ISSN (Print): 2077-9879 ISSN (Online): 2617-2658
Eurasian
Mathematical Journal
2021, Volume 12, Number 3
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EURASIAN MATHEMATICAL JOURNAL
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V.I. Burenkov, M. Otelbaev, V.A. Sadovnichy
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c
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VICTOR IVANOVICH BURENKOV (to the 80th birthday)
On July 15, 2021 was the 80th birthday of Victor Ivanovich Burenkov, editor-in-chief of the Eurasian Mathematical Journal (together with V.A.
Sadovnichy and M. Otelbaev), professor of the S.M. Nikol'skii Institute of Mathematics at the RUDN University (Moscow), chairman of the Dis- sertation Council at the RUDN University, research fellow (part-time) at the Steklov Institute of Mathematics (Moscow), honorary academician of the National Academy of Sciences of the Republic of Kazakhstan, doctor of physical and mathematical sciences(1983), professor (1986), honorary professor of the L.N. Gumilyov Eurasian National University (Astana, Kazakhstan, 2006), honorary doctor of the Russian-Armenian (Slavonic) University (Yerevan, Arme- nia, 2007), honorary member of sta of the University of Padua (Italy, 2011), honorary distinguished professor of the Cardi School of Mathematics (UK,2014), honorary professor of the Aktobe Regional State University (Kazakhstan, 2015).
V.I. Burenkov graduated from the Moscow Institute of Physics and Technology (1963) and com- pleted his postgraduate studies there in 1966 under supervision of the famous Russian mathematician academician S.M. Nikol'skii.He worked at several universities, in particular for more than 10 years at the Moscow Institute of Electronics, Radio-engineering, and Automation, the RUDN University, and the Cardi University. He also worked at the Moscow Institute of Physics and Technology, the University of Padua, and the L.N. Gumilyov Eurasian National University. Through 2015-2017 he was head of the Department of Mathematical Analysis and Theory of Functions (RUDN University).
He was one of the organisers and the rst director of the S.M. Nikol'skii Institute of Mathematics at the RUDN University (2016-2017).
He obtained seminal scientic results in several areas of functional analysis and the theory of partial dierential and integral equations. Some of his results and methods are named after him:
Burenkov's theorem on composition of absolutely continuous functions, Burenkov's theorem on con- ditional hypoellipticity, Burenkov's method of molliers with variable step, Burenkov's method of extending functions, the Burenkov-Lamberti method of transition operators in the problem of spec- tral stability of dierential operators, the Burenkov-Guliyevs conditions for boundedness of operators in Morrey-type spaces. On the whole, the results obtained by V.I. Burenkov have laid the ground- work for new perspective scientic directions in the theory of functions spaces and its applications to partial dierential equations, the spectral theory in particular.
More than 30 postgraduate students from more than 10 countries gained candidate of sciences or PhD degrees under his supervision. He has published more than 190 scientic papers. His monograph Sobolev spaces on domains became a popular text for both experts in the theory of function spaces and a wide range of mathematicians interested in applying the theory of Sobolev spaces. In 2011 the conference Operators in Morrey-type Spaces and Applications, dedicated to his 70th birthday was held at the Ahi Evran University (Kirsehir, Turkey). Proceedings of that conference were published in the EMJ 3-3 and EMJ 4-1.
V.I. Burenkov is still very active in research. Through 2016-2021 he published 20 papers in leading mathematical journals.
The Editorial Board of the Eurasian Mathematical Journal congratulates Victor Ivanovich Bu- renkov on the occasion of his 80th birthday and wishes him good health and new achievements in science and teaching!
EURASIAN MATHEMATICAL JOURNAL ISSN 2077-9879
Volume 12, Number 3 (2021), 29 41
THE FUNCTOR OF IDEMPOTENT PROBABILITY MEASURES AND MAPS WITH UNIFORMITY PROPERTIES OF UNIFORM SPACES
A.A. Borubaev, D.T. Eshkobilova Communicated by R. Oinarov
Key words: idempotent measure, uniform space, uniformly continuous map.
AMS Mathematics Subject Classication: 54E15, 54B30, 18B30.
Abstract. In the present paper we 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. Consequently, the space of idempotent probability measures with a compact support is a locally compact Hausdor space if and only if the original space is such.
DOI: https://doi.org/10.32523/2077-9879-2021-12-3-29-41
1 Introduction
Idempotent mathematics is a new branch of mathematical sciences, rapidly developing and gaining popularity over the last two decades. It is closely related to mathematical physics. The literature on the subject is vast and includes numerous books and innumerable body of journal papers. An important stage of development of the subject was presented in the book Idempotency edited by J. Gunawardena [12]. This book arose out of the well-known international workshop that was held in Bristol, England, in October 1994.
The next stage of development of idempotent and tropical mathematics was presented in the book Idempotent Mathematics and Mathematical Physics edited by G.L. Litvinov and V.P. Maslov [23].
The book arose out of the international workshop that was held in Vienna, Austria, in February 2003.
In [25] achievements of the International Workshop on Idempotent and Tropical Mathematics and Problems of Mathematical Physics were presented, which was held at the Independent University of Moscow, Russia, on August 25-30, 2007.
Idempotent mathematics is based on replacing the usual arithmetic operations with a new set of basic operations, i. e., on replacing numerical elds by idempotent semirings and semields. A typical example is the so-calledmax-plus algebra Rmax.
Many authors (S.C. Kleene, S.N.N. Pandit, N.N. Vorobjev, B.A. Carre, R.A. Cuninghame-Green, K. Zimmermann, U. Zimmermann, M. Gondran, F.L. Baccelli, G. Cohen, S. Gaubert, G.J. Olsder, J.-P. Quadrat, V.N. Kolokoltsov and others) used idempotent semirings and matrices over these semirings for solving some applied problems in computer science and discrete mathematics, starting from the classical paper by S.C. Kleene [20].
The modern idempotent analysis (or idempotent calculus, or idempotent mathematics) was founded by V.P. Maslov and his collaborators [24]. Some preliminary results are due to E. Hopf and G. Choquet, see [8], [16].
Idempotent mathematics can be treated as the result of a dequantization of the traditional math- ematics over numerical elds as the Planck constant h tends to zero taking imaginary values. This
30 A.A. Borubaev, D.T. Eshkobilova
point of view was presented by G.L. Litvinov and V.P. Maslov [26]. In other words, idempotent mathematics is an asymptotic version of the traditional mathematics over the elds of real and complex numbers.
The basic paradigm is expressed in terms of an idempotent correspondence principle. This prin- ciple is closely related to the well-known correspondence principle of N. Bohr in quantum theory.
Actually, there exists a heuristic correspondence between important, interesting, and useful construc- tions and results of the traditional mathematics over elds and analogous constructions and results over idempotent semirings and semields (i. e., semirings and semields with idempotent addition).
A systematic and consistent application of the idempotent correspondence principle leads to a variety of results, often quite unexpected. As a result, in parallel with the traditional mathematics over elds, its shadow, idempotent mathematics, appears. This shadow stands approximately in the same relation to traditional mathematics as classical physics does to quantum theory.
The notion of idempotent (Maslov) measure nds important applications in dierent areas of mathematics, mathematical physics and economics (see the survey article [24] and the bibliography therein). Topological and categorical properties of the functor of idempotent measures were studied in [31], [34]. Although idempotent measures are not additive and the corresponding functionals are not linear, there are some parallels between topological properties of the functor of probability measures and the functor of idempotent measures (see, for example [28], [30], [31]) which are based on a natural equiconnectedness structure on both functors.
The theory of uniform spaces in present time has become logically justied, far advanced due to the fundamental papers of A. Weil, N. Bourbaki, Yu.M. Smirnov, H. Inassaridze, V.A. Efremovich, A.A. Borubaev, J. Isbell, B.A. Pasynkov, P. Samuel, V.V. Fedorchuk, A.Ch. Chigogidze, Z. Frolik, V. Kulpa, and others ([7], [13], [14]).
Despite of independent character of a uniform space it is closely connected to the theory of topological spaces and between them there is a deep analogy. So the problem of dening and research of uniform analogies of most important classes of topological spaces and continuous maps has become not only of current interest but produces ne instruments for studying topological spaces themselves. The rst problem appeared in nding a uniform analogue of the paracompactness.
American mathematician M.D. Rice was the rst to determine uniformly paracompact spaces. But unfortunately this class does not include even the class of metric spaces. Later in [3] there were considered some generalizations of metric, normed, and unitary spaces.
Recently, in [33] the action of the functor exp (the construction of taking of a hyperspace of a given space) on Π-complete spaces and Π-complete maps were investigated. In [10] it was shown that the functor of idempotent probability measures extends from the category Comp of compact Hausdor spaces and their continuous maps to the category Unif of uniform spaces and uniformly continuous maps. The present paper may be considered as a continuation of [10]. In [11] it was established that the functor of idempotent probability measures with compact supports transforms perfect maps into perfect maps and preserves the weight and completeness index of uniform spaces.
The present paper develops the mentioned papers and in it we show that the functor of idempotent probability measures with compact supports preserves the uniform openness of maps, the weight and the completeness index of uniform spaces. The preserving of the completeness index by the functor of idempotent probability measures with compact supports yields that the space of idempotent probability measures with compact supports is a locally compact Hausdor space if and only if the original space is such.
The functor of idempotent probability measures and uniformity properties 31
2 Idempotent probability measures
Recall [24] that a setS equipped with two algebraic operations: addition ⊕and multiplication, is said to be a semiring if the following conditions are satised:
• the addition⊕ and the multiplication are associative;
• the addition⊕ is commutative;
• the multiplication is distributive with respect to the addition⊕: x(y⊕z) =xy⊕xz
and
(x⊕y)z =xz⊕yz for all x, y, z∈S.
A unit of a semiring S is an element 1 ∈ S such that 1x = x1 = x for all x ∈ S. A zero of the semiring S is an element 0 ∈ S such that 0 6= 1 and 0⊕x = x⊕0 = x for all x ∈ S. A semiring S is called an idempotent semiring ifx⊕x=xfor allx∈S. A (an idempotent) semiringS with neutral elements 0 and 1 is called a (an idempotent) semield if every nonzero element of S is invertible. Note that diods, quantales and inclines are examples of idempotent semirings [24].
Let R = (−∞, +∞) be the eld of real numbers and R+ = [0, +∞) be the semiring of all nonnegative real numbers (with respect to the usual addition and multiplication). Consider the map Φh: R+→S =R∪ {−∞} dened by the equality
Φh(x) = hlnx, h >0.
Let x, y ∈ X and u = Φh(x), v = Φh(y). Put u ⊕h v = Φh(x+y) and uv = Φh(xy). The image Φh(0) =−∞ of the usual zero 0is a zero 0 and the image Φh(1) = 0 of the usual unit 1 is a unit 1 in S with respect to these new operations. Thus, S obtains the structure of a semiring R(h) isomorphic to R+.
A direct check shows that u⊕hv →max{u, v}ash→0. The convention −∞ x=−∞
allows us to extend ⊕ and overS. It can easily be checked that S forms a semiring with respect to the addition u⊕v = max{u, v} and the multiplication uv = u+v with zero 0 = −∞ and unit 1 = 0. Denote this semiring by Rmax; it is idempotent, i. e., u⊕u =u for all its elements u. The semiringRmax is actually a semield. The analogy with quantization is obvious; the parameter hplays the role of the Planck constant, so R+ can be viewed as a quantum object andRmax as the result of its dequantization. This passage to Rmax is called the Maslov dequantization (for details, see [21], [22], [28]).
Let X be a compact Hausdor space, C(X) be the algebra of all continuous functions on X with the usual algebraic operations. OnC(X) the operations⊕ and are dened by the equalities ϕ⊕ψ = max{ϕ, ψ} and ϕψ = ϕ+ψ, where ϕ, ψ ∈ C(X). For a number λ ∈ R the symbol λX denotes the constant function, dened as λX(x) = λ, x∈X.
Recall [34] that a functional µ: C(X)→R is said to be an idempotent probability measure onX if it satises the following properties:
(1) µ(λX) = λ for all λ∈R;
(2) µ(λϕ) =λµ(ϕ) for all λ∈R and ϕ∈C(X); (3) µ(ϕ⊕ψ) = µ(ϕ)⊕µ(ψ) for all ϕ, ψ ∈C(X).
32 A.A. Borubaev, D.T. Eshkobilova
For a compact Hausdor space X by I(X) we denote the set of all idempotent probability measures on X. Since I(X) ⊂ RC(X), we consider I(X) as a subspace of RC(X). The family of sets of the form
hµ; ϕ1, . . . , ϕn; εi={ν∈I(X) : |ν(ϕi)−µ(ϕi)|< ε, i= 1, . . . , n}
is a base of open neighbourhoods of a given idempotent probability measure µ∈I(X) according to the induced topology, where ϕi ∈ C(X), i = 1, . . . , n, and ε > 0. It is obvious that the induced topology and the pointwise convergence topology on I(X)coincide.
Let X,Y be compact Hausdor spaces and f: X →Y be a continuous map. It is easy to check that the mapI(f) : I(X)→I(Y) dened by the formula I(f)(µ)(ψ) =µ(ψ◦f) is continuous. The construction I is a normal functor acting in the category of all compact Hausdor spaces and their continuous maps. Therefore, for each idempotent probability measure µ∈ I(X) one may dene its support:
suppµ=\
A⊂X :A=A, µ∈I(A) .
Consider functions of the type λ : X → [−∞, 0]. On a given set X we dene the max-plus- characteristic function ⊕χA: X→Rmax of a subsetA⊂X by the rule [32]
⊕χA(x) =
(0 at x∈A,
−∞ at x∈X\A. (2.1)
For a singleton {x} we will write ⊕χx instead of ⊕χ{x}.
Let F1, F2, . . . ,Fn be a disjoint system of closed sets of a space X, and a1, a2, . . . ,an be non- positive real numbers. We call [32] the function
⊕χaF1, ..., an
1, ..., Fn(x) =
a1 at x∈F1, . . . ,
an at x∈Fn,
−∞ at x∈X\ Sn
i=1
Fn
(2.2)
the max-plus-step-function dened by the sets F1, F2, . . . , Fn and the numbers a1, a2, . . . ,an. Note that
⊕χaA(x) =a ⊕χA(x) =
(0a at x∈A,
−∞ at x∈X\A =
(a atx∈A,
−∞ atx∈X\A
for a setA in X and a non-positive number a. Consequently, for a disjoint system of closed sets F1, F2, . . . ,Fn in a space X, and non-positive real numbers a1, a2, . . . ,an we have
⊕χaF1, ..., an
1, ..., Fn(x) = ⊕χaF1
1(x)⊕⊕χaF2
2(x)⊕ . . . ⊕⊕χaFn
n(x).
In the case when F1,F2, . . . ,Fn are singletons, say Fi ={xi}, i= 1, . . . , n, we have
⊕χa{x1, ..., an
1}, ...,{xn} =⊕χa{x1
1}⊕⊕χa{x2
2}⊕ . . . ⊕⊕χa{xn
n}. (2.3)
The notion of density for an idempotent measure was introduced in [21], where the main result on the existence on densities for arbitrary measures was proved. A more detailed exposition is given
The functor of idempotent probability measures and uniformity properties 33 in [22] the rst systematic monograph on the idempotent analysis. Later on paper [1] appeared, where further investigations of densities were done. Let µ∈I(X). Then we can dene the function dµ: X →[−∞, 0]by the formula
dµ(x) = inf{µ(ϕ) : ϕ∈C(X) such thatϕ≤0and ϕ(x) = 0}, x∈X. (2.4) The function dµ is upper semicontinuous and is called the density of µ. Conversely, for each upper semicontinuous functionf: X →[−∞,0]withmax{f(x) : x∈X}= 0we can dene the idempotent measureνf by the formula
νf(ϕ) =M
x∈X
f(x)ϕ(x), ϕ∈C(X). (2.5)
Note that a function f: X → R is said to be upper semicontinuous if for each x ∈ X, and for every real number r which satises f(x) < r, there exists an open neighbourhood U ⊂X of x such that f(x0)< r for all x0 ∈ U. It is easy to see that functions dened by (2.1) or by (2.2) are upper semicontinuous.
Put US(X) =
λ: X →[−∞, 0]
λ is upper semicontinuous and there exists a
point x0 ∈X such that λ(x0) = 0 . Then we have
I(X) = (
M
x∈X
λ(x)δx : λ∈US(X) )
.
Obviously L
x∈X
⊕χx0(x)δx = δx0, i. e. for the max-plus-characteristic function ⊕χx0 formula (2.5) denes the Dirac measure δx0 supported on the singleton {x0}. The set of all Dirac measures on a Hausdor compact space X we denote by δ(X). It is easy to notice that the space X and the subspace δ(X) ⊂ I(X) are homeomorphic. This phenomenon gives us the opportunity to consider X as subspace of I(X).
LetA be a closed subset of a compact Hausdorf spaceX. It is easy to check that ν∈I(A) if and only if {x∈X : dν(x)>−∞} ⊂A. Hence,
suppν ={x∈X: dν(x)>−∞}.
It is evident that suppν = {x1, . . . , xn} if and only if the density dν of ν has form (2.3) for the singletons {x1}, . . . , {xn} and for some non-negative numbers a1, . . . , an with ai > −∞, i = 1, . . . , n, and max{a1, . . . , an} = 0. In this case ν is said to be an idempotent probability measure with a nite support. The subset of I(X) consisting of all idempotent probability measures with nite supports we denote byIω(X) [32].
3 Uniformity. Preliminaries
Given a set X, a partial ordering ⊂ can be dened on the powerset P(X) by the subset inclusion, turning (P(X),⊂) into a lattice. Dene a lter F on X as a non-empty subset of P(X) with the following properties:
F1) if A and B are in F, then so is their intersection (F is closed under nite intersection);
F2) ifAis inF andAis a subset ofB, thenB is inF, for all subsetsB ofX (F is upward-closed).
34 A.A. Borubaev, D.T. Eshkobilova
A lter base is a subset B ofP(X) with the properties thatB is non-empty and the intersection of any two members ofB includes (as a subset) a member of B (B is downward directed).
In the sequel, the notations E−1 and E◦F are to be understood as relations on X. Recall that if E ={(x, y)} is a relation on X, i. e., a subset of X×X, then the inverse relation E−1 is dened to be the subset (y, x) of X×X. If E and F are relations on X, then their composition E ◦F is dened to be the set of all pairs (x, z) such that, for some y ∈ X, (x, y)∈ E and (y, z)∈F.
The following denition was given in its present form by Bourbaki [5].
Denition 1. The diagonal of a setX×Xis the subset∆ ={(x, x)|x∈ X}. A diagonal uniformity on a set X is a lterE on X×X consisting of subsets of X×X called entourages or surroundings such that:
E1) if E ∈ E then ∆⊂E;
E2) if E ∈ E then there exists an entourage F ∈ E such thatF ⊂ E−1; E3) if E ∈ E then there exists an entourage F ∈ E such thatF ◦F ⊂ E.
Note that by properties F2) and E2) it follows that E ∈ E implies thatE−1 ∈ E.
A base Bfor a diagonal uniformity E onX is a lter base forX×X that satises conditionsE1) E3)of Denition 1.
Let E be an entourage,A⊂ X. Put
E[x] ={y: (x, y)∈E}, E[A] = [
x∈A
E[x].
If (X,E) is a uniform space, the topology T of the uniformity E, or the uniform topology, is the topology in which the family of sets B(x) = {E[x] : E ∈ E} is a neighbourhood base at x∈ X [6].
ConditionE3)of Denition 1 has been described by Kelley [19] as a vestigial form of the triangle inequality. It imposes a stronger structure than a topology on X. Given any cover of X, one can always dene a topology by taking nite intersections and arbitrary unions. Take now a family of subsets ofX×X each containing the diagonal. Add all nite intersections to the family, and dene a lter by taking supersets. This lter may fail to satisfy condition E3) of Denition 1. See [18] for an example.
Let E and F be diagonal uniformities on X and Y, respectively. A functionf: X →Y is said to be uniformly continuous if and only if for each F ∈ F, there is some E ∈ E such that (x1, x2)∈ E ⇒(f(x1), f(x2))∈F.
It follows easily that every uniformly continuous function is continuous.
The notion of uniform equivalence is akin to that of topological equivalence or homeomorphism, but there is no shorter condition for it.
The uniform spaces X and Y are said to be uniformly equivalent if there exists a bijection f: X →Y such that both f and f−1 are uniformly continuous [6].
A topological space X is said to be uniformizable if there is a (compatible) uniformity E onX such that the topology of this uniformity coincides with the topology of X [6].
The following theorem is well known.
Theorem 3.1. A topological space is uniformizable if and only if it is completely regular.
Recall that a completely regular space is a space in which a closed set and a point disjoint from it can be separated by a continuous function. A Tychono space is a completely regular space in which one-point sets are closed.
The functor of idempotent probability measures and uniformity properties 35 We begin by recalling the denition of a Cauchy lter in a uniform space.
Cauchy lter: a lter F in a uniform space(X, E)is called a Cauchy lter if for each entourage E of the uniformityE there exists a member W of the lter F such that W ×W ⊂E [6].
A uniform space is called complete if and only if every Cauchy lter in the space converges to a point in the space [6].
Uniform completion of a uniform space: The uniform completion of a uniform space (X,E) is a pair(f, (X∗,E∗)), where (X∗,E∗)is a complete uniform space and f is a uniform embeddingX into X∗ as a dense subspace [6].
We say [15] that X is totally bounded or precompact if for every entourageU there is a nite set F such that U[F] = X. Precompact spaces have the following properties: subspaces and products of precompact spaces are precompact spaces.
If E is a compatible totally bounded uniformity on X then its completion is a compactication of X, the Samuel compactication of(X, E) [15].
Let X be a Tychono space, β X be the Stone-Cech compact extension ofX. We dene [11, 15]
the subspace
Iβ(X) = {µ∈I(βX) : suppµ⊂X},
whose elements we call [17], [29] idempotent probability measures with compact supports. We supply Iβ(X) with the induced topology from I(βX). Then Iβ(X) is a Tychono space. That is why Theorem 3.1 immediately yields
Corollary 3.1. Iβ(X) is uniformizable.
Recall that Corollary 3.1 was obtained in [10].
We note that the density of a topological spaceX is dened as the least power|A|of everywhere dense subsets A⊂ X and it is denoted by d(X).
Lemma 3.1. For an arbitrary innite Tychono space X we have d(Iβ(X))≤d(X). Proof. The proof uses a modied construction suggested in [2].
4 The functor of idempotent probability measures and maps with uniformity properties
Let f: X → Y be a continuous map of Tychono spaces and β f: β X → β Y be its the maximal extension (it is unique). Then I(βf)(Iβ(X))⊂Iβ(Y). Put
Iβ(f) =I(βf)|Iβ(X).
Thus, the operation Iβ is a functor acting in the category T ych of Tychono spaces and their continuous maps [17], [29].
For a Tychono space X consider an idempotent probability measure µ= L
x∈X
λ(x)δx∈Iβ(X) and a nite system {U1, . . . , Un} of open sets Ui ⊂X such that suppµ∩Ui 6=∅, i= 1, . . . , n, and suppµ⊂
n
S
i=1
Ui. Dene the set [32]
hµ; U1 . . . , Un; εi= n
ν =M
x∈X
γ(x)δx ∈I(X) : suppν∩Ui 6=∅,suppν⊂
n
[
i=1
Ui,
and |λ(x)−γ(y)|< εat the points x∈suppµ∩Ui and y ∈suppν∩Ui, i= 1, . . . , no
. (4.1)
36 A.A. Borubaev, D.T. Eshkobilova
Theorem 4.1. [32] The sets of type (4.1) form a base of the pointwise convergence topology inIβ(X). Theorem 4.1 immediately yields the following two statements.
Corollary 4.1. [10] Let X be a Tychono space and E be a diagonal uniformity on X such that the topology of this uniformity coincides with the topology of X. Then the family
BI =n [
α
hµ;E[Aα1], . . . , E[Aαn];ε1i × hν; E[B1α], . . . , E[Bkα];ε2i :
Aαi ∈ E, i= 1, . . . , n; Bjα ∈ E, j = 1, . . . , k; ε1 >0, ε2 >0;
[
α
µ; E[Aα1], . . . , E[Aαn]; ε1
=[
α
ν; E[B1α], . . . , E[Bαk]; ε2
=Iβ(X)o
forms a base of the uniformity on Iβ(X) such that the topology of this uniformity coincides with the pointwise convergence topology on IβX.
For a diagonal uniformityE on a Tychono spaceX we denote byIβ(E)the uniformity, generated by the base BI.
Corollary 4.2. [10] If a map f: (X, E) → (Y, F) is uniformly continuous, then the map Iβ(f) : (Iβ(X), Iβ(E)) → (Iβ(Y), Iβ(F)) is also uniformly continuous.
Theorem 4.2. [10] The functor Iβ: Tych→Tych may be lifted on (i. e., may be extended to) the category Unif of uniform spaces and uniformly continuous maps.
Proof. The proof is based on Corollaries 3.1 and 4.2.
For a Tychono space X dene a mapδ: X →Iβ(X)as δ(x) = δx, x∈X.
Proposition 4.1. [10] For a uniform space(X, E)the mapδ: (X, E)→(Iβ(X), Iβ(E))is a uniform embedding.
Proposition 4.2. [10] If i: (X, E)→ (Y, F) is a uniform embedding, then Iβ(i) : (Iβ(X), Iβ(E))
→ (Iβ(Y), Iβ(F))is also a uniform embedding.
Theorem 4.3. [10] A uniform space (X, E) is precompact if an only if the uniform space (Iβ(X), Iβ(E)) is precompact.
Proof. The property of precompact spaces (see page 35) and Proposition 4.1 yield the necessity. Let (X, E) be a precompact space. Then its completion (X∗, E∗) is a compact Hausdor space. By virtue of Proposition 4.2 one can embed the space (Iβ(X), Iβ(E))into the compact Hausdor space (Iβ(X∗), Iβ(E)∗) = (I(X∗), Iβ(E)∗). Therefore, (Iβ(X), Iβ(E)) is precompact.
The Samuel compactication of a uniform space(X, E)may be dened as a completion of(X,Ep), where Ep is the maximal precompact uniformity, lying in E [27]. For a uniform spaces (X, E) its Samuel compactication is denoted by SE(X). The Samuel compacticationSE may be extended to a covariant functor S: Unif→Comp.
There is a natural transformation T: S ◦Iβ → I ◦ S, which components TE dene as follow- ing. The identity map (X, E) → (X, Ep) is uniformly continuous. Consequently, the embedding iE: (X, E) → SE(X) is uniformly continuous. Hence Iβ(iE) : (Iβ(X), Iβ(E)) → I(SE(X)) is also uniformly continuous. But the operation E → Ep, which aligns each uniformity E to the precompact uniformity Ep, is a covariant functor. Therefore Iβ(iE)p: (Iβ(X), Iβ(Ep)) → I(SE(X)) is uniformly continuous. This map extends to the mapTE: SIβ(E)(Iβ(X))→I(SE(X))of completions.
The functor of idempotent probability measures and uniformity properties 37 Theorem 4.4. [10] If a uniformity E is precompact, then the map TE is a homeomorphism.
Proof. If a uniformity E is precompact then the map iE: (X, E)→ SE(X) is a uniform embedding.
By virtue of Theorem 4.3 the mapsIβ(iE)and Iβ(iE)p coincide. So, according to Proposition 4.2 the mapIβ(iE)is an uniform embedding. Thus, TE is homeomorphism being a completion of the uniform embedding of an everywhere dense subspace.
A continuous map f: X → Y of a topological space X onto a topological space Y is said to be perfect if f is closed and for each y∈ Y the preimage f−1(y) is compact.
Theorem 4.5. [11] Let f: X → Y be a continuous map. The map Iβ(f) : Iβ(X)→Iβ(Y)is perfect if and only if f: X → Y is perfect.
A family A of diagonal entourages of a uniform space (X,E)forms a base of the uniformity E if for each E ∈ E there exists A∈ A such thatA ⊂ E. The least capacity of bases of the uniformity E is called the weight of the uniformity E and is denoted by w(E).
The denition immediately yields the following statement.
Lemma 4.1. Let (X,E) be a uniform space and (Y, E|Y) be its subspace, where E|Y = {E∩ (Y × Y) : E ∈ E}. Then w(E|Y)≤ w(E).
Theorem 4.6. We have w(Iβ(E)) =w(E).
Proof. Proposition 4.2 and Lemma 4.1 imply that w(Iβ(E))≥ w(E).
Since X is a completely regular space (see Theorem 3.1), it has a compact Hausdor extension b X and a uniformity Eb X such that Eb X|X =E and w(E) =w(Eb X).
One can easily notice that
Iβ(X) = {µ∈ I(β X) : supp µ⊂ X} ∼={µ∈ I(b X) : supp µ⊂ X}=Ib(X).
Therefore one can choose open sets U1 . . . , Un from a base in Theorem 4.1. Then the sets hµ; U1 . . . , Un; εi dened by (4.1) form a base in Iβ(X).
Let now B be a base of the uniformity such that |B| =w(E). Then BI =n [
α∈A
hµ; E[Aα1], . . . , E[Aαn]; ε1i × hν;E[B1α], . . . , E[Bkα]; ε2i :
Aαi ∈ B, i= 1, . . . , n; Bjα ∈ B, j = 1, . . . , k; |A|<∞; ε1 >0, ε2 >0;
[
α∈A
µ;E[Aα1], . . . , E[Aαn];ε1
= [
α∈A
ν; E[B1α], . . . , E[Bαk]; ε2
=Iβ(X)o
forms a base of the uniformity on Iβ(X). Evidently, |BI|=|B|. Hence, w(Iβ(E))≤w(E).
Note that the H-completeness of uniform spaces is ner than their completeness.
Let(X, E)be a uniform space and H ⊂ E be an arbitrary system of uniform entourages. A lter F on the set X is said to be a Cauchy H-lter in (X, E) if for each entourage E ∈ E there exists a member W ∈ F such thatW ×W ⊂E.
Denition 2. [4] Let (X,E) be a uniform space and H ⊂ E. The space (X, E) is said to be H- complete and the systemHis said to be complete if each CauchyH-lterF has at least one adherent point, i. e. ∩{F: F ∈ F } 6=∅.
38 A.A. Borubaev, D.T. Eshkobilova
Denition 3. [4] The least cardinal number τ is said to be the completeness index of a uniform space (X, E) if there exists such system H ⊂ E that |H| = τ and (X, E) is H-complete uniform space.
The completeness index of a uniform space (X, E) is denoted byic(E). Theorem 4.7. We have ic(Iβ(E)) =ic(E).
Proof. Proposition 1.2.19 [4] and Proposition 4.1 yield that ic(Iβ(E))≥ ic(E).
Let us show that the inverse inequality holds. Let ic(E) ≤ τ. Then by Theorem 1.2.22 [4]
(X, E) can be mapped onto some complete uniform space (Y, F) of weight w(F) ≤ τ by a perfect map f. Then by Theorem 4.5 the map Iβ(f) : Iβ(X) → Iβ(Y) is perfect, and by Theorem 4.6 one has w(Iβ(F)) ≤ τ. Applying again Theorem 1.2.22 [4] we have ic(Iβ(E)) ≤ τ. So, we have ic(Iβ(E))≤ ic(E) as a result.
Proposition 1.2.18 [4] and Theorem 4.7 yield the following statement.
Corollary 4.3. A uniform space (Iβ(X), Iβ(E)) is a uniformly locally compact Hausdor space if and only if the uniform space (X, E) is a uniformly locally compact Hausdor space.
To consider uniformly open maps below we recall the following equivalent denition of uniformity.
Denition 4. [4] Let X be a non-empty set. A family U of coverings of the set X is said to be a uniformity (in term of coverings) on X if the following conditions are satised:
(C1) if α∈ U and α renes a covering γ of X then γ ∈ U;
(C2) for any pair α, β ∈ U there exists γ ∈ U that is renes both α and β; (C3) for every α∈ U there exists β ∈ U such that β strongly star-renes α;
(C4) for any pairx,yof distinct elements ofX there existsα∈ U such that no element ofαcontains bothx and y.
A pair (X, U) consisting of a set X and a uniformity U on it is said to be a uniform space.
Denition 5. [4] A uniform continuous map f: (X, U) → (Y, V) of a uniform space (X, U) to a uniform space(Y, V)is said to be uniformly open iff transforms every open uniform coveringα∈ U to an open uniform covering f(α)∈ V.
Put
I(α) ={hµ; U1, . . . , Un;εi: Ui ∈α, i= 1, . . . , n; ε >0}
and
BI,U ={I(α) : α∈ U }.
Since the system BI satises conditions(B1) (B3)of Proposition 1.1.2 [4] it is a base for some uniformity on Iβ(X). Let UI be the uniformity generated by BI,U.
Theorem 4.8. Let f: (X, U) → (Y, V) be a uniformly continuous map. The map Iβ(f) : (Iβ(X), UI)→(Iβ(Y), VI) is a uniformly open if and only if f is uniformly open.
The functor of idempotent probability measures and uniformity properties 39 Proof. Since f =Iβ(f)|X then the uniform openness of Iβ(f) implies that f: (X,U) →(Y, V) is a uniformly open map.
Let f: (X, U)→ (Y, V) be a uniformly open map. Take any open uniform covering G ∈ UI of Iβ(X). We have to show thatIβ(f)(G)∈ VI. There existsI(α)∈ BI,U that is renes G, i. e. every hµ; U1, . . . , Un;εi ∈ I(α) has a G∈ G such that hµ; U1, . . . , Un;εi ⊂ G. Sincef(α)∈ V we have
hIβ(f)(µ); f(U1), . . . , f(Un);εi ∈ I(f(α))∈ VI
and
hIβ(f)(µ); f(U1), . . . , f(Un);εi ⊂ Iβ(f)(G).
Thus,I(f(α))renes Iβ(f)(G). Consequently, Iβ(f)(G)∈ VI.
Acknowledgments
The authors express deep gratitude to the unknown reviewer for useful advices and the editor of the paper for corrections.
40 A.A. Borubaev, D.T. Eshkobilova
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Altai Asylkanovich Borubaev
Institute of Mathematics of National Academy of Sciences of Kyrgyz Republic, 265 Chuy Avenue, Bishkek, 720071, Kyrgyz Republic
e-mail: [email protected]
Dilrabo Turaxanovna Eshkobilova Termez State University,
43 F. Khodjaev St, Termez, 190111, Uzbekistan e-mail: [email protected]
Received: 28.10.2020
