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1ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №1
УДК 515.12
https://doi.org/10.52754/16948645 2023 1 281
SOME PROPERTIES UNIFORM SPACE AND ITS HYPERSPACE
Beshimov Ruzinazar Bebutovich, Dr Sc, professor,
rbeshimov@mail.ru Safarova Dilnora Teshaboevna, teacher, safarova.dilnora87@mail.ru National University of Uzbekistan Tashkent, Uzbekistan
Abstract. In this paper, we will study the connection between a uniformly connected, uniformly pseudocompact, P — precompact and its hyperspace. It is proved that if a uniform space {X,U) is uniformly
pseudocompact iff (expc X,expc U) isuniformly pseudocompact. It is also shown that if a uniform space (X,U)
is P — precompact, then a uniform space (expc X,expcU) is P — precompact.
Key words: Hyperspace, uniformity, uniform space, uniformly connected, uniformly pseudocompact, P — precompact.
НЕКОТОРОЕ СВОЙСТВА РАВНОМЕРНОЕ ПРОСТРАНСТВО И ЕГО ГИПЕРПРОСТРАНСТВО
Бешимов Рузиназар Бебутович, д.ф.-м.н., профессор,
rbeshimov@mail.ru Сафарова Дилнора Тешабоевна, преподаватель, safarova.dilnora87@mail.ru Национальный университет Узбекистана Ташкент, Узбекистан
Аннотация. В этой статье мы изучим связь между равномерно связным, равномерно псевдокомпактным, P — предкомпактом и его гиперпространством. Доказано, что если равномерное пространство (X, U) равномерно псевдокомпактно тогда и только тогда, когда (expc X,expc U)
равномерно псевдокомпактно. Также показано, что если равномерное пространство {X, U) P —
предкомпактно, то равномерное пространство (expc X,expc U) P — предкомпактно.
Ключевые слова: гиперпространство, равномерное пространство, равномерность, равномерное связное пространство, равномерное псевдокомпактное пространство, P — прекомпактное пространство.
Introduction (Введение)
In [1], the connection between a finally compact, pseudocompact, extremely disconnected, К -space and its hyperspace is studied. It is proved: if the uniform space (X,U) is uniformly
paracompact, then (exp^X,exp си) is uniformly paracompact, if the uniform space (X,U) is uniformly R -paracompact, then uniform space (exp^ X,expc U) isuniformly R -paracompact. In
[2] the properties of space of the G -permutation degree, like: weight, uniform connectedness and index boundedness are studied. It was proved the G -permutation degree proserves the uniformly connected and index bounded. In the work [3] are established that the functor of idempotent probability measures with a compact support transforms open maps into open maps and preserves the weight and the completeness index of uniform spaces.
Definition 1 [4]. Let X be a nonempty set. A family u of coverings of a set X is called uniformity on X if the following conditions are satisfied:
(Р1) If aeU and « is inscribed in some cover ¡3 of the set X, then ^eU. (Р2) For any a1 e U, a2 e U there exists a<=u, which is inscribed in a1 and «2. (Р3) For any aeU, there exists ^eU strongly star inscribed in a .
(P4) For any x, ^ of a pair of different points of X, there exists a e U such that no element of a contains both x and y .
A family u consisting of a set X satisfying conditions (P1) - (P3) is called a pseudo-uniformity on X; and the pair (X,U) is a pseudo-uniform space.
A family u consisting of a set X satisfying conditions (P1) - (P4) is called a uniformity on X; and the pair (X, U) is a uniform space.
Proposition 1 [4]. For any uniformity of u on X, the family rU = {O c X: for each x e O exists a<=u such that a(x)cO} is a topology on X and the topological space (X,rU) is a 71-space.
The topology of tv is called the topology generated or induced by the uniformity of U.
Let (X,U) be a uniform space and expX the set of all nonempty closed subsets of the space (X,rU). For each aeU, put p(«) = {(«'): a'ca], where (a') = {F e expX: F c ^a'andFn A ^ 0for each Aea'J.
Proposition 2 [8]. If b is the base of a uniform space (X,U), then p(m~) = {p(a): a e rn1} forms a base of some uniformity expU on expX.
A uniform space (expX, expU) iscalled a hyperspace of closed subsets of a uniform space (X,U) ,and uniformity expU is called Hausdorff uniformity on expX.
Remark 1 [8]. Let expc X be the set of all nonempty compact subsets of the uniform space (X,U) .For each a<=U, put k (a) = {(a') : a' c a anda'- finit^.
Note that K(a) is the cover of the set expc X.
Corollary 1 [5]. Let (X,U) be a uniform space. Then w(U) = w(expU).
Corollary 2 [5]. If the uniform space (X,U) is metrizable, then its hyperspace (expX,expU) isalso metrizable.
Theorem 1. If (X,U) is a uniform space and aeU is a cover of (X,U).Then the following equality is true [(«')] , where (a') e P(a) and P(a) e expc U.
A uniform space (X,U) is called uniformly connected, and uniformity u is connected if any uniformly continuous mapping f:(X,U(D,U) of the uniform space (X,U) into any discrete uniform space (D,UD) is constant.
A finite sequence {A1, A2,..., Axj of subsets of a set x is called linked if A n, AI+1 of each i =1,2,..., n-1.
Definition 2 [4]. A uniform space (X,U) is called uniformly linked if for any cover a e U there exists a natural number n, such that to any points x,y e X one can choose a linked sequence {A,A2,..., a, such that k < n , x e Ax, y e Ak.
Proposition 3 [4]. For a uniform space (X ,U) ,the following conditions are equivalent:
(1) The uniform space (X,U) is uniformly connected.
(2) The uniformity of u does not contain disjoint covers consisting of at most one element.
(3) For any ae 7/ and for any point xeX, [Jan(x) = X.
И-1
(4) For any aeU and for any points of x,yeX there exists a finite linked sequence {4,4,..., Asuch that x e 4, y e Ak .
Теорема 2. A uniform space (X,U) is uniformly linked if and only if the uniform space (expc X,expc U) isuniformly linked.
It follows from Proposition 3 that every uniformly linked uniform space (X,U) is uniformly connected.
Corollary 3. A uniform space (X,U) is uniformly connected if and only if a uniform space (exp^, X,expc U) is uniformly connected.
Definition 3 [4]. A uniform space (X,U) is called uniformly pseudocompact if every uniformly continuous real-valued function defined on (X,U) is bounded.
Every Tychonoff pseudocompact space X with universal uniformity U* is uniformly pseudocompact. Conversely, if a universal space (X,U) is uniformly pseudocompact, then its topological space is pseudocompact.
A uniform space (X, U) is uniformly pseudocompact if for every countable centered open
oo
cover a = {Vj : i eM} of the uniform space (X, <v/) the intersection Q ] non-empty.
7 = 1
Theorem 3. A uniform space (X,U) is uniformly pseudocompact if and only if a uniform space (exp^. X,expc U) is uniformly pseudocompact.
A cover у of a uniform space (X,U) is said to be uniformly star-finite if there exists a uniform cover aeU such that y(B) intersects only a finite number of elements of у for any Bea [6].
A cover у of a uniform space (X,U) is called uniformly point-finite if for each x e X the set {a e M: x e Aa ey} isfinite.
Let us give examples of the property P of uniform covers of uniform spaces:
(1) covers of brevity < n ;
(2) star-finite covers;
(3) point-finite covers;
(4) finite covers;
(5) covers of power < г, т > K0.
A uniform space (X,U) is called P - precompact if the uniformity U has a base B
consisting of covers with property P .
Theorem 4. A uniform space (X,U) is P - precompact if and only if a uniform space
(expсX,expc U) is P - precompact, where properties P is a uniformly point-finite cover of
uniform space.
Теорема 5. A uniform space (X,U) is P - precompact if and only if a uniform space (expcX,expc U) is P - precompact, where properties P is a uniformly star-finite cover of uniform space.
Reference
1. Beshimov, R.B. Some Topological Properties of a Functor of Finite Degree[Теxt]/ Beshimov R.B., Safarova D.T., // Lobachevskii Journal of Mathematics, 2021, 42(12), стр. 27442753.
2. Beshimov, R.B. Index boundedness and uniform connectedness of space of the G-permutation degree^xt]/ Beshimov R.B., Georgiou D.N., Zhuraev R.M. // Applied General Topology, 2021, 22(2), стр. 447-459.
3. Borubaev, A.A. The functor of idempotent probability measures and maps with uniformity properties of uniform spaces^xt]/ Borubaev A.A., Eshkobilova D.T. // Eurasian Mathematical Journal, 2021, 12 (3), 29-41.
4. Борубаев, А.А., Равномерная топология. [Текст]/ Борубаев А.А.// Бишкек. Илим, 2013.-338 с.
5. Michael, E. Topologies on spaces of supsets^xt]/ Michael. E. // Trans. Amer. Math. Soc. - 1951. - № 1 (71). - P. 152-172.
6. Борубаев, А.А. Равномерные пространства и равномерно непрерывные отображения. [Текст]/ Борубаев А.А. // - Фрунзе: Илим, 1990.-172 с.
