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МАТЕМАТИЧЕСКИЕ МЕТОДЫ В ЕСТЕСТВЕННЫХ НАУКАХ MATHEMATICAL METHODS IN NATURAL SCIENCES
Scientific article
DOI: 10.18287/2541-7525-2023-29-2-40-47
Submited: 27.02.2023 Revised: 04.04.2023 Accepted: 30.06.2023
T.F. Zhuraev
Tashkent State Pedagogical University named after Nizami, Tashkent, Uzbekistan E-mail: [email protected]. ORCID: https://orcid.org/0009-0005-5379-3862
M.V. Dolgopolov
Samara State Technical University, Samara, Russian Federation E-mail: [email protected]. ORCID: https://orcid.org/0000-0002-8725-7831
EQUIVARIANT PROPERTIES OF THE SPACE Z(X) FOR A STRATIFIABLE SPACE X
ABSTRACT
In this paper, we prove the action of the compact group G defined by the stratified space X is continuous to the space Z(X) being a stratified space containing the self-stratified space X as a closed subset. An equivariant analogue of some results of R. Cauty concerning A(N)R(S) - spaces is proved. It is presented that the orbit space Z(X)/G by the action of the group G is a S space.
Key words: equivariant maps; stratified space; group actions; orbit space; invariant set; homotopy density; dimension; absolute extensor; neighborhood extensor; covariant functor; probabilistic measures.
Citation. Zhuraev T.F., Dolgopolov M.V. Equivariant properties of the space Z(X) for a stratifiable space X. Vestnik Samarskogo universiteta. Estestvennonauchnaia seriia = Vestnik of Samara University. Natural Science Series, 2023, vol. 29, no. 2, pp. 40-47. DOI: http://doi.org/10.18287/2541-7525-2023-29-2-40-47. (In Russ.)
Information about the conflict of interests: authors and reviewers declare no conflict of interests.
© Zhuraev T.F., 2023
Tursunboy F. Zhuraev — Doctor of Physical and Mathematical Sciences, associate professor of the Department of Mathematics, Tashkent State Pedagogical University named after Nizami, 27, Bunyodkor Street, Tashkent, 700100, Uzbekistan.
©c Dolgopolov M.V., 2023
Mikhail V. Dolgopolov — associate professor, Candidate of Physical and Mathematical Sciences, Department of Higher Mathematics, Samara State Technical University, 244, Molodogvardeyskaya Street, Samara, 443100, Russian Federation.
Introduction
In the category of stratifiable spaces and continuous images, we include one construction belonging to the test space [1; 2] that defines the covariant functor in this category. This construction defines the functor that allows each stratified space X to be immersed in a closed manner into some other space Z(X), which is the stratified space with "good" functorial, geometric and topological properties.
A stratifiable spaces can be defined of as topological space that is divided into smooth manifolds. Then a stratification in this context [3] is a structure associated with the emergence of closed sets, which is locally a decomposed space. This is what [4] refer to as a "germinal stratification". Each decomposed space causes a bundle of germs-stratifications, hence the concepts are consistent. Another notion of stratification can be found in [4], where the boundary conditions are slightly different. Stratifiable spaces always admit
tangent bundle. They are relevant because they are really singular, while the usual vector bundle is not [5]. Families of examples and applications [6; 7] arise from smooth equivariant vector bundles. The next main motivation for developing stratifiable vector bundles is to use them for quantization purposes. In particular, in the Kostant-Suriot-Weyl quantization picture, three components of the initial data are required: a symplectic manifold, a complex linear bundle with a connection, and polarization, all of which satisfy various compatibility conditions [8]. Therefore, it is relevant to consider and define theorems for equivariant properties of spaces over a stratifiable space.
Let X be a stratified (briefly, S—space) space. For each open subset U of the space X and any point x G U of the set U we put:
a) n(U, x) = mm{m : x G Um}, where U = |Jc^=1 Uk;
b) Ux = Un(U, x)\(X\{x})n(U,x).
Obviously, the set Ux is an open neighborhood of the point x and Ux c U. The set Ux has the following properties:
1°. Ux is an open neighborhood of the point x; 2°. If Ux n Vy = 0 and n(U, x) < n(V,, y) then y G U. 3°. If Ux n Vy = 0 then x G V or y G U.
Let X be the topological space, |F(X)| be a complete simplicial complex whose vertices are points in the space X, i.e. |F(X)0| = X. The space |F(X)| has a weak topology. Now we define the topology on the space |F(X)|, the bases of open sets of which we denote by Z(X) consists of W open in F(X), satisfies following conditions:
01. W n X is an open in X;
02. F(W n X^ c W;
i.e. rZ(x) = {W G r\F(X)| : W satisfies the conditions o1-o2}.
Condition o2 means that every simplex a G F (X) is contained in W if all vertices of this simplex a lie entirely in W n X.
1. Main results
For the subset A c X, the set F(A) is a subcomplex of the full complex F(X) and Z(A) is a subspace of the space Z(X). Obviously, Z(A) is closed in Z(A) if A is closed in X.
For each n G Z+ = N U {0} we put Zn (X) = |F(X)n| Zn(X) is a subspace of Z (X). Then Zo(X) = X and Z (X) = Un=0 Zn (X). It is easy to see that for any n G Z+ the subspace Zn(X) is closed in Z(X). Let us introduce the following notation: T(A) = {a G F(X)\F(A) : a n A = 0}; M (A) = {x G Z (X) : exists a G F (A) such that x(a) > 0}; Tn(A) = T(A) n (F(X)n\F(X)n-1); Mn(A) = Z(A) U (M(A) n Zn(X);
For each e G (0,1)T(A) and for each n G N, define the set:
M (A,e) = Unez+ Mn(A,e),
where Mo(A,e) = Z(A) = F(A)| and Mn(A,e) = Z(A) U {a(e(a)) n n-1(Mn-i(A,e)) : a G Tn(A)}. Then the equality M(A,e) n X = A holds.
For each open set U of the space X, the set M(U,e) is open in Z(X). In this case, the family B(M) = = {(U,e) : U is open in e e (0,1)(U)} is an open base of the space Z(X).
Therefore, if for every n G N and every e G (0, i)Tl(A)UT2(A)u---UTn(A) the set Mn(A,e) is defined, then the family
B(M) = {M1, (U,e)} : U is open in X and e G (0,1)Tl(U)} is an open base for Z1(X), i.e. the following holds.
Lemma [9]. Families {M(U,e): U is open in X and e G (0,1)T(U)} and {M1 (U,e) : U is open in X and e G (0,1)Tl(U)} is the base of the space Z(X), (respectively, the space Z1(X)).
In the work [2] R. Cauty claimed that for the space Z(X) the following are true: a) Each continuous map f : A ^ Y, where A is a closed subset of the stratified space X, has a continuous extension to all X with values in Z(Y): that is, the following diagram holds
A f Y X f Z(Y)
fA = f. X, Y G S.
b) The stratifiable space X is AR(S) (ANR(S)) if and only if X is a retract (respectively, a neighborhood retract) of the space Z(X).
Definition [6]. A topological space L is called hyper-connected (respectively, m -hyper-connected) if for each i e N, there is a mapping hi : Ll x ai-1 ^ L satisfying a, b and c (respectively, a, b and d):
a) t e an-1 and ti = 0 implies hn(x, t) = hn-1(5i • x,Si • t) for each x e Ln and n = 2, 3,...
b) For each x e Ln the mapping t ^ Ln(x,t) maps the sets an-1 to L continuously;
c) For each x e L and a neighborhood U of x, there is a neighborhood V of x such that |J°=1 hi(Vi x x ai-1) C U and V C U;
d) For each x e L and a neighborhood U of the point x, there is a neighborhood V of the point x such that |Jn=1 hi(V1 xai-1) C U and V C U, where an-1 = {t e Rn : YT=1 ti = l,ti > 0} - (n-l) is a dimensional simplex a Si : An ^ An-1 mapping defined by the formula 5i(a1, a2,..., an) = (a1: a2,..., ai-1, ai+1,..., an) i = l,n, i.e. 5i - "forgetting" i-th coordinate of the product.
A space L is said to be a locally hyper-connected if for each point x e L, there exists a neighborhood V of the point x such that V is hyper-connected.
In the paper [2] R. Cauty proved that X e A(N)R(S) if and only if X is hyperconnected (respectively, locally hyperconnected).
Theorem 1. For an arbitrary ¿"-space X, the space Z(X)\X is a AR(S) space.
Proof. Let n e N. We construct mapping hn(z1,..., zn,t) : (ZX\X)n x an 1 ^ ZX\X assuming hn(z1,Z2,..., Zn)(t1,t2,... ,tn) = J2 n=1 Ziti, where (z1,z2,..., Zn) e (ZX\X )n, (t1,t2,... ,tn) e an -1,
En=1 ti = l,ti > 0.
It is easy to show that hn((z1,..., zn) x t) e Z(X )\X.
Now we show that the space Z(X)\X is hyperconnected.
a) Let t e an-1, t = (t1,t2,... ,ti-1,0,ti+1,ti+2,..., tn). Then
hn(z,t) = hn((z1,z2, . . ., zn)(t1, t2,...,ti-1, 0, ti+1,ti+2, . . . , tn) = (t1z1 + t2z2 + . ..+ti-1zi-1 + ti+1zi+1 + + . . .tnzn) = hn-1((z1, z2,.. .,zi-1 ,zi, zi+1,...,zn)(t1,t2,.. .,tn)) = hn-1(Siz,6it).
b) We fix zo e (Z (X )\X)n, zo = (z0°, z0,..., z0n), z0° e ZX\X.
Hence z0 = J2lk=1 mik, xL Eli=1 mik = mik > _ _ _
Let t e an-1, then t ^ hri(zo, t) = J2 ni=1 tiz0 = lk=1 Vkxk +12 E lk2=1 i^kx2k + ... + triY!kn=1 № x1; Let_us put ti^j = aij, ti > 0, Hj > 0, aij > 0, i = l,n, j = ¿1,..., ln, Eij aij = l. Hence, we get J]rn=1Y!jj=1 aijxj. Consider the set X0 = {xj : i,j}.
In this case, point h(z0,t) e Z(X0), i.e. there is a simplex a lying in Z(X0) whose vertices consist of points of the set X0. On the other hand, if we consider the simplex an-1 with vertices z0,...,z1 , i.e. a^1 = {z<0,z^,... ,z1 }. The mapping hn(z,t) with continuity in the argument t or the mapping t ^ hn(z,t) completely covers the simplex an-1, i.e. the mapping t ^ hn(z,t) as a homeomorphism maps an-1 to z1-1. Hence, the mapping t ^ hn(z,t) is continuous.
c) Let z0 e Z(X)\X and Uz0 be an arbitrary neighborhood of the point z0 in Z(X)\X. Consider suppz0 = = {x1, x2,..., xk} the support of the point z0 of the space Z (XX )\X. Then z0 e (x1,x2,..., xk} = a. By the definition of topology in the space Z(X)\X, the set V1 = ^p|UZ0 is open. Consider a set V of the form {z e aHUZ0 = V1 : segment [z,z0] C V1}. Obviously, the set V is open and convex. By definition, the following takes place: V C V1 C UZ0. Note that if z e (Z(X)\X)n, z = (z1,z2,... ,zn) and suppzi C A, A C X, then supp hn(z,t) C A. If V is convex, then the maps hn(z,t) by definition maps Vn x an-1 to V. Therefore, the following holds: U^L1 hn(Vn x an-1) C U. Hence, the space Z(X)\X is hyperslash. By virtue of R. Cauty's theorem [1], we obtain that Z(X)\X is AR(S). Theorem 1 is proved.
Theorem 2. The finite product of A(N)R(S) spaces is A(N)R(S) spaces. Theorem 2 is proved in [6].
Let X be a topological space, G is a topological group d : G x X ^ X is a continuous mapping such that
(1) 9(g,9(h,x)) = 9(gh,x) for all h e G and x e X;
(2) 0(e, x) = x for all x e X, where e is the unit of the group G.
The mapping 9 is called the action of the group G on the space X. The space X with a fixed action 9 of the group G is called a G-space.
A set A is called invariant under the action of the group G (or G-invariant) if G(A) = A, where G(A) = = {g(x) : g e G,x e A}.
For g e G, we define the mapping 9g : X ^ X by the formula 9g(x) = g(x) = 9(g,x). By virtue of (1) or (2) we have 9g o 9h = 9gh and 9e is the identity mapping lX of the space X into itself. Thus, 9g o 9g-1 = = 9e = lX = 9g-1 • 9g therefore, for g e G, the mapping 9g is a homeomorphism of the space X onto itself.
An action 0 is called effective if Ker0 = e (i.e., the mapping 0 is injective), where Ker0 = {g G G : g(x) = = x} for any x G X the kernel of the action of 0, and is almost effective if KerO - is a discrete subgroup of the group G. Obviously, the kernel Ker0 is a normal divisor of the group G and is closed in G.
We note, that some of the supporting statements were considered in the articles [10-14].
Definition [15]. Let X and Y be G—spaces.
The mapping ^ : X — Y is called an equivariant mapping (or G-mapping) if ^ commutes by actions, i.e. y(g(x)) = g(y(x)) for all g G G and all x G X.
For a fixed group G, the class of G-spaces is a class of objects of a certain category, whose morphisms are called equivariant maps.
An equivariant mapping ^ : X — Y that is a homeomorphism is called the G -equivalence of G -spaces.
Note that if we denote by Homeo(X) the group (with respect to, composition) of all homomorphisms of the space X onto itself. The mapping g — 0g defines a homeomorphism 0 : G — Homeo(X).
Let X be some G space, and let x G X. The set Gx = {g G G : g(x) = x} of elements of the group, for which x is a fixed point, is obviously a closed subgroup of the group G. This subgroup Gx is called the stationary subgroup (or the stabilizer of the point x).
On the other hand, note that ker 0 is exactly p|x£X Gx, i.e. ker 0 = p| x Gx.
The action of the group G on the space X is called free if for any point x G X the subgroup Gx is trivial. An action is called semi-free if the stationary subgroup Gx of any point x G X is either trivial or is the whole of G. Take x G X. The subspace G(x) = {g(x) : g G G} is called the orbit of the point x (with respect to the action of the group G). Note that G(x) c X for any x G X and for points x and y the sets G(x) and G(y) either do not intersect each other or coincide, i.e. G(x) G(y) = 0 or G(x) = G(y) for any x, y G X. By X\G = {G(x) : x G X} we denote the orbit set of G—space X. Let n = nx : X — X/G— be a natural mapping, associating the point x and the orbit x* = G(x). Then X\G is endowed with the quotient topology in the usual way (i.e., the set U c X\G is open if and only if n-1(U) is open in X ), and the resulting topological space is called the orbit space. Note that if U c X is open, then the set G(U) = |JgeG g(U) is open, since each of the sets g(U) = 0g(U) (recall that 0g : X — X is a homeomorphism).
Therefore, for an open U c X, the set n-1n(U) = G(U) is also open, which by definition means that the set n(U) is open in X\G. Hence the projection n : X — X/G is a continuous open map.
Theorem [15]. Let the group G be compact and X is some G— space. Then
(1) the space X \ G is Hausdorff;
(2) The projection n : X — X/G is a closed map;
(3) The projection n : X — X/G is a proper mapping (that is, the preimage of any compact set is compact);
(4) The compactness of the space X is equivalent to the compactness of the space X\G;
(5) The local compactness of the space X is equivalent to the local compactness of the space X\G.
Let X be a stratified G -space that is the topological group G acts on the space X, i.e. there is a
continuous mapping (G,X) : G x X — X defined by the formula: (g,x) = gx. On the test space Z(X), the action of the group G is defined as follows: (G, Z(X)) : G x Z(X) — Z(X) g G G, z G Z(X), z = J2ki=1 m^xi, mi = 1, mi > 0 (g,z) = g ■ z = ^i=1 mig(xi). Thus, the space Z(X) is a G— space. It is easy to see that the space X in the G space Z(X) is an invariant G-subset, i.e. if x G X, then g(x) = x G X.
Thus, the following holds.
Theorem 3. A continuous action of the group G defined on the space X extends continuously to the entire space Z(X). Take the point z = Yli=1 mixi, mi > 0, ^i=1 mi = 1, ( k k then Gz = < gz : g G G , gz = g J2 mixi = J2 migxi I i=1 i=1
Obviously, Gz = Gm,ix[+...+mkxk = {m{gx1 + ... + mkgxk : g G G}.
Note that Z(X/G) = Z(X)/G and is invariant in Z(X/G).
X X/G x — Gx ^ i ^ i
Z (X )—G—Z (X )/G z — Gz In his paper [9] Cauty proved the following
Lemma 1.2 [1]. Let X be a topological stratified space. If Y G S and A c Y — is closed and f : A — X is continuous, then the mapping f has a continuous extension f : Y — Z (X).
Lemma 1. Let X be a topological stratified G— space, A c Y a closed G— invariant subset, f : A — X an equivariant continuous mapping, when the mapping f has a continuous equivariant extension f : Y — Z(X).
Proof. Let A be a closed subset of the space X. We put W = X\A. W' = {x e W : x e Uy,y e A and U is open in X} and m(x) = mrAx{n(U,y) : y e A and x e Uy}. Obviously, W' С W and for every x e W' there is m(x) < n(W,x) < то.
Let W = Y\A, W' = {x e W : x e Uy,y e A and U is open in X}. Consider the open covered W* = = {Wx : x e W} subspace W. Since the subspace W is paracompact, there exists a locally finite G— cover V inscribed in W* .
For any v e V, we fix a point (vertex) xv e W such that gxv = xgv, where g e G. If a point xv e W' we fix such a point (vertex) av e A and an open set Sv,av e Sv such that xv e (Sv)a and n(Sv,av) = m(xv) and gav = agv. If xveW' we put av fixed a0 e A. Let {Pv : v e V} be partition of unity subordinate to V and Pv(x) = Pgv(gx). The required continuation F : Y ^ Z(X) is defined as follows:
F(x) = f f(x); if x e A F (x) 1 EPv(x) ■ f (av), if x e W
a) Now we show that the mapping F : Y ^ Z(X) is equivariant, i.e. gF(x) = F(gx), where g e G.
1) If the point x e A, due to the invariance of the set, we have, gF(x) = gf(x) = f(gx) = F(gx).
2) Let the point x e W then we have gF(x) = g22 (Pv(x) ■ f (av)) = 22 Pv(x) ■ gf (av) = Pv(x)f (agv).
F (gx) = J2 Pgv g(x) ■ f (agv) = J2 Pv (x)f (agv).
Hence, gF(x) = F(gx) i.e. the mapping F is equivariant.
b) Due to the -invariance of the closed set A and the simpliciality of a certain mapping F : Y ^ Z(X), the mapping F(x) is continuous.
Lemma 1 is proved.
By Lemma 1 and Theorem 1, we have
Theorem 4. The space X e G A(N)R(S) if and only if there is a G—retraction r (neighborhood) G— space Z( X) on the G— space X.
Lemma 2. Let X e A(N)R(S). Then there is a G— retraction Rn : O(Xn) ^ Xn such that G С Sn and n e N such that O(Xn) is a neighborhood Xn to Z(Xn).
Proof. Let X be ANR(S)— space. It follows from the results of R. Cauty that there is a retraction r : U ^ X, where U is a neighborhood of the space X in Z(X). We put V = (rn )~L(Un), where V С (Z(X))n, rn : Un ^ Xn, Un С (Z(X))n.
Now we define the mapping ф : Z(Xn) ^ (Z(X))n as follows: z e Z(Xn), z = ^k=1 mx,, xi = = (x\,x\,..., xin). We put ф^) = (22 mix\ mixl2, ..., 22 mixin). Obviously, ф(z) e (Z(X))n. It is easy to check that the mapping ф : Z(Xn) ^ (Z(X))n is continuous. We put Rn = rn оф and ф-1(у) = O( Z(Xn)).
Hence Rn : Z(Xn) ^ Xn. Now we show that Rn is an equivariant mapping, i.e. the equality R^(gz) = = gRn (z) holds.
Rn(gz) = rn^(gz)) = гп(ф(222 mi gx)) = rn (ф(22 mig(xii . ..,xin))) =
bn V
У.П (
= rn(ф(22mi(xig(i) xig(n)))) = rn((Emi xig{j))) = mi xj)) = g(rn22m xj) = gRn(z).
Hence, the mapping Rn is equivariant. It is easy to check that Rn is a continuous retraction.
Lemma 2 is proved.
Theorem 5. Let X e A(N)R(S). Then Xn e G - A(N)R(S), where G С Sn - is a subgroup of the group of all permutations.
Proof. Let X e A(N)R(S) and Y be a stratified G—space, A its closed invariant G— subset, f : A ^ Xn is an arbitrary continuous G—mapping. Let O = ф-1(й^(Xn))). We put Fg = Forn, rn = Rn— the Cartesian product of retraction Rn defined by Lemma 2, F : Y ^ Z(Xn) mapping defined in Lemma 1.
Then the mapping Fg is a G extension, since Fg is the composition of two G mappings F and rU. Obviously, Fg is an extension of the mapping f .The theorem is proved.
This theorem implies
Corollary 1. If X is a G— space and X e A(N)R(S) then X e G — A(N)R(S).
By virtue of Lemma 1, we can also assert.
Corollary 2. Let X be a topological stratifiable G—space. If Y is a stratified G—space A is an invariant G—space, f : A ^ X\G is an equivariant mapping. Then f has an equivariant continuous extension F : Y ^ Z (X/G).
Corollary 1 implies.
Corollary 3. If X e G — A(N)R(S), then X/G e G — A(N)R(S).
Fig. Illustrations to theorems: a — z0(x) - x; b — zi(x) — segments, vertices — point X (one-dimensional simplices); c — z2(x) — triangles, vertices — points X (two-dimensional simplices; d — z3(x) — tetrahedra,
vertices - points X (three-dimensional simplices) Рис. Иллюстрации к теоремам: a — z0(x) — x; b — zi(x) — отрезки, вершины — точки X (одномерные симплексы); c — z2(x) — треугольники, вершины — точки X (двумерные симплексы; d — z3(x) — тетраэдры, вершины — точки X (трехмерные симплексы)
Definition [13]. The set A с X is called homotopically dense in X if there exists a homotopy h(x,t): X x [0,1] ^ X such that h(x, 0) = idX and h(X x (0,1]) С A.
Theorem 6. For any stratified space X and for any n e N + subspace Z(X)\Zn(X) is homotopically dense in Z(X).
Proof. Let X is the stratified space and n e N+. Fixing the point z0 e Z(X)\Zn(X), where z0 e< x1,x2, ...,xn+1 > and
zo = ml xi + m<°x2 + ... + m0n+1xn+i,
thus suppz0 = {x1,x2, ...,xn+1}. We will construct the homotopy h(z,t): Z(X) x [0,1] ^ Z(X), assuming h(z,t) = tz0 + (1 - t)z.
By virtue of the convexity of the space Z(X) for any z e Z(X) and t e [0,1] the point h(^,t) belongs to Z(X), that is h(p,t) e Z(X), 4z e Z and Ш e [0,1]. If t = 0, then h(/л, 0) = z, that is h(/л, 0) = idZ(X).
If t > 0 and t < 1, then h(/,t) = tz0 + (1 — t)z belongs to Z(X)\Zn(X) because the carriers supph(/,t) of point h(t) consist of at least n + 1 points, that is
h(t) = tzo + (1 — t)z =
= t(m0{xi + m°2x2 + ... + m0n+1xn+i) + (1 — t)(m01x'1 + m°2x'2 + ... + m0kx'k) e Z(X)\Zri(X),
the point h(/,t) carrier consists of points z and z0 carriers, and it consists of different (n +1) points. So the point h(z, (0,1])CZn(X) and h(z, (0,1]) с Z(X)\Zn(X), which was required to be proved. The theorem is proved.
Conclusion
In this paper we consider that the functor Z is open, normal and monadic in the category of stratified spaces and continuous maps to itself. The dimensional properties of the space Z(X) for the stratified space X are also studied, the subfunctor of the functor Z with the corresponding nested dimension is determined for each n.
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DOI: 10.18287/2541-7525-2023-29-2-40-47
УДК 515.12 Дата: поступления статьи: 27.02.2023
после рецензирования: 04.04.2023 принятия статьи: 30.06.2023
Т.Ф. Жураев
Ташкентский государственный педагогический университет имени Низами, г. Ташкент, Узбекистан
E-mail: [email protected]: https://orcid.org/0009-0005-5379-3862
М.В. Долгополов
Самарский государственный технический университет, г. Самара, Российская Федерация E-mail: [email protected]. ORCID: https://orcid.org/0000-0002-8725-7831
ЭКВИВАРИАНТНЫЕ СВОЙСТВА ПРОСТРАНСТВА Z(X) ДЛЯ СТРАТИФИЦИРУЕМОГО ПРОСТРАНСТВА X
АННОТАЦИЯ
В этой статье доказано, что действие компактной группы О, определяемой стратифицированным пространством X, непрерывно для пространства Z (X), являющегося стратифицированным пространством, содержащим самостратифицированное пространство X как замкнутое подмножество.
Доказан эквивариантный аналог некоторых результатов Р. Коти относительно A(N)R(¿^-пространств. Также показано, что орбитальное пространство Z (X )/G под действием группы G является пространством S.
Ключевые слова: эквивариантные отображения; стратифицированное пространство; действия группы; орбитальное пространство; инвариантное множество; гомотопическая плотность; размерность; абсолютный экстензор; окрестностный экстензор; ковариантный функтор; вероятностные меры.
Цитирование. Zhuraev T.F., Dolgopolov M.V. Equivariant properties of the space Z(X) for a stratifiable space X // Вестник Самарского университета. Естественнонаучная серия. 2023. Т. 29, № 2. С. 40-47. DOI: http://doi.org/10.18287/2541-7525-2023-29-2-40-47.
Информация о конфликте интересов: авторы и рецензенты заявляют об отсутствии конфликта интересов.
© Жураев Т.Ф., 2023
Турсунбой Файзиевич Жураев — доктор физико-математических наук, доцент кафедры математики, Ташкентский государственный педагогический университет имени Низами, 700100, Узбекистан, г. Ташкент, ул. Юсуф Хас Хажиб, 103.
© Долгополов М.В., 2023 Михаил Вячеславович Долгополов — доцент, кандидат физико-математических наук, кафедра высшей математики, Самарский государственный технический университет, 443086, Российская Федерация, г. Самара, ул. Молодогвардейская, 244.
Литература
[1] Borges C.R. On stratifiable spaces // Pacific Journal on Mathematics. 1966. Vol. 17, No. 1. Pp. 1-16. DOI: https://doi.Org/10.2140/PJM.1966.17.1.
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[7] Borsuk K. The theory of retracts. Warsawa: [Pan?stwowe Wydawn. Naukowe], 1971. 251 p. Available at: https://archive.org/details/theoryofretracts0000bors.
[8] Hall Brian C. Quantum Theory for Mathematicians. Part of the book series: Graduate Texts in Mathematics. Vol. 267. New York: Springer. 2013. DOI: https://doi.org/10.1007/978-1-4614-7116-5.
[9] Cauty R., Guo Bao-Lin, Sakai K. The huperspaces of finite subsets of stratifiable spaces // Fundamenta Mathematicae. 1995. Vol. 147. Issue 1. Pp. 1-9. DOI: http://dx.doi.org/10.4064/fm_1995_147_1_1_1_9.
[10] Zhuraev T.F. Equivariant analogs of some geometric and topological properties on stratified spaces X // West. Kirg. Nat. University Named after Bolasagyn Zhasup. 2014. No. 1. P. 23-27.
[11] Александров П.С., Пасынков Б.А. Введение в теорию размерности. Москва: Наука, 1973, 575 с. URL: https://djvu.online/file/jSWvCi71nbeFs.
[12] Жураев Т.Ф. Некоторые геометрические свойства функтора вероятностных мер и его субфункторов: дис. ... канд. физ.-мат. наук. Москва: Московский государственный университет, 1989. 90 с.
[13] Banakh T., Radul T., Zarichniy M. Absorbing sets in infinite-dimensional manifolds. Львов: VNTL Publishers, 1996. Т. 1. 232 с.
[14] Жураев Т.Ф. Размерность паракомпактных а-пространств и функторы конечной степени // ДАН Узбекистана. 1992. № 4. С. 15-18.
[15] Бредон Г. Введение в теорию компактных групп преобразований. Москва: Наука, 1980. URL: https://libcats.org/book/508918.
