CC BY
7
INVERSION-CLOSED ALGEBRAS OF FUNCTIONS ON A UNIFORM
SPACE
Chekeev Л.А.1, Kasymova T.D2, Rakhmankulov B.Z.3 Email: [email protected]
'Chekeev Asylbek Asakeevich — Habilitated Doctor of Mathematics, Full Doctor of Mathematics, Doctor of
Physical and Mathematical Sciences, Professor; 2Kasymova Tumar Dzhapashevna — PhD Mathematics, Candidate of Physical and Mathematical Sciences,
Associate Professor; 3Rakhmankulov Bakhtiar Zulpukarovich — senior lecturer, DEPARTMENT OF ALGEBRA, GEOMETRY, TOPOLOGY AND TEACHING OF MATHEMATICS, FACULTY OF MATHEMATICS AND INFORMATICS, KYRGYZNATIONAL UNIVERSITY, BISHKEK, REPUBLIC OFKYRGYZSTAN
Abstract: for any inversion-closed algebra A in sense [22, '8, '9] on Tychonoff space X there is a uniformity U such, that A coincides with an algebra Cu (X) [7] of all U — continuous functions on some uniform space uX . For a ring structure of algebra Cu (X) and subring C* (X) the uniform analogues of the Stone Theorem [26] and Gelfand-Kolmogoroff Theorem ['5] have been proved. For a uniform space uX the concept of Z — completeness is considered, which is equivalent to the realcompactness in the category ZUnif [8], determined by means of
realcompactification Uu X [7], various characterizations of Zu — complete uniform spaces are established.
Keywords: algebra, uniformity, U — open, U — closed sets, normal base, COZ — mapping, COZ —fine space, ring, ideal, maximal ideal, compactification.
ИНВЕРСНО-ЗАМКНУТЫЕ АЛГЕБРЫ ФУНКЦИЙ НА РАВНОМЕРНОМ ПРОСТРАНСТВЕ Чекеев А.А.1, Касымова Т.Д.2, Рахманкулов Б.З.3
'Чекеев Асылбек Асакеевич — хабилитированный доктор математики, доктор физико-математических
наук, профессор;
2Касымова Тумар Джапашевна — кандидат физико-математических наук, доцент; 3Рахманкулов Бахтияр Зулпукарович — старший преподаватель, кафедра алгебры, геометрии, топологии и преподавания высшей математики, факультет математики и информатики, Кыргызский национальный университет, г. Бишкек, Кыргызская Республика
Аннотация: для любой инверсно-замкнутой алгебры A в смысле работ [22, '8, '9] на тихоновском пространстве X существует равномерность U такая, что A совпадает с алгеброй Cu (X) [7] всех U — непрерывных функций на некотором равномерном
пространстве UX. Для кольцевой структуры алгебры Cu (X) и подкольца C* (X) доказаны равномерные аналоги теорем Стоуна [26] и Гельфанда-Колмогорова ['5]. Для равномерного пространства UX рассмотрено понятие Z — полноты, которое
эквивалентно реалкомпактности в категории ZUnif [8], определенное посредством
реалкомпактификации Uu X [7], установлены различные характеристики Zu — полных равномерных пространств.
Ключевые слова: алгебра, равномерность, U — открытые, U — замкнутые множества, нормальная база, COZ — отображение, COZ — тонкое пространство, кольцо, идеал, максимальный идеал, компактификация.
УДК 515.12
1. Introduction and preliminaries
In [7] by zero-sets a normal base Zu of all uniformly continuous functions of the uniform space UX Wallman compactification ДX = c(X,ZU) [1, 12, 16, 18, 19, 21, 22, 24, 25] and Wallman realcompactification V X = v(X, Z) [18, 19, 22, 24, 25] had been constructed. A normal base Z is a separating, nest-generated intersection ring (s.n.-g.i.r.) on X [25] and the compactification ДX is a Д — like compactification [24]. The uniformities on X , by which
ДиX and V X are completions, had been described. The properties of Д — like
compactification are similar in many respects with Stone-Cech compactification [7, Th.2.6, Cor. 2.7, Th. 2.8] and it is closely linked with the ring of all u continuous functions and the ring of all bounded
u continuous functions [7, Th. 2.6 (3)]. The compactification ДиX and realcompactification
v X yielded a number of results on the partial order and relations of dimensions in the lattice of all compactifications [7, Part 3].
Under an algebra A on X [22, 18, 19] is meant a subalgebra of C(X), containing all
constants, separating points and closed sets, and it is closed in C(X) with respect to uniform convergence (i.e. if sequence of functions '[./^j' ^ of A is uniformly converging to function f , then f g A) and inversion (i.e. f g A and Z (f) = 0, then 1/ f g A ). If A is an
algebra on X , then A is a set of all bounded functions of A , it is a ring and H (A ) is a structure space of algebra A , i.e. it is a space of maximal ideals of the algebra A with the Stone topology [26] and H(A) is the intersection of all cozero-sets in H (A ), containing X. H (A ) is a compactification of X and a ring A is isomorphic to the ring C(H (A )) (A * = C (H (A *)) ) in terms [19, 22]). By Lemma 1.4 [25], traces on X of zero-sets of H(A ), which are zero-sets of functions from A , form s.,n.-g.i.r. Z(A*) = {Z (f) : f G A*} . From [22] it follows, that A precisely is a family of all continuous functions of//(y? ) into two-point compactification 00"+00 J of a real line К , which are real-valued on X . Then Z (A) = Z (A * ) ={Z П X : Z G Z (H (A * ))}, where Z (H (a * )) = {z (f): f G C(H (a * ))} and Z (A) is s.n.-g.i.r..
Thus, if A is an algebra on X , then Wallman compactification c(X, Zu) is H(A ) and Wallman realcompactification v(X, Z ) is H(A), but an algebra A cannot be isomorphic to some algebra C (Y) [25, Ex. 3.13], [22, Ex.1.22], [19, Ex. 9.7].
In [7] it had been proved, that for uniform space uX a set Cu (X) of all U — continuous functions on uX forms an algebra [7, Prop. 2.2] and Z(uX) = ^Z^ f ^ : f G Cu(X coincides with Zu = ^Z^ f ^ : f g U(uX[7, item (3) of Lemma 2.4]. For the ring C (X) of all bounded u — continuous functions fiuX = H(C*(X)) and
Vu X = H (Cu (X)) [7, Th. 2.8, Th. 2.12].
For an algebra A on X it has been proved, that A = Cu (X) (A = C(X) for fine uniformity uy on X ) for some uniformity u on X (Theorem 2.2, Corollaries 2.4, 2.5) and the
equality H(A*) = Pu (H(A)) (H(A*) = P(H(A))) is fulfilled (Theorem 2.6, Corollary 2.7). Since a ring C (X) (C* (Y)) is isomorphic to a ring
C(H(C*(X))) = C(P„X) (C(H(C*(Y))) = C(PVY)), it has been proved, that
the isomorphism of rings C* (X) and C* (Y) is equivalent to the homeomorphism of P — like
compactifications PuX and PY (Theorem 2.8) and it is equivalent to the COZ —
homeomorphism of the uniform spaces uX and vY with the first countability axiom (Corollaries 2.9, 2.10). For COZ — fine uniform spaces a similar results for the rings of all bounded uniformly
continuous functions have been obtained (Corollary 2.11). Maximal ideals of a ring C* (X) (
C (X)) corresponding to the points of P — like compactification PuX immediately have been described (Theorem 2.12). From Theorem 2.14, in particular, if it follows, that the isomorphism of rings C (X) and C (Y) is equivalent to the homeomorphism of P — like
compactifications PuX and P Y and it is equivalent to the COZ — homeomorphism of the
uniform spaces uX and with vY the first countability axiom (Theorem 2.16, Corollaries 2.17, 2.18). As a set of all uniformly continuous functions of COZ — fine spaces is a ring, so for COZ — fine spaces the similar results are fulfilled (Corollary 2.19).
A construction of uniformity for realcompactification v X [7, Th. 2.12] implies a determination of Z — complete uniform space (Definition 3.1). Theorem 3.3 establishes the various
characterizations of the Z — complete uniform spaces. When u = uf is a maximal uniformity,
u J
then Z — complete space is a realcompact space [10, Th. 3.11.11]. But on a realcompact non-Lindelof space X there is a uniformity u such that a uniform space uX is not Z — complete
(Lemma 3.4, Corollary 3.5), and Z — completeness for any uniformity u is equivalent to Lin-
delofness (Theorem 3.6). Theorem 3.7 clarifies Theorem 2.14 [7] and establishes the various
characterizations of the realcompactification V X . Theorem 3.8 and Corollaries 3.10, 3.11 are the
uniform analogue of Hewitt Theorem [11, 3.12.21 (g)].
Denotations and basic properties of uniform spaces and compactifications from [1, 3, 11, 17, 27].
We denote by U(uX)(U (uX) ) the set of all (bounded) uniformly continuous functions on the uniform space uX . The natural uniformity on uX , generated by U (uX) , be u is the
smallest uniformity on X with respect to its all functions from U (uX) are uniformly continuous. Evidently, that U С U . Samuel compactification S X is a completion of X with respect to the uniformity U . Z is a ring of zero-sets of functions from U (uX) or
U (uX) and CZ, is a ring of cozero-sets of functions from U (uX) or U {uX). CZ, consists of complements of sets of Z and, vice versa. We note, that all sets of CZ, (Z,,)
u u u u
coincide with the set of all u — open (u — closed ) sets in sense of M. Charalambous [4, 5]. Z is
the base for closed sets topology and forms s.n.-g.i.r. on X [25], implying that it is a normal base [1, 12, 21]. The topology of a uniform space is generated by its uniformity and in case of compactum X we always use its unique uniformity. The restriction of a uniformity from uniform space uY to its
subspace X is denoted u . A uniform space uX which has a base of all uniform coverings of \л
cardinalities ^ T is said to be T — bounded [3]. For a uniform space uX its completion is denoted by uX [3, 11, 23]. We denote the set of all natural numbers by In , Ж is the real line, uniformity on Ж is induced by the ordinary metric, (Q is the set of rational numbers and Wq = Uщ. | ;
I = [Ojl] is a unit segment; for X d l^and a family T of subsets F in Y we denote
X Л T = {x a F : F e <F } and \X~\Y be a closure X in I . A sign □ is the end of
any proof. A fillter T is said to be countably centered if P) {Fn ^ Ф 0 for any countable
subfamily \F \ of the filter (f . For a fine uniformity Uf on Tychonoff space X [11,23]
v n ) «ej^ J
every continuous function is uniformly continuous, hence U (ujX) = C (X)
(U (ufX) = C (X) ) is the set of all (bounded) continuous functions on X and
Zu = Z(X) is the set of all zero-sets, CZU = CZ(X) is the set of all cozero-sets on
X [11, 17]. Every maximal Z — filter on Z is Z — ultrafilter. For any function
f g C(X) it is traditionally Z(f) = f 1 (0). A covering of u — open sets is said to be u — open covering and a covering of cozero-sets is said to be cozero covering.
Definition 1.1. A mapping f: uX ^ vY is said to be COZ — mapping, if
f—1 (CZV ) С CZU (or f_1 (Zv ) с Zu ) [13, 14]. A mapping f: uX ^ vY of a
uniform space uX into Tychonoff space Y is said to be Z — continuous, if
f -1 (CZ(X)) С CZU(or f -1 (Z(Y)) С Zu) [10].
Evidently, that every uniformly continuous mapping is a COZ — mapping and the converse, generally speaking, incorrectly [4, 5]. Also, every Z — continuous mapping f: uX ^ Y is COZ — mapping of f : uX ^ VY for any uniformity v on Y .
If Y is Lindelof or (Y, P) is a metric space, then its COZ — mapping is Zu — continuous
(see, for example, [4, 5]). If I = M or Y = I , then the COZ — mapping is
said to be u — continuous function and the COZ — mapping f: uX — I said to be u — function [4, 5].
We denote the set of all (bounded) u — continuous functions on the uniform space
uX as
C (X) (C* (X)) and Z(uX) be a ring of zero-sets functions from C (X) or C* (X) and
CZ (uX) consists of complements of sets of Z (uX) and, vice versa.
Definition 1.2. [14] A uniform space uX is said to be Alexandroff space if its each finite u — open covering is uniform.
Definition 1.3. [14] A mapping
f: uX — vY is said to be a COZ — homeomorphism, if f is COZ — mapping of uX onto vY in a one-to-one way, and the inverse mapping
f 1 : vY —^ uX is COZ — mapping. A two uniform spaces uX and vY are COZ — homeomorphic if there exists a COZ — homeomorphism of
uX onto vY.
Definition 1.4. [13, 14] A uniform space uX is said to be COZ —fine, if each COZ — mapping f: uX — vY is uniformly continuous.
Theorem 1.5. [13, 14] For a uniform space uX the next conditions are equivalent:
(1) uX is a COZ — fine space;
(2) if f is COZ — mapping of uX into a metrizable uniform space vY, then f: uX — vY is uniformly continuous;
(3) uX is a M - fine and proximally fine space;
(4) uX is separable M - fine [20] and proximally fine space;
(5) X is a proximally fine Alexandroff space;
(6) for every mapping f of uX into metrizable space vY , if f : uX — vY is
uniformly continuous, then f: uX — vfY is uniformly continuous, where vy is a fine
uniformity on Y.
Remark 1.6. Information about of (separable)
M - fine and proximally fine uniform spaces, see,
for example, [13, 14, 20].
2. Algebra of continuous functions
Definition 2.1. Let A be an algebra on
X . A function f '. X —> M. is said to be A) — continuous, if preimage under f of each closed (open) subset of K is in Z( A) ■ For an algebra A on X we denote by C^ (X) a set of all Z(A) — continuous functions on X. Evidently, that C^ (X) C C(X). Theorem 2.2. Let A be an algebra on
X . Then on X there is a uniformity u such that
Cn (X) = Cu (X).
Proof. Let U be the weakest uniformity on such that any function of A is uniformly continuous [23, Ch. I., Th. 3]. Evidently, that Zu = Z(A) . The uniformity U^ has a base of all
_ z
countable U — open coverings and U c U [7, Prop. 2.1]. From the item (2) of Lemma 2.4 [7] it follows CA (X) = U(UX) c U(U^X) = Cu (X) . From the item (3) of Lemma 2.4 [7] it follows, hence, any U — continuous function is Z (A) — continuous, i.e.
Cu(X)<zCA(X).n
z
Remark 2.3. In [19, 6.5 (a)] the statement: "a family U forms a uniformity", was noted without proof.
Corollary 2.4. Lei A be an algebra on X. Then is an algebra on X too.
Proof follows from the equality = CM(X) and Prop. 2.2. [7]. □
Corollary 2.5. Let A be an algebra on X . Then A = C^ (X) .
Proof. As CA(X) = Q(X) , then Z(A) = Z(uX) . Hence from the Theorem 4.3 [24] it follows, that A = CU (X) = C^ (X) . □ Theorem 2.6. Let A be an algebra on
X . Then A — C (X) for some uniformity U on
X if and only if H(A*) — Pu (H(A)).
Proof. Let A — CU (X) for some uniformity U on
X. Then
(3uX — H (C*(X)) — H (A*) [7, Th. 2.6 (3)]. From the item (3) of Theorem 2.12 [7] it follows, that vuX — H(A) and X C VuX C PuX . Let Z[ — VuX A Z(@uX). Then Pu(VUX) = ®(VUXZU) = PUX [24, Th. 2.9], i.e. H(X) — Pu(HA). Conversely, let H(A ) — Pu (H(A)) for some uniformity U on
X. Then
Pu (H (A)) — ®(X,Z:), where ZJ — H (A) a Z(H(A)). As Z(H( A*)) aJ = Z( A) [25, Th. 2.2], then Z( A) = X a Z" = Zu ■ By Theorem 2.2 and Corollary 2.5 we obtain A = Cu (X) . □
Compactification PuX coincides with Stone-Cech compactification PX if and only if U — Uj- [7]. So, Theorem 2.6. implies the next corollary.
Corollary 2.7. [19, 1.2 (i), (ii)] Let A be an algebra on X . Then A — C(X) if and only if
H (a*) — P(H (A)).
If C* (X) and C* (Y) are isomorphic for a uniform spaces UX and vY , then H (C*(X)) — PuX and H(C (Y)) — PVY are homeomorphic, and vice versa.
U\/s 'U \ V
Thus, we have the next theorem.
Theorem 2.8. For a uniform spaces uX and vY their compactifications PuX and PY
are homeomorphic if and only if C* (X) and C* (Y) are isomorphic.
From Theorem 2.5 and Corollaries 2.6, 2.8 of the paper [9] the next results are fulfilled. Corollary 2.9. Let uX and vY be the first-countable uniform spaces. Then uX is COZ
homeomorphic to vY if and only if C* (X) is isomorphic to C* (Y) .
Corollary 2.10. Let uX and vY be the first-countable COZ —fine uniform spaces. Then uX is uniformly homeomorphic to vY if and only f C (X) is isomorphic to C* (Y) .
For COZ — fine uniform spaces uX and vY Cu (X) = U (uX) both C*(Y) = U*(vY) [13, 14] and rings U* (liX) , U* (vY) are isomorphic to the rings U*(uX) , U*(vY) , respectively.
Corollary 2.11. Let uX and vY be a complete the first-countable COZ —fine uniform spaces. Then uX is uniformly homeomorphic to vY if and only if U (uX) is isomorphic to
U*(vY).
For C we can utilize the ring isomorphism f I—> Puf °f CM (^0
C (PMX) [7, Th. 2.8(4)] to characterize the maximal ideals of C* (X) in puX terms.
Theorem 2.12. The maximal ideals of C* (X) are in the one-to-one correspondence with the points of puX and are given I^ = ^ f g C (X) : Pu f (p) = 0 ^ for p is a point of
puX.
For the relate Z — filteres and Z — ultrafilteres to the ring C (X) we consider the
u u u s
mapping Z : Cu (X) — Zu (Z (f ) = f"1(0) g Zu for f G Cu (X)). The next
result is analogue [17, 2.3] and shows that the image of an ideal (maximal ideal) under Z is a Z —
filter (Z — ultrafilter) and that the preimage of a Z — filter (Z — ultrafilter) is an ideal u u u
(maximal ideal).
Proposition 2.13. (a) If I is a proper (maximal) ideal in Cu (X), then Z(I) = |Z (f) : f g I ^ is a Zu — filter (Zu — ultrafilter) on uX.
(b) If F is a Z — filter (Z — ultrafilter) on uX, then Z ] = { f g Cu (X) : Z (f) g F ^ is an (maximal) ideal in Cu (X).
Theorem 2.14. The maximal ideals of Cu (X) are in the one-to-one correspondence with the points of p X and are given
Jp={fE Q(J) : P £ [Z(/)]AX}/0r P isaP°int°f PuX ■
Proof. It is analogically to the proof of Theorem 1.30 [27]. □
Corollary 2.15. A compactification ßu X is homeomorphic to the space M-(uX ) of maximal ideals of the ring Cu ( X ) .
Proof. □ It is analogically to the proof of Corollary 1.31 [27].
Theorem 2.16. For a uniform spaces uX and vY their compactifications ßuX and ßvY are homeomorphic if and only if C (X) and Cv (Y) are isomorphic.
Corollary 2.17. Let uX and vY be the first-countable uniform spaces. Then uX is COZ— homeomorphic to vY if and only if Cu (X) is isomorphic to Cv (Y ).
Corollary 2.18. Let uX and vY be the first-countable COZ— fine uniform spaces. Then uX is uniformly homeomorphic to vY if and only if Cu (X) is isomorphic to Cv (Y ) .
For coz-fine uniform spaces uX and vY Cu ( X ) = U (uX ) and CV(Y) = U(vY) [13, 14] and the rings U(uX) , U(vY) are isomorphic to the rings
U(ÜX) , U(vY) , respectively.
Corollary 2.19. Let uX and vY be a complete the first-countable COZ— fine uniform spaces. Then uX is uniformly homeomorphic to vY if and only if U (uX ) is isomorphic to
U (vY ).
Remark 2.20. Note, that Theorem 2.12 is a uniform analogue of Stone Theorem [26] and Theorem 2.14 is a uniform analogue of Gelfand-Kolmogoroff Theorem [15].
3. On Z — complete uniform spaces.
Definition 3.1. A uniform space uX is said to be Z — complete if it is COZ —
homeomorphic to a closed uniform subspace of a power of M .
Remark 3.2. By analogue with [16], an ideal I С C (X) is said to be fixed, if
D-Z(/) = n{-Z(/): and if z(/) is a countably centered Zu ~
ultrafilter, then a maximal ideal i is said to be real ideal.
Theorem 3.3. For uniform space uX the following conditions are equivalent:
(1) uX is Zu — complete;
(2) X is complete with respect to the uniformity u ;
(3) uX = ßuX ;
(4) each countably centered Zu — ultrafilter is convergent;
(5) each point in X is the limit of the unique countably centered Z — ultrafilter on uX.
(6) every real maximal ideal in Cu ( X ) is fixed.
Proof, (l) ■ i '■ uX —^ ® be a COZ — homeomorphism of the uniform space
uX onto a closed uniform subspace X = CZ МГ with the uniformity U = UK ,
where М.Г is a power of M^Wg,). A uniform space W^M is — bounded and
complete [3], hence u'X' is also ÎSn — bounded and complete [3]. Then X' is complete with
tz
respect to the uniformity U m (see Proposition 2.1 [7]). From the item (1') of Lemma 2.4 [7], it follows, that the uniform spaces uzX and u X' are uniformly homeomorphic, so X is
X and U ^
m
2
complete with respect to the uniformity U .
(2) ^ (3) . It follows from items (1), (2) of Theorem 2.12 [7].
(3) ^ (4) ^ (5). It follows from items (1), (8) of Theorem 2.14 [7]. (5) ^^ (6). It is obvious, (see Remark 3.2).
(2)^(l). Let\CU(X)| = T. By the item 2 of Lemma 2.4 [7],
C (X) = U(u^X), hence, the uniform space U X is uniformly homeomorphically
embedded into M , i.e. the uniform space uX is COZ — homeomorphically embedded into M . From item (2) it follows, that uX is COZ — homeomorphic to a closed uniform subspace of
WpT.D
Lemma 3.4. [24] If p c Z(X) is a filter closed under countable intersections and n p = 0 , then on Tychonoff space X there is a base for closed sets, which is s.,n.-g.i.r. and there is a uniformity U such, that p £ U X.
Proof. [25, Lemma 3.5]. We put F = |Z £ Z(X) ! Z £ p or Z n P = 0 for some
P £ p^ . Then F is s.,n.-g.i.r. and the Wallman compactification G)(X, F ) is ß — like [24] compactification. All countable cozero coverings, consisting of the elements of family CF = {X \ Z: Z £ F }, form the uniformity U on X [7, Prop. 2.1]. Therefore,
CO(X,F) = ßuX, v(X, F) = UUX and p is a free countably centered Zu — ultrafilter on the uniform space uX , i.e. p £ VUX. □
Corollary 3.5. If X is a realcompact and non-Lindelöf space, then there exists a uniformity U on X such, that uX is not Z — complete.
Proof. If X is a realcompact and non-Lindelöf, then there is a filter p C Z(X) closed under countable intersections and n p = 0 [11, 3.8.3]. By Lemma 3.4, on X there exists a uniformity
u such, that X ^ U, X , i.e. the uniform space uX is not Z — complete. □
The next theorem characterizes the Tychonoff Lindelöf spaces by means of uniform structures. Theorem 3.6. Tychonoff space X is Lindelöf if and only if uX is Z — complete for any
uniformity u on X.
Proof. If Tychonoff space X is Lindelöf, then, evidently, that uX is Z — complete ([11, 3.8.3], item (2) of Theorem 3.3).
Let uX be z — complete for any uniformity u on the Tychonoff space X . We assume, that X is non-Lindelöf space. Then on X there is a filter p C Z(X) closed under countable
40
intersections such, that Op — 0 [10, 3.8.3]. Then, by Lemma 3.4, on X there is a uniformity
U such that X ^ u, X , i.e. the uniform space UX is not Z — complete. Contradiction. □ The next theorem characterizes the Wallman realcompactifications.
Theorem 3.7. For a Wallman realcompactifications u X the following conditions are equivalent:
(1) uX is an intersection of all cozero-sets in pu X which contains X;
(2) uUX is a completion of X with respect to U z ;
(3) any COZ — mapping f: uX — vY into a Z — complete uniform space vY has a uniformly continuous extension Vuf '. UUX —^ vY;
(4) any U — continuous function f '. uX —^ ^as a unifonnfy continuous extension
(5) If a countable u — closed sets family in
uX has empty intersection, then their closures in
u X have empty intersection.
(6) The equality IP)jewZj I = P)jeM I Z^ I is fulfilled for any countable family of
U — closed sets Z 5 in uX . I 'JzeN
(7) each point of u
X
is the limit of the unique countably centered Zu — ultrafilter on uX. Proof, (l) ^ (2). It follows by the items (1), (3) of Theorem 2.12 [7].
(2)^^ (3). Let f : UX —^ vY be an arbitrary COZ — mapping of uniform space UX into the Z — complete uniform space
vY
. By the item (1') of Lemma 2.4 [7] the mapping f : UZaX —^ v^Y is uniformly continuous. As v^Y is a complete uniform space the mapping f can be uniformly continuously extended over u X to the mapping u f : u X —> vY .
(3)^>(4). It is obvious, as W^IR is Zy —complete.
(4) ^^ (5). From Definition 2.10 [7] and Theorem 2.14 for any U — continuous function feCu(X) Z (vj) = is fulfilled. LetZ; = f~x (0); Z £ N , and
Pi {z. j- ^ = 0. Then CC = ^X \ Zi j- is a countable U — open covering and a e llZ0) and for a system EXJX = \EXu (X \ Z. by the item (3)
U Exua = VUX is fulfilled, where
Exu (X \ Z. ) = U X \ [ Zt \X=VU X \Z (u f). Suppose p be a countably
centered Z - ultrafilter such that p £ vX \ \jEx„a . Then p Ex„ (X \ Z ) ,
H 1A 14 14 \ I /
hence, p £ Z(vufj) = [Z(/j)J ^ , i.e. Z. =Z P for all IE N. We obtain
a contradiction, as . ^Z ^ 0. So, Ex OL is a uniform covering of V X. Thus,
Ï £ in Î ti ti
ÖExua= uuXor f]^[Z,\x=0.
(5) ^ (6). It is obvious.
It is obvious. (7)CÎ>(2). From Lemma 2.11 [7]. □
Theorem 3.8. If uX and vY are Z — and Z — complete uniform spaces, respectively. Then uX is COZ — homeomorphic to vY if and only if C (X) and Cv (Y) are isomorphic.
Proof. If a Z — complete uniform space uX is COZ — homeomorphic to Z — complete uniform space vY , then the rings CM (X) and Cv (Y) are isomorphic. It is obvious.
Let the rings C (X) and C (Y) be isomorphic. We fix an isomorphism i : C (X) —^ C ( Y) . Then an isomorphism i induces a homeomorphism
7 : H (C*(X)) — H (C*(Y)) . Each point X g X is identically with unique countably
centered real Z — ultrafilter p (item (6'), Th.3.7), which corresponds to the maximal real ideal
/M g H (C*(X)) . Then i (/M ) = is a maximal real ideal of H (C* (Y)), and vice
versa. Thus, under an isomorphism all maximal real ideals of the ring C (X) are transferring into
the maximal real ideals of the ring Cv (Y) , and vice versa. Then i : U^X —^ VpY is
Z Z
uniform homeomorphism, where U , V (see [7, Prop.2.1]) are the Wallman precompact uniformities [7, Remark 2.9] on X and Y, respectively. As Zu = Z Z and Zv = Z Z [7,
Lemma 2.4(3)], then the uniform spaces uX and vY are COZ — homeomorphic. □
Remark 3.9. Every fixed maximal real ideal / of a ring C (X) corresponds to a fixed
maximal real ideal / P| Cu (X^) of a ring Cu (X^) and this correspondence is one-to-one (it can be prove similar to [17,4.7]).
Remark 3.10. For the uniformity U of uniform space uX the equalities Ü^X = U^X [7, Th. 2.12(1)] and VUX = G)(ßuX,Z(ßuX) a VUX) [25, Th.2.9, Th.3.9] are fulfilled, hence, Wallman realcompactifications of u X and M„ coincide. By item (4) of Lemma 2.4 [7] for separable M — fine uniform space uX [18] U — Um and Cu (X) = U(uX ) = U(ÜX ). Thus, it takes place Corollary 3.11. If UX
and vY are complete separable M —fine uniform spaces. Then uX is uniformly homeomorphic to vY if and only if U (uX ) and U (vY ) are isomorphic.
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