2 → F X k  subfunctors of the normal functor inductively closed (p.i.c.) if the mapping is inductively , ,  k of G finite = G degree G is a normal functor of degree . Then their intersection k . closed for any Tikhonov space X and a positive integer k. Theorem 30[16-17]. Every continuous finitely open functor F preserving the empty set and preimages is a p.i.c. functor. Corollary 31[17]. Every finitary normal functor, in  1 2 G G be normal subfunctors , Proposition 38. Let 1 2 of the normal functor F in Comp. Then 14

7. following classes:  − ( G ) ( G ) = (  G G )  a) paracompact spaces; b) p-paracompact  1 2 1 2 spaces; kP in c)  -paracompact spaces; d) stratified spaces; e) metrizable spaces. ( ) F X  Theorem 39[11,17]. Let F be a normal subfunctor of Comp. Then F is a  -p.i.c.-functor. Definition 40[18]. A normal space X is said to be weakly countable-dimensional if X is the union of a countable family of X such that dim The next two statements are trivial. Proposition 41[18]. Every closed subspace of a weakly countable-dimensional space is a weakly countable-dimensional space. Proposition 42[18]. Every normal space that is the union of a countable family of its closed weakly countable- dimensional subspaces is weakly countable-dimensional. A completely regular space X is called a cirrus ( p− space)[19] if there exists a countable family coverings of the space X by sets covered in its Stone–Cech extension betaX such that X  . A space X is called a exists a  − discrete family N and covered C consisting of closed countably compact sets such that if c C C U  F U   U is open in X . A space that has a  − locally finite network is called a  − space[19]. those. X is a countable sum of locally iF , each of which is a network of the space X . 1T − space X is called a stratifiable (or lace) space (in short, a G-space) [19] if each open set U can be associated with a sequence { subsets in such a way that the following conditions are satisfied: U U  for { : }= n U n N U  U V  Theorem 43[11]. Let F-p.i.c. be a functor of finite degree transforming finite sets into finite-dimensional compact sets, let X be a weakly countable-dimensional space, and let X belong to one from the following classes: −  spaces; c)  -paracompact spaces; d) stratified spaces; e) metrizable spaces. ( ) F X  space. Corollary 44[11]. Let F be a normal finitary functor of finite degree, in particular, the functor expk, X be a weakly countable-dimensional space, and let X belongs to one of the Then is a weakly countable-dimensional space.  for each i. i X < Corollary 45[11].. Let F be a normal finitary functor of finite degree, in particular, the functor , X be a weakly countable- dimensional space, and let X belong to one of the following classes: −  spaces; c)  -paracompact spaces; d) stratified spaces; e) metrizable spaces. Then is a weakly countable-dimensional space. Recall that for : F Comp Comp → F  for any Tikhonov space X. closed subsets i a) paracompact spaces; b) p-paracompact U a normal functor of n , by F they denote the  -   compact subfunctor , defined as: =13 ( , b X U ) X n n ( ) = F X  F X ( ) for each x k k =1  −  space[19] if there F = F Thus, .  k k =1 Theorem 46[17]. If X is a weakly countable- dimensional space that is p-paracompact (in particular, metrizable) or stratifiable M , then countable-dimensional. Proposition 41 and Theorem 46 give Corollary 47[17]. If X is a weakly countable- dimensional space that is either p-paracompact (in particular,  N and  P X ( )  for some F , then C is weakly , where ( ) F X metrizable) or stratifiable, then is a weakly countable- finite families  dimensional space for an arbitrary closed subfunctor F . Applications Theorem [2]. Every locally convex subfunctor F of a nP preserves the property of the space to be an ( ) A N R -compact space and the property of a compact space to be a Q − manifold or a Hilbert brick. Theorem [4]. Let X ( ) ( ) A N R M P A  } X  U : n N open functor n   each n N V a) ; b) n if U ; c) , then be a weakly countable- space, and let F − be a nP for all n . dimensional n n locally ( ) F X ( ) convex A N R  ( ) ( A N R  subfunctor M M of . Then ( ) ( ) ) . In particular, nP X a) paracompact spaces; b) p-paracompact . fP X ( ) Corollary [4]. Separable spaces , c c fP X complete metric if and only if X itself is separable and metrizable by a complete metric. Corollary [8]. Let A be a Z − set of a Q − variety. Then the set A has a neighborhood homeomorphic to ( ) ,( ) f n P X ,( ) f n P X , and is metrizable by a Then is a weakly countable-dimensional 15

8. an open subset of the space Q . From this corollary we can conclude that if a Z − set A is connected in Q , then OA is a neighborhood of the Q ; A of a Hilbert cube Q has a closed neighborhood homeomorphic to the cube Q itself. Lemma [5-6]. Let A OA Q ; c fP and locally convex subfunctors of the functor following pairs are homeomorphic: nP the , ( ) ( ), ( ) F Q F S ; ( , ) Q S . those. any connected Z − set cube Q and OA ( ) ( ), F Q BdQ ( , Q BdQ ( ) n P X ( ) P X ; ). ( )\ P X ( ) \ P X Note P X Q − variety for any infinite compact set X n N  . In the particular case ( ) \ ( ) f n P Q P Q variety. A subspace P X ( ) P X . For an infinite compact set X expX , the subspace exp ( ) exp X  ( ) exp X   − − ( ) exp X  a F-subspace of the space expX, the subspace ( ) F X  that the ( ) X subspaces ,  be a Z − set and id , where Q → Q ( ) \ P and are a f n , f . If OA −is close to O A ,  −close ( ) and any ( ) n P Q Q ( )\ P Q ( ) P Q − is a Q − ( ) P X  is convex, then there is a retraction : r Q id . , ( ) \ P Q ( )\ , to , f Q is G− set in Has the following Lemma [5-6]. For any compact set X , if the mapping ( ) P X → ( ): ( ) ( ) P f P X P X → : ( ) P X  →  − is close to f X  id : id , then X and a functor expnX  expnX then  − close and ( ) P X – retraction of ( ) X ( ) ( ) exp and  2 , where n( ) k X P X ( ) are open in expX, is open in , the monad [6] . c c f n P fP , fP , P is a countable union of compacta in expX , i.e. Theorem [5-6]. Functors , , , , f n compact. On the other hand, X ( ) X c nP and F – preserves the property of the layer in the −  nP , mappings being Q − , F − are locally convex subfunctors of the functor ( )   F  exp is dense everywhere in expX. a is f− manifolds, where , and 2 is n nP . an open set in . c c f n P fP , fP , P Corollary [5-6]. Functors , , nP is the n− T , , f n The normal subfunctor of the functor c f nP and F –preserves a) ) ( f P  ( ) f n P  nP , P  ; where F − is a locally convex subfunctor of the functor ; fP ,  ( ) ( ) f f P P P     ,( ) ( ) f n n P P     , where F − locally convex subfunctors of the functor Theorem [5]. Let F −be a continuous functor with finite supports preserving separable ( ) −  n - varieties. In particular, )   SP 2 th symmetric degree functor . The component P → of the X  f c f f ( ) P ( n T SP : f 2 2 f n  , 2 natural transformation is defined by the n  c f f c f f f P ( ) P ( ) F ( ) equality , 2 n 2 n 2 2 2 nP n 1 n     T ( , x x ,..., x ) = , X 1 2 n xi i =1 c c f n c P fP , nP P nP , b) Functors , , nP , , f n From the results of the main part it follows that the functor  − − and F .. . pic – preserves varieties.  ( n P  In particular, and all its subfunctors are functors. SP fP ,  −  n c ( ) Consequently, .. . pic functor, i.e. functors P and F − are where F is a locally convex subfunctor of the functor the functor is P also a f n  ,  − c c   nP . fP , pic nP ,   c c ) F ( ) , , f n c f n c .. . P , P, - functors, , nP . A N R−spaces. Then and  are everywhere f Note that the spaces 2 preserves varieties. dense subsets of the Hilbert cube Q and are preserved by the c f n F = P P fP F f Theorem [5]. For functors , , F ( ) reduced functors F , then ( ) , , f n and is 2 16

9. are Q − n ( ) P X ( ) \ P X SP X ( )  f homeomorphic to the Hilbert cube Q i.e. ; F ( ) Q is open in i.e. 2 varieties. On the other hand, by virtue of Theorem [13], for any infinitely compact set X and for a natural number n ( ) SP X , where F is one of the following F ( ) Q  N and , is a Z − set in ( ) set X ( ) SP X . In this case we have n P X c f n c P nP , P nP , the compact set . functors: locally convex functors , , , , , f n For an infinite  compact we set n c SP fP . fP and On the other hand, for these functors F , by Theorem ( ) = ( ) = F F Q , n SP X ( ) =  n =1 Theorem 2. For any infinite compact set X and any number n N  ( ) \ ( ) P X SP X ( ) SP X Z   − − − set in ( ) f Q 5.5[6]-5.8[6],  holds and 2 natural , the subspace F ( ) = ( ) = F Q Q is a Q − manifold and a subspace P X . ( ) = SP Q ( ) , A N R if X A N R  ( ) P X  Hilbert cube Q the following holds: ( ) SP Q  − is a Q − manifold, since the ( ) SP Q  ( ) = k SP Q SP Q  Q − varieties, since for < k ( ) SP Q . Theorem 3. For any infinite compact set X and for any n N  , the subspace P  ( ) P X  Proof. Let X be an infinite compact set and n N  . Note that the space ( ) P X  ( , ): ( ) [0,1] h t P X   ( , ) =(1 ) h t t t    − +  supp n  . =0 t ( ,0) =(1 0) 0 h   − +  ( ,0) = P X h id  >0 t ( , ) =(1 ) h t t t    − +  ( , ) supph t  points. those. ( (0,1]) h   From this Theorem 3 it follows that the compact set ( ) n . Therefore, ( F the following holds: n f ( ) \ F Q ) ; S 2 n Q It is known that and ; ( )\ F Q F ( ) S .   n ( ) SP X SP X ( ) ( )  . Note that SP and the For any compact set X we have: n . For the functor   n n SP X ( ) SP X ( ) ( ). P X n n Theorem 1. For any infinite compact set X and for any n N  , the subspace ( ) \ P X dense in ( ) P X . Proof. Let X be an infinite compact space. In this ( ) P X Q ; = ... x k xk > ,, =1 i i k n m  , i j m m i j    We construct ( , ): ( ) [0,1] h t P X   → ( , ) =(1 ) h t t t    − +  1) n SP X ( ) n n SP Q ( ) ( ) \ SP Q is homotopy subspace is open in ; SP Q ( ) \ n n k ( ) SP Q 2) are ( ) n k n  . We fix n m  + N case  . Take the measure n 1  + m is open in , where 0 1 k i m >0 ( ) X , , is homotopy dense in n =1 . . the homotopy ( ) P X  P X ( )  AR with a flat slope , since is convex and locally convex. We construct the desired  0 t =0 = → ( )  2( ) n P   P X For , homotopy flat X ( ,0) =(1 0) h  ( ,0) = h  −  +    0 . those. 0 , where . 0 0 id 0= 2 P X ( ) those. >0 SP X  n+ ( ) \ P X SP X t For , If , then ( , ) = (1 t  ( , ) supph −  +   n h t ) t ( ) 1) ( )   = , so how − points. This 0 . those. 0  t consists of than (  . ( ) n means that the subspace dense in ( ) P X is homotopy If , then  P X ( ) n+ . , since 0 n n contains more than ( 1) SP X ( ) Since the compact set is closed in distinct ( ) P X , it follows that for an infinite compact set X and a N  P ( ) X . ¦  n n the subspace ( ) \ P X SP X ( ) natural number n is a Z − set in nP X P X ( )  . those. the space 17

10.  − −set. We noted that for any compact ( ) SP X  матем. наук. Москва, 1991. Т.46. Вып.1 277. с.41-80. Щепин Е.В. Функторы и несчетные степени компактов Успехи мат. наук. Москва, 1981. Т.36. 3. с. 3-62. Жураев Т.Ф. Некоторые подфункторов функтора Рвероятностных мер. Москва, МГУ, 1989. 60 с., Деп. в ВИНИТИ АН СССР 05.07.89. 4471- В89. Жураев Т.Ф. Некоторые функтора вероятностных мер и его подфункторов: Дис. канд. физ.-мат.–М: МГУ, 1989. 90 с. Заричный М.М. Абсолютные экстензоры и геометрия умножения монад в категории компактов. Мат. сб. 1991, 182, 9, с. 1261-1280. Богатый С.А., Федорчук В.В. Теория ретрактов и бесконечномерные многообразия Итоги науки и техники. Алгебра. Топология. Геометрия, 1986. Т.24. с. 195-270. Чепмэн Т. Лекции о -многообразиях. Москва: Мир, 1981.– 170 с. Bessaga C., Pelczunski A. Selected topics in infinity dimensional topology. PWN, Warsaw, 1975, 58, 353 р. Banakh, T. Radul, M. Zarichnyi Absorbing sets in infinite – dimensional Manifolds, Math Studies Monogh. Ser. V.1. VNTL Publishers, 1996, 232 p. Жураев Т.Ф., Геометрические и топологические свойства пространств, являющихся ковариантных функторов. Ташкент.2022, Доктор.-дис. Мат. инс. Романовского АН Узбекистана, 198 с. Banakh T.O. Topology of probability measure spaces. I. Matem Stud. Proc. Lwow Math. Soc. 5 , 1995, р. 65-87. Ayupov Sh. A. Zhuraev T. F. On projectively inductively closed subfunctors of the functor P of probability measures. Journal of Mathematical Sciences. 2020. Vol. 245(3). pp. 382– 389. Zhuraev T.F. On projectively quotrient functors. Comment. Math. Univ. Corolinae, 2001. V.42. 3. р.р. 561-573. Zhuraev T.F. On paracompact spaces and projectively inductively closed functors. Applied General Topology. 2002. Vol.3(1). pp. 33-44. ZhuraevT.F. On paracompact inductively closed functors, and dimension. Mathematica Aeterna, International Journal for Pure and Applied Mathematics. Bulgariya. 2015. 6. pp. 175-189. Zhuraev T.F. On dimension and Z ( )  set X the following holds: From Theorem 3 it follows Corollary 1. For any infinite compact set X and for any n N  , the compact set ( ) P X  Corollary 2. For any infinite compact set X , the ( ) SP X   − P X is a • n P X ( ) . • геометрические свойства n is a Z − set in n SP X ( ) • геометрические свойства . • −set in  − Z ( )  .. . pic P X subspace is a . • − functors Using the given properties of and theorem [10] (Problem 16 § 1.2), the following can be easily proven.  − P , , f n P SP nP holds: • • .. . pic F : nP , Theorem 4. For functors • n c c c nP , P and locally convex ,SP,P, , , f n • subfunctors значениями некоторых f aF ( ) ; Q ; • 2 • ; aF ( ) Q . aF X ( ) F X where is the one-point Alexandrov compactification ( ) . • of the space • n SP For a functor of symmetric degree we have: X =[0,1] • spaces, projectively 1. For the segment = ) SP X I n ; those. SP the space  n n ( ; is a standard simplex of dimension ([0,1])   ; = X S is a sphere in n CP isa projective  − .. . pic n n • functors. Mathematica Aeterna, International Journal for Pure and Applied Mathematics. Bulgariya 2014, 6, pp.577-596. Федорчук В.В. Слабосчетномерные пространства Усп. мат. Наук, 2007, Т. 62, 2 (374), с.109-164. Энгелькинг Р. Общая топология. Москва. Мир, 1986. 752 с. 2 3 R 2. If 2 ) , then • n SP S dimensions n ; ( ; (complex) space • S− is a circle in 1 2 2 S M − ; = S− circle 2 3. If X 1 ) ; R , then R ; 3 2 SP S ( exp Miyobius sheet in 2 R R X 4.  ; If S in , then − 4 3 exp S exp S is a set consisting of no more than sphere of dimension 3 in , where 3 3 ; 2 2 5. If X is a circle in R expcS B , then , where expcS consists of 2 S R  . 2 R is a closed circle in connected continua of the circle REFERENCES • Федорчук В.В. Ковариантные функторы в категории компактов, абсолютные ретракты и Q-многообразия Успехи матем. наук. Москва, 1981. Т.36. Вып.3.с. 177-195. Федорчук В.В. Вероятностные меры в топологии. Успехи • 18