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14URAL MATHEMATICAL JOURNAL, Vol. 6, No. 2, 2020, pp. 108-116
DOI: 10.15826/umj.2020.2.011
THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE
Tursun K. Yuldashev^, Farkhod G. Mukhamadiev"^
National University of Uzbekistan, 700174, Tashkent, Uzbekistan
ttursun.k.yuldashev@gmail.com, ttfarhod8717@mail.ru
Abstract: In this paper, the local density (Id) and the local weak density (Zwd) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree SPn and the subfunctor of permutation degree SPQ, P is the cardinal number of topological spaces. Let X be an infinite Ti-space. We prove that the following propositions hold.
(1) Let Yn C Xn; (A) if d (Yn) = d (Xn), then d (SPnY) = d (SPnX); (B) if Iwd (Yn) = Iwd (Xn), then Iwd (SPnY) = Iwd (SPnX).
(2) Let Y C X; (A) if Id (Y) = Id (X), then Id (SP nY) = Id (SP nX); (B) if wd (Y) = wd (X), then wd (SPnY) = wd (SPnX).
(3) Let n be a positive integer, and let G be a subgroup of the permutation group Sn. If X is a locally compact Ti-space, then SPnX, SPQX, and expn X are fc-spaces.
(4) Let n be a positive integer, and let G be a subgroup of the permutation group Sn. If X is an infinite T1 -space, then nn w (X) = nn w (SPnX) = nn w (SPqX) = nn w (expn X).
We also have studied that the functors SPn, spq, and expn preserve any fc-space. The functors SP2 and spq do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite Ti-space X coincides with the densities of the spaces Xn, SPnX, and expn X. It is also shown that the weak density of an infinite Ti-space X coincides with the weak densities of the spaces Xn, SPnX, and expn X.
Keywords: Local density, Local weak density, Space of permutation degree, Hattori space, Covariant functors.
1. Introduction
In mathematical research in the modern world, a special place is occupied by the study of the topological properties of objects in various topological spaces. Research in general topology is topical, where the properties of topological spaces and their continuous mappings, operations on topological spaces and their mappings, as well as the classification of topological spaces are studied. This section of general topology uses concepts such as neighborhood, closure, compactness, density, separability, cardinal number, n-base of sets, sum, intersection, Tikhonov product, and others. An overview of the main stages in the development of set-theoretic topology is given in [1]. Some cardinal properties of topological spaces related to weak density were studied in [4]. In [5], some cardinal properties of Hattori spaces and their hyperspaces were studied. In [2, 6], some properties of topological spaces related to local density and local weak density in various topological spaces were studied.
Along with the concepts of local t-density and local weak t-density in various topological spaces, we are interested in such concepts as hereditary Souslin number, hereditary density, hereditary n-weight, hereditary Shanin number, hereditary pre-Shanin number, hereditary caliber, hereditary precaliber, hereditary weak density, hereditary Lindelof number, and hereditary extent of topolog-ical spaces.
Denote by P the cardinal number of topological spaces. Let SPn be a functor of permutation degree, and let SPg be a subfunctor of the functor of permutation degree.
At the Prague topological symposium in 1981, V.V. Fedorchuk posed the following general problem in the theory of covariant functors [8] and thus created a new direction of research in this area of topology.
Problems. Let P be a geometrical or topological property, and let F be a covariant functor. If X has the property P, does F(X) have the same property P? The opposite problem: for which functors F(X) the space X has the property P if F(X) has this property?
In [2], it was proved that the property of local density and the property of local weak density coincide for stratifiable spaces. These cardinal numbers are preserved under open mappings and are inverse invariant of a class of closed irreducible mappings.
In our present work, we prove that the following propositions are true for an infinite Ti-space X:
(1) if (Y) = (X) for Y c X, then (SPnY) = (SPnX);
(2) if (Y) = (X) for Y c X, then (SPnY) = (SPnX);
(3) if X is a locally compact T1-space, n is a positive integer, and G is a subgroup of the permutation group Sn, then SPnX, SP^X, and expn X are k-spaces;
(4) if X is an infinite T1-space, n is a positive integer, and G is a subgroup of the permutation group Sn, then
nnw (X) = nnw (SPnX) = nnw (SPgX) = nnw (expnX).
We also prove that the functors SP2 and SP,3 do not preserve Hattori spaces on the real line. In addition, we prove that the density of an infinite T1 -space X coincides with the densities of the spaces Xn, SPnX, and expn X. We show that the weak density of an infinite T1-space X coincides with the weak densities of the spaces Xn, SPnX, and expn X.
2. Auxiliary material
Recall some notation, definitions, and statements that are widely used in this paper. The permutation group of X is the group of all permutations (one-to-one and onto mappings X ^ X). The permutation group of a set X is usually denoted by S(X). If X = {1,2,3,..., n}, then S(X) is denoted by Sn.
Let Xn be the nth power of a compact set X. The permutation group Sn of all permutations acts on the nth power Xn as the permutation of coordinates. The set of all orbits of this action with quotient topology is denoted by SPnX. Thus, points of the space SPnX are finite subsets (equivalence classes) of the product Xn. Thus, two points (x1, x2,... , xn), (y1, y2,..., yn) € Xn are equivalent if there is a permutation a € Sn such that y» = x^). The space SPnX is called the n-permutation degree of a space X. An equivalent relation by which we obtain the space SPnX is called the symmetric equivalence relation. The nth permutation degree is always a quotient of Xn. Thus, the quotient mapping is denoted as nn: Xn ^ SPnX, where nn ((x1, x2,..., xn)) = [x = (x1, x2,..., xn) ] is an orbit of the point x = (x1, x2,..., xn) € Xn.
The concept of permutation degree has generalizations. Let SP^X be any subgroup of the group Sn. Then it also acts on Xn as the group of permutations of coordinates. Consequently, it generates a G-symmetric equivalence relation on Xn. This quotient space of the product Xn under the G-symmetric equivalence relation is called the G-permutation degree of the space X and is denoted by SP^X. The operation SP^ is also a covariant functor in the category of compact sets and is said to be a functor of G-permutation degree. If G = Sn, then SPg = SPn. If the group SP^X consists of only one element, then SP^X = Xn.
Let X be a Ti-space. The collection of all nonempty closed subsets of X is denoted by exp X. The family B of all sets of the form
O (Ui, ..., Un) = {F : F € exp X, F C .U Ui, F n U = 0, i = 1, 2,...,n}
generates a topology on the set exp X, where U1,..., Un is a family of open sets of X. This topology is called the Vietoris topology. The set exp X with the Vietoris topology is called the exponential space or the hyperspace of X [9]. Let X be a T1-space. Denote by expn X the set of all closed subsets of X such that expn X = {F € exp X : | F | < n}.
We use the following notation:
expw X = U{expn X : n = 1, 2,...}, expc X = {F € exp X : F is compact inX}.
It is clear that expn X C expw X C expc X C exp X for any topological space X. Moreover, if G1 C G2 for subgroups G1 and G2 of the permutation group
nn( (X1,X2, . . . ,®n) ) = [x = (X1,X2, € Xn,
then we have the following chain of factorizations of functors [9]:
Xn — SP^ X — SP£2 X — SPnX — expn X.
A subset D of a topological space X is called a dense set in X if [D] = X. Define the density d (X) of X by d (X) =min{|D| : D is a dense subset of X} [7].
We say that the local density of a topological space X is t at a point x if t is the smallest cardinal number such that x has a neighbourhood of density t in X. The local density at a point x is denoted by Id (x). The local density of a topological space X is defined as the supremum of all numbers Id (x) for x € X: Id (X) = sup{ld (x): x € X} [2, 6]. It is known that Id (X) < d (X) for any topological space.
Example. Let R be the real line with discrete topology. In the discrete topological space (R, Td), every point x € R has the one-point neighbourhood {x}. It follows that Id (R, Td) = 1. On the other hand, the boundary set of any set is empty in a discrete space, and hence the only dense set is the space itself. This means that d (R, Td) = |R| = c. Then 1 = ld (R, Td) < d (R, Td) = c.
We say that the weak density of a topological space is t > if t is the smallest cardinal number such that there exists a n-base coinciding with t centered systems of open sets, i.e., there is a n-base B = IJ{Ba : a € A}, where Ba is a centered system of open sets for every a € A, |A| = t .
The weak density of a topological space X is denoted by wd (X). If d (X) = t > then wd (X) < t . Similarly, if Y is dense in a topological space X, then wd (Y) = wd (X) [3]. The following theorem and proposition were proved in [3].
Theorem 1. Let {Xa : a € A} be a family of topological spaces such that wd (Xa) < t > for every a € A, where |A| < 2T. Then wd (IlaeA Xa) < t.
Proposition 1. Assume that X and Y are topological spaces and there exists a continuous "onto" mapping f: X Y. Then wd (Y) < wd (X).
A topological space X is called a locally weak t-dense space at a point x € X if t is the smallest cardinal number such that x has a neighbourhood of weak density t in X. The local weak density at a point x is denoted by Iwd (x). The local weak density of a topological space X is defined as the supremum of all numbers lwd(x) for x € X: Iwd (X) = sup{lwd (x): x € X} [2, 6]. If X is a space of local density t and f: X — Y is an open continuous "onto" mapping, then Y is a space of local density t [12]. The quotient mapping nn: Xn — SPnX is a clopen continuous onto mapping [13].
The following two statements are from [11].
Proposition 2. If X is a topological space, then expn X is dense in exp X.
Proposition 3. X is separable if and only if exp X is separable.
These propositions imply that, for any infinite T1-space X, we have
Iwd (X) = Iwd (Xn) = Iwd (SPn X).
The following theorem was proved in [4].
Theorem 2. Let X be an infinite T1-spa^ce. Then wd (X) = wd (expn X) = wd (exp X).
To substantiate our results, we also use the following notation and definitions from [7].
An uncountable cardinal number t is a caliber of a topological space if every family of cardinality t consisting of nonempty open sets contains subfamily of the same cardinality with nonempty intersection. The caliber of a topological space X is denoted by k (X).
The cardinal number min{T: t + is a caliber of X} is called the Shanin number of X and is denoted by sh (X).
A cardinal number t > is called a precaliber of a space X if every family of cardinality t consisting of nonempty open subsets of X contains a subfamily of cardinality t with finite intersection. Define
pk (X) = {t + : t is a precaliber of X}.
The cardinal number psh (X) = min{T + : t is a precaliber of X} is called the pre-Shanin number. We always have c (X) < psh (X) < sh (X) < d (X).
The Lindelof number l(X) of X is defined as l(X) = min{T : every open cover of X has a refinement of cardinality < t} + If l(X) = i.e., every open cover has a countable refinement, we say that X is a Lindelof space.
The notion of cellularity (Souslin number) c (X) of X is defined as c (X) =min{T : every family of pairwise disjoint nonempty open subsets of X has cardinality < t} + If c(X) = we say that X has the countable chain condition (Souslin property).
The spread s(X) and the extent e(X) are defined as follows: s(X) =sup{|D| : D is a discrete subset of X} + and e(X) =sup{|D| : D is a discrete closed subset of X} + respectively.
For a metrizable space X, we have l (X) = d (X) = c (X) = s (X) = e (X).
For a cardinal function we define the corresponding hereditary cardinal function h^> = sup{^(Y) : Y c X}. For example, we have the hereditary Souslin number hc (X), the hereditary density hd (X), the hereditary n-weight hnw (X), and the hereditary Shanin number hsh (X). Similar symbols we use to denote the hereditary pre-Shanin number, the hereditary caliber, the hereditary precaliber, the hereditary weak density, the hereditary Lindeloof number, and the hereditary extent of the space X, respectively: hpsh (X), hk (X), hpk (X), hwd (X), hl (X), and he (X).
It is easy to see that the hereditary Souslin number hc (X) of a space X coincides with its spread s (X).
Definition 1. A topological space is a k-space if it is a quotient image of some topological space Y.
Recall that a topological space is locally compact if, for every x € X, there exists a neighbourhood U of x such that [U] is a compact subspace of X.
In 2010, Hattori defined [10] the following topology on R. Let R be the real line and A C R. The topology t(A) on R is defined as follows:
(1) for each x € A, {(x — e, x + e) : e > 0} is the neighbourhood base at x;
(2) for each x € R\A, {[x, x + e) : e > 0} is the neighbourhood base at x.
The space (R, t (A)) is called [5] a Hattori space. Let te be the Euclidean topology on R. Note that, for any A, B C R, we have A D B if t (A) C t (B), in particular, t (R) = te C t (A) and t (B) C t(0) = ts. We set Ptop (R) = {t (A) : A C R} and define a partial order < on Ptop (R) by the inclusion: t (A) < t (B) if t (A) C t (B).
3. Main results
Theorem 3. Let X be an infinite T1-space, and let Yn be dense in Xn. Then SPnY is also dense in SPnX.
Proof. Let Yn be a dense subset of Xn, and let SPnU be an arbitrary open set from SPnX. Since the mapping nn : Xn ^ SPnX is continuous, the set (nn)-1 (SPnU) c Xn is open. Thus, taking into account the density of Yn in Xn, we conclude that «)-1 (SPnU) n Yn = 0. Therefore, there exists y € Yn such that y € «)-1 (SPnU). Then <(y) € SPnU (and <(y) € SPnY). Hence, we have SPnU n SPnY = 0 for every open set SPnU. This means that the set SPnY is dense in SPnX. Theorem 3 is proved. □
Corollary 1. If X is an infinite T1-sp^ce and Yn is a subset of Xn such that d (Yn) = d (Xn), then d (SPnY) = d (SPnX).
Proposition 4. Assume that X is an infinite T1-spa^ce, n is a positive number, and G1 and G2 are subgroups of the permutation group Sn such that G1 c G2. Then
d (X) = d (Xn) = d (Sfg1 X) = d (SPg2 X) = d (SPnX) = d (expn X).
Proof. Let X be an infinite T1-space. Taking into account that
Xn ^ SP(n1 X ^ SP(52 X ^ SPnX ^ expn X
and the fact that continuous mappings do not increase the density of topological spaces, we directly obtain the inequalities
d (X) > d (Xn) > d (SP,n1 X) > d (Sfg2X) > d (SPnX) > d (expnX).
By Propositions 2 and 3, we get d (X) = d (expn X, and hence
d (X) = d (Xn) = d (SPg1 X) = d (SPg2 X) = d (SPnX) = d (expn X).
Proposition 4 is proved. □
Theorem 4. Let X be an infinite T1-sp^ce, and let Yn be a locally dense set in Xn. Then SPnY is also locally dense in SPnX.
Proof. The set Yn is locally dense in Xn. By definition, for any point y € Yn, there exists a neighbourhood Oy c Xn such that Oy is dense in Xn. Then Theorem 3 implies that SPn(Oy) is also dense in SPnX. On the other hand, the quotient mapping nn : Xn ^ SPnX is an open mapping. Therefore, SPn(Oy) is a neighbourhood of the point nn(y) € SPnY. Then SPnY is locally dense in SPnX. Theorem 4 is proved. □
Corollary 2. If X is an infinite T1-space and Y C X is such that Id (Y) = Id (X), then
ld (SPnY) = ld (SPnX).
Theorem 5. Let X be an infinite T1-space. Then wd (X) = wd (SPnX).
Proof. First, we will show that wd (SPnX) < wd(X). Suppose that wd(X) = t > Then wd (Xn) = t by Theorem 1. The space SPnX is a continuous image of the space Xn. Proposition 1 implies that wd (SPnX) < t.
Now we will prove that wd (SPnX) > wd (Xn). To this end, assume that wd (SPnX) = t > This means that there exists SPnB = U {SPnBa : a € A, |A| = t} and this is a n-base in SPnX, where SPnBa = {SPnUf : s € Aa} is a centered system of nonempty open sets for every a € A. We set
Ba = { (nn)-1 (SPnUf) : s € A«}, B = U{Ba : a € A}.
Let us show that Ba is a centered system of nonempty open sets in Xn for every a € A. For every finite subfamily {SPnUf. of SPnB«, we have n^SP^f. = 0. Then
0 = (< )-1 (ni=1 sp nUa) = ni=1( (nn )-1 (SPnUSi)).
This shows that Ba = { (n n )-1 (SPnUsa) : s € Aa} is also a centered system of nonempty open sets in Xn. Now, we show that B is a n-base in Xn. Since
SPnB = u{SPnBa : a € A, |A| = t}
is a n-base of SPnX, for every open subset SPnU of SPnX, there exists SPnUsa € SPnB« C SPnB such that SPnUsa C SPnU. Since the quotient mapping nn : Xn — SPnX is open and onto, we have
(<)-1 (SPnUsa) C (nn)-1 (SPnU). This means that B is a n-base in Xn. Therefore, we have wd (Xn) < t. Theorem 5 is proved. □
Corollary 3. If X is an infinite T1-spa^ce and Y C X is such that wd (Y) = wd (X), then
wd (SPnY) = wd (SPnX).
Theorem 6. Let X be an infinite T1-spa^ce, and let Yn be locally weakly dense in Xn. Then SPnY is locally weakly dense in SPnX.
Proof. Suppose that X is an infinite T1 -space and Yn C Xn is locally weakly dense. Then, for every point y € Yn, there exists a neighbourhood Oy such that Oy is weakly dense in Xn. According to Theorem 5, SPn(Oy) = {nn(y') : y' € Oy} is also weakly dense in SPnX. This means that, for every point (y) € SPnY, there exists SPn(Oy) such that it is weakly dense in SPnX. This shows that SPnY is locally weakly dense in SPnX. Theorem 6 is proved. □
Corollary 4. If X is an infinite T1 -space and Yn C Xn is such that Iwd (Yn) = Iwd (Xn), then Iwd (SPnY) = lwd (SPnX).
Proposition 5. Assume that X is an infinite T1-spa^ce, n is a positive number, and G1 and G2 are subgroups of the permutation group Sn such that G1 C G2. Then
wd (X) = wd (Xn) = wd (SP^ X) = wd (SP£2X) = wd (SPnX) = wd (expn X).
Proof. Let X be an infinite Ti-space. Taking into account that
Xn ^ SP^ X ^ SPg2 X ^ SPnX ^ expn X
and the fact that continuous mappings do not increase the weak density of topological spaces, we directly obtain the inequalities
wd(X) > wd (Xn) > wd (SP(g1 X) > wd (SPg2X) > wd (SPnX) > wd(expnX).
According to Theorem 2, wd (X) = wd (expn X). Hence, we get
wd(X) = wd (Xn) = wd (SP(g1 X) = wd (SPg2X) = wd (SPnX) = wd(expnX).
Proposition 5 is proved. □
Proposition 6. Assume that X is a locally compact T1-spa^ce, n is a positive integer, and G is a subgroup of the permutation group Sn. Then SPnX, SP^X, and expn X are k-spaces.
Proof. Let X be a locally compact T1 -space. Then Xn is a locally compact space for each n € N. The spaces SPnX, SP^X, and expn X become quotient images of the space Xn. Therefore, SPnX, SPgX, and expn X are k-spaces. Proposition 6 is proved. □
Corollary 5. The functors SPn, SPg, and expn preserve any k-space.
Proposition 7. Assume that X is an infinite T1-spa^ce, n is a positive integer, and G is a subgroup of the permutation group Sn. Then nnw (SPnX) = nnw (X).
Proof. It was proved in Proposition 4 that d (SPnX) = d (X), n € N. It is known that any dense set M c X can be a n-net of this space. Hence, we have nnw (SPnX) = nnw (X). Proposition 7 is proved. □
Corollary 6. Assume that X is an infinite T1 -space, n is a positive integer, and G is a subgroup of the permutation group Sn. Then
nn w(X) = nn w(SPn X) = nnw(SPgX) = nnw(SPg1 X) = = nnw(SPg2 X) = nnw(expn X) = nnw(expw X) = nnw(exp X).
Theorem 7. Let A be a subset of R such that int (R\A) = 0. Then the following nonequalities hold for the Hattori space (R, t (A)) and the functor of permutation degree SP2:
(1) s (R, t(A)) = s (SP2(R, t(A)));
(2) hd(R, t(A)) = hd (SP2(R, t(A)));
(3) hnw (R, t(A)) = hn (SP2(R, t(A)));
(4) hsh(R, t(A)) = hsh (SP2(R, t(A)));
(5) hc(R, t(A)) = hc (SP2(R, t(A)));
(6) hk (R, t(A)) = hk (SP2(R, t(A)));
(7) hpk(R, t(A)) = hpk (SP2(R, t(A)));
(8) hpsh (R, t(A)) = hpsh (SP2(R, t(A)));
(9) hwd (R, t(A)) = hwd (SP2(R, t(A)));
(10) hl (R, t(A)) = hl (SP2(R, t(A)));
(11) he(R, t(A)) = he (SP2(R, t(A))).
Proof. It is known that the space SP2X contains the squared Hattori space X2. However, X2 contains a discrete set of cardinality c. The other nonequalities can be easily checked. Theorem 7 is proved. □
Corollary 7. The functor SP2 does not preserve Hattori spaces on the real line.
Corollary 8. Let A be a subset of R such that int (R\A) = 0, and let G be an arbitrary subgroup of the group S3. Then the following nonequalities hold for the Hattori space (R, t (A)) and the functor of permutation degree SP,3:
(1) s (R, t (A)) = s (SP3(R, t(A))) ;
(2) hd (R, t (A)) = hd (SP^(R, t(A))) ;
(3) hnw (R, t (A)) = hnw (SP,3(R, t(A))) ;
(4) hsh (R, t (A)) = hsh (SP3(R, t(A))) ;
(5) hc (R, t (A)) = hc (SP3(R, t(A))) ;
(6) hk (R, t (A)) = hk (SP3(R, t(A))) ;
(7) hpk (R, t (A)) = hpk (SP|(R, t(A))) ;
(8) hpsh (R, t (A)) = hpsh (SP|(R, t(A))) ;
(9) hwd (R, t (A)) = hwd (SP^(R, t(A))) ;
(10) hl (R, t (A)) = hl (SP3(R, t(A))) ;
(11) he (R, t (A)) = he (SP|(R, t(A))) .
Corollary 9. The functor SP,3 does not preserve Hattori spaces on the real line.
REFERENCES
1. Aleksandrov P.S., Fedorchuk V. V., Zaitsev V.I. The main aspects in the development of set-theoretical topology. Russian Math. Surveys, 1978. Vol. 33, No. 3. P. 1-53. DOI: 10.1070/RM1978v033n03ABEH002464
2. Beshimov R. B., Mamadaliev N.K., Mukhamadiev F. G. Some properties of topological spaces related to the local density and the local weak density. Math. Stat., 2015. Vol. 3, No. 4. P. 101-105. DOI: 10.13189/ms.2015.030404
3. Beshimov R. B. A note on weakly separable spaces. Math. Morav., 2002. Vol. 6. P. 9-19. DOI: 10.5937/MatMor0206009B
4. Beshimov R. B. Some cardinal properties of topological spaces connected with weakly density. Methods Funct. Anal. Topology, 2004. Vol. 10, No. 3. P. 17-22. http://mfat.imath.kiev.ua/article/?id=251
5. Beshimov R. B., Mukhamadiev F. G. Cardinal properties of Hattori spaces and their hyperspaces. Questions Answers Gen. Topology, 2015. Vol. 33, No. 1. P. 43-48.
6. Beshimov R. B., Mukhamadiev F. G., Mamadaliev N. K. The local density and the local weak density of hyperspaces. Int. J. Geom., 2015. Vol. 4, No. 1. P. 42-49.
7. Engelking R. General Topology. Berlin: Heldermann Verlag, 1989. 529 p.
8. Fedorchuk V. V. Covariant functors in the category of compacts, absolute retracts, and Q-manifolds. Russian Math. Surveys, 1981. Vol. 36, No. 3. P. 211-233. DOI: 10.1070/RM1981v036n03ABEH004251
9. Fedorchuk V. V., Filippov V. V. Topology of Hyperspaces and its Applications. Moscow: Mathematica, Cybernetica, 1989. Vol. 4. 48 p. (in Russian)
10. Hattori Y. Order and topological structures of posets of the formal balls on metric spaces. Mem. Fac. Sci. Eng. Shimane Univ. Ser. B: Math. Sci., 2010. Vol. 43. P. 13-26.
11. Michael E. Topologies on spaces of subsets. Trans. Amer. Math. Soc., 1951. Vol. 71, No. 1. P. 152-182. DOI: 10.1090/S0002-9947-1951-0042109-4
12. Mukhamadiev F. G. Some cardinal and topological properties of the n-permutation degree of a topological spaces and locally T-density of hyperspaces. Bull. Nat. Univ. Uzbekistan: Math. Nat. Sci., 2018. Vol. f, No. f. Art. no. ff. P. 30-35. https://uzjournals.edu.uz/mns_nuu/volf/issf/ff
13. Wagner C.H. Symmetric, Cyclic, and Permutation Products of Manifolds. Warszawa: PWN, 1980. 48 p.
