OF PRODUCTS OF TWO
G
-SPACES
NATELLA ANTONYANReceived 5 September 2005; Revised 27 December 2005; Accepted 8 January 2006
LetGbe any Hausdorfftopological group and letβGXdenote the maximalG -compactifi-cation of aG-TychonoffspaceX. We prove that ifXandY are twoG-Tychonoffspaces such that the productX×Y is pseudocompact, thenβG(X×Y)=βGX×βGX.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
LetGbe any Hausdorfftopological group and letβGXdenote the maximalG -compac-tification of aG-TychonoffspaceX (i.e., a TychonoffG-space possessing aG -compac-tification). Recall that a completely regular Hausdorfftopological space is called pseudo-compact if every continuous function f :X→Ris bounded.
In this paper, we prove that if X and Y are twoG-Tychonoffspaces such that the productX×Yis pseudocompact, thenβG(X×Y)=βGX×βGX(seeTheorem 2.2). This is aG-equivariant version of the well-known result of Glicksberg [16], which for Ga locally compact group was proved earlier by de Vries in [10]. Note that even in the case of a locally compact acting groupG, our proof is shorter than that of [10, Theorem 4.1]. It follows fromProposition 2.7that the equalityβG(X×Y)=βGX×βGXdoes not imply, in general, the pseudocompactness ofX×Y even ifXandY both are infinite (cf. [16, Theorem 1]).
Theorem 2.10says that if a pseudocompact groupGacts continuously on a pseudo-compact spaceX, thenβGX=βX.
Let us introduce some terminology we will use in the paper.
Throughout the paper, all topological spaces are assumed to be Tychonoff(i.e., com-pletely regular and Hausdorff). The letter “G” will always denote a Hausdorff(and hence, completely regular) topological group unless otherwise stated.
For the basic ideas and facts of the theory ofG-spaces or topological transformation groups, we refer the reader to [5,7,11]. However, we recall below some more special notions and facts we need in the paper.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 93218, Pages1–9
By aG-space we mean a TychonoffspaceX endowed with a continuous actionG× X→X of a topological groupG. A continuous map ofG-spaces f :X→Y is called a G-map or an equivariant map if f(gx)=g f(x) for allx∈Xandg∈G.
IfX is aG-space andSa subset ofX, thenG(S) denotes theG-saturation ofS, that is,G(S)= {gs|g∈G, s∈S}. In particular,G(x) denotes theG-orbit{gx∈X|g∈G} ofx. IfG(S)=S, thenSis said to be an invariant set. The orbit space endowed with the quotient topology is denoted byX/G.
For a closed subgroupH⊂G, byG/H we will denote theG-space of cosets{gH|g∈ G}under the action induced by left translations.
On any product ofG-spaces we always consider the diagonal action ofG.
AG-compactification of aG-spaceXis a pair (b,bX), whereb:X→bXis aG -homeo-morphic embedding into a compactG-spacebXsuch that the imageb(X) is dense inbX. UsuallybX alone is a sufficient denotation. We will say that twoG-compactifications b1X and b2X are equivalent if there exists a G-homeomorphism f :b1X→b2X such that f(b1(x))=b2(x) for allx∈X. Clearly, the equivalence ofG-compactifications is an equivalence relation in the class of allG-compactifications ofX. We will identify equiva-lentG-compactifications; any class of equivalentG-compactifications will be denoted by the same symbolbX, wherebXis anyG-compactification from this equivalence class. An order relation in the family of allG-compactifications is defined as follows:b1Xb2X if there exists aG-map f :b2X→b1Xsuch that f b2=b1. It is easy to see thatb1X and b2X are equivalent if and only if b1Xb2X andb2Xb1X. We will writeb1X=b2X wheneverb1Xandb2X are equivalentG-compactifications. In a standard way, one can show that each nonempty family ofG-compactifications ofX has a least upper bound with respect to the order. In particular, if aG-spaceXhas aG-compactification, then there exists a largestG-compactificationβGXwith respect to the order;βGX is called the maximalG-compactification ofX.
A continuous real-valued function f :X→Ron aG-spaceXis said to beG-uniform if for anyε >0, there exists a neighborhoodU of the identity element in Gsuch that |f(gx)−f(x)|< εfor allx∈X,g∈U.
AG-spaceX is said to beG-Tychonoffif for any closed setA⊂X and any pointx∈ X\A, there exists aG-uniform function f :X→[0, 1] such thatf(x)=0 andA⊂ f−1(1). It is evident that each continuous function on a compactG-space isG-uniform, and hence every compactG-space isG-Tychonoff. Since an invariant subspace of aG -Tych-onoff space is again G-Tychonoff, we see that if aG-space has a G-compactification, then it isG-Tychonoff. The converse is also true (see, e.g., [1,2]). Thus, aG-space is G-Tychonoffif and only if it admits aG-compactification, and in particular, a maximal G-compactification. In [8,9], it was proved that ifGis a locally compact group, then ev-ery TychonoffG-space isG-Tychonoff. The local compactness ofGis essential here (see [18]).
and f ∈K. IfK is equicontinuous at each pointz0∈Z, then we will say that it is an equicontinuous set.
If additionallyZ is a G-space for a groupG, then one can define the following (in general not continuous) action ofGonC(Z,R):
(gψ)(z)=ψg−1z, ψ∈C(Z,R),z∈Z,g∈G. (1.1)
IfGis locally compact, then this action is continuous, otherwise it may be discontinuous (see, e.g., [7, Chapter I, Section 2.1]). However, the following result is true.
Lemma 1.1. LetZbe aG-space andKan invariant equicontinuous subset ofC(Z,R). Then the closureKis also an invariant set and the restriction of the action (1.1) toG×Kis con-tinuous.
Proof. For everyg∈G, define the mapg∗:C(Z,R)→C(Z,R) by settingg∗(ψ)=gψ,
wheregψis defined as in (1.1). First we show thatg∗is a continuous map.
Indeed, letCbe a compact set inZ,Uan open set inR, andM(C,U)= {ψ∈C(Z,R)| ψ(C)⊂U}. Since all the sets of the formM(C,U) constitute a subbase of the compact-open topology ofC(Z,R) andg−1
∗ (M(C,U))=M(g−1C,U), we infer thatg∗is
continu-ous.
Now chooseϕ∈K andh∈Garbitrary. One needs to show thathϕ∈K. LetV be a neighborhood ofgϕ. Since the above-defined maph∗is continuous, the seth−∗1(V)= h−1V is a neighborhood ofϕ. Consequently,h−1(V)∩K= ∅, which is equivalent to V∩hK= ∅. ButhK=KbecauseK is invariant. Hence,V∩K= ∅, as required. Thus, the proof that the closureKis an invariant subset is complete.
Next we observe that the closure of an equicontinuous set is again equicontinuous [17, Chapter 7, Theorem 14]; soKis an equicontinuous invariant subset ofC(Z,R).
Now the continuity of the restriction of the action (1.1) toG×Kfollows easily from the continuity of the evaluation mapω:K×Z→Rdefined byω(ψ,z)=ψ(z),ψ∈K, z∈Z (see, e.g., [17, Chapter 7, Theorem 15]). We refer the reader to [2, Lemma 2] for
more details.
We will need this lemma in the proof ofTheorem 2.2.
In what follows, we will need also the following two characterizations of the maximal G-compactificationβGXestablished in [8] (see also [4]).
Proposition 1.2. LetGbe a group andXaG-Tychonoffspace. Then the following hold. (1) EachG-mapf :X→Bto a compactG-space has a uniqueG-extensionF:βGX→B. (2) LetbXbe aG-compactification ofXsuch that everyG-map f :X→Bto a compact
G-space has aG-extensionF:bX→B. ThenbXis equivalent toβGX.
Proposition 1.3. LetGbe a group andXaG-Tychonoffspace. Then the following hold. (1) Each boundedG-uniform function f :X→Rpossesses a unique continuous
exten-sionF:βGX→R.
(2) IfbXis aG-compactification such that each boundedG-uniform functionf :X→R
2. Main results
Lemma 2.1. LetGbe any group,XaG-space, andAa denseG-subset ofX. Assume that f :X→Ris a continuous map such that the restriction f|A:A→RisG-uniform. Then f isG-uniform as well.
Proof. Define the map f:X→C(G,R) by setting f(x)(g)= f(gx),x∈X,g∈G. The continuity of ffollows from the fact that the compact-open topology is proper (see [14, Theorem 3.4.1]).
It is easy to see that theG-uniformness of f is just equivalent to the equicontinuity of the image f(X) inC(G,R). Since the restriction f|AisG-uniform, we infer that the set f(A) is equicontinuous. But closure of an equicontinuous set is again equicontinuous
[17, Chapter 7, Theorem 14]; so f(A) is equicontinuous. By continuity of f, f(X)⊂ f(A), yielding that f(X) is also equicontinuous. Hence, f isG-uniform. Theorem 2.2. LetGbe any group and letXandY beG-Tychonoffspaces such thatX×Y is pseudocompact. ThenβG(X×Y)=βGX×βGY.
Proof. According toProposition 1.3, it suffices to prove that every boundedG-uniform function f :X×Y→Rhas a continuous extensionF:βGX×βGY→R.
The idea is first to extend f to a boundedG-uniform functionϕ:βGX×Y→R, and then to extend in a similar wayϕto obtain the desired extensionF. In the nonequivariant case, this is due to Todd [21].
Define the mapf:X→C(G×Y,R) by setting
f(x)(g,y)=f(gx,g y) ∀x∈X, (g,y)∈G×Y. (2.1)
Continuity of ffollows from the fact that the compact-open topology is proper (see [13, Theorem 3.1]).
Claim 2.3. The image f(X) is an equicontinuous set inC(G×Y,R).
Proof of the claim. Letε >0 and (g0,y0)∈G×Y. We have to show that there exist neigh-borhoodsUofg0andVofy0such that
f(x)(g,y)−f(x)g0,y0< ε ∀x∈X,g∈U, y∈V. (2.2)
Since f is aG-uniform function, one can choose a neighborhoodUof the unity inG such that
f(tx,ty)−f(x,y)<ε
3 ∀(x,y)∈X×Y,t∈U. (2.3)
Then
f(x)(g,y)−f(x)g0,y0=f(gx,g y)−fg0x,g0y0
≤f(gx,g y)−fgx,g0y0+fgx,g0y0−fgx,g y0
+fgx,g y0−fg0x,g0y0.
It follows from (2.3) that for allx∈Xandg∈Ug0, we have f
gx,g y0−fg0x,g0y0< ε
3. (2.5)
It is known that the formula ϕ(y)=sup
x∈X
f(x,y)−fx
,g0y0, y∈Y, (2.6)
defines a continuous functionϕ:Y→R(see [15, Lemma 1.3]).
Sinceϕ(g0y0)=0, we conclude that there is a neighborhoodV ofg0y0inY such that ϕ(v)< ε/3 for allv∈V. Hence, one has
f(x,v)−f
x,g0y0<ε
3 ∀v∈V,x∈X. (2.7)
By continuity of the action onY, there exist neighborhoodsOandW ofg0 and y0, respectively, such thatOW⊂V andO⊂Ug0. Consequently, ifg∈Oandy∈W, then g y∈V andg y0∈V. Hence, (2.7) yields for allx∈X
f(gx,g y)−f
gx,g0y0<ε
3, f
gx,g y0−fgx,g0y0< ε
3. (2.8) Now, (2.4), (2.5), and (2.8) imply for allg∈Ug0andy∈Wthat
f(x)(g,y)−f(x)g0,y0< ε
3+ ε 3+
ε
3=ε, (2.9)
as required. Thus, f(X) is indeed an equicontinuous set, and the proof of the claim is
complete.
Now we continue with the proof ofTheorem 2.2. ConsiderG×Y as aG-space en-dowed with the actionh∗(g,y)=(gh−1,hy). Then the induced action (1.1) becomes the following action:
(hψ)(g,y)=ψgh,h−1y ∀ψ∈C(G×Y,R), g,h∈G,y∈Y. (2.10)
We claim that fis algebraically equivariant, that is,h f(x)= f(hx) for allx∈Xand h∈G. Indeed, if (g,y)∈G×Y, then we have
h f(x)(g,y)=f(x)gh,h−1y=f(ghx,g y)=f(hx)(g,y)=h f(x)(g,y),
(2.11)
which means thath f(x)=f(hx).
Consequently, f(X) is an invariant subset of C(G×Y,R). ByLemma 1.1 and the above claim, the closureT= f(X) also is an invariant subset ofC(G×Y,R), and the
restriction of the action (2.10) toG×Tis continuous.
Further, since f(X) is a bounded subset ofC(G×Y,R), it follows from the Arzela-Ascoli theorem [13, Theorem 6.4] thatTis compact.
Define the mapφ:βGX×Y→Rby the formulaφ(z,y)=F(z)(e,y), where (z,y)∈ βGX×Y andeis the unity ofG. Clearly,φis bounded.
Since the evaluation mapω:T×(G×Y)→Rdefined byω(ψ,t)=ψ(t),ψ∈T,t∈ G×Y, is continuous (see, e.g., [17, Chapter 7, Theorem 15]), we infer that φis also continuous.
If (x,y)∈X×Y, thenφ(x,y)=F(x)(e,y)=f(x)(e,y)= f(x,y), showing thatφ ex-tends f. Since f isG-uniform, it follows fromLemma 2.1thatφisG-uniform.
Since the product of a pseudocompact space and a compact space is pseudocompact (see, e.g., [14, Corollary 3.10.27]),βGX×Y is a pseudocompactG-space. Consequently, by the same way, one can prove that the boundedG-uniform functionφ:βGX×Y→R extends to a continuous functionF:βGX×βGY→R, which is the desired extension of
f. This completes the proof.
Remark 2.4. ForGa locally compact group,Theorem 2.2was proved earlier by de Vries in [10] in a different way. IfG, as a topological space, is ak-space (i.e., a quotient image of a locally compact space) andXis a pseudocompactG-space, thenβGX=βX(see [10, Lemma 5.5]). Hence,Theorem 2.2follows in this case directly from the classical result of Glicksberg [16] (this is just [10, Corollary 5.7]).
In the following lemma, we just list two known important cases when the product of two pseudocompact spaces is pseudocompact.
Lemma 2.5. The productX×Yof two spaces is pseudocompact, if at least one of the follow-ing conditions is fulfilled:
(1)Xis a pseudocompactk-space andY is a pseudocompact space;
(2)Xis a pseudocompact topological group andY is a pseudocompact space.
Proof. For the first statement, see, for example, [14, Theorem 3.10.26]. The second one is
proved in [20, Corollary 2.14].
Corollary 2.6. LetGbe any group,H a closed subgroup ofGsuch thatG/H is compact, and letXbe a pseudocompactG-Tychonoffspace. ThenβG(G/H×X)=G/H×βGX.
The following simple result shows that the converse ofTheorem 2.2is not true even if XandY both are infinite (cf. [16, Theorem 1]).
Proposition 2.7. LetGbe any group,Ha closed subgroup ofGsuch thatG/His compact, and letXbe a Tychonoffspace endowed with the trivial action ofG. ThenβG(G/H×X)= G/H×βX.
Proof. Evidently,G/H×βX is a G-compactification ofG/H×X. Hence, according to
Proposition 1.3, it suffices to prove that every boundedG-uniform function f :G/H× X→Rhas a continuous extensionF:G/H×βX→R.
Define a function f:X→C(G/H,R) by f(x)(t)= f(t,x), where (t,x)∈G/H×X. Then fis continuous, and it follows from theG-uniformness of f that the image f(X) is an equicontinuous set inC(G/H,R). Besides, the set f(X)(t0)= {f(x)(t0)|x∈X}is
F:βX→ f(X)⊂C(G/H,R). DefineF:G/H×βX→RbyF(t,z)= f(z)(t). The
com-pactness ofG/H insures thatFis continuous (see, e.g., [14, Theorem 3.4.3]). It remains
only to observe thatFextends f.
Recall that aG-spaceXis called free if for everyx∈X, the equalitygx=ximplies that g=e, the unity ofG.
Below, we will need the following well-known result.
Lemma 2.8. LetGbe a compact group andXa freeG-space. Then (G×X)/GisG -homeo-morphic toX, whereGacts on the orbit space (G×X)/Gaccording to the ruleh∗G(g,x)= G(gh−1,x).
Proof. The desiredG-homeomorphism f : (G×X)/G→Xis defined as follows:
fG(g,x)=g−1x ∀(g,x)∈G×X, (2.12)
whereG(g,x) stands for theG-orbit of the pair (g,x).
It is easy to verify that f is continuous and bijective. The closedness of f follows from that of the mapG×X→X, (g,x)→g−1x(see [5, Chapter I, Theorem 1.2]). If the action ofGonXis not trivial, thenProposition 2.7is no longer true. Namely, we have the following proposition.
Proposition 2.9. LetGbe an infinite, compact, metrizable group andXa finite-dimension-al, paracompact, noncompact, freeG-space. ThenβG(G×X)=G×βGX.
Proof. Suppose the contrary, thatβG(G×X)=G×βGX. Passing to the orbit spaces, we have
G×βGX
G =
βG(G×X)
G . (2.13)
Using the formula (βGZ)/G=β(Z/G) (see [4, Corollary 4.10]), we get βG(G×X)
G =β
G×X G
. (2.14)
Hence,
G×βGX
G =β
G×X G
. (2.15)
It is known that a finite-dimensional, paracompact, freeG-space has a freeG -compac-tification and in this caseβGXis also a freeG-space (see [3, Proposition 3.7]). Conse-quently, by virtue ofLemma 2.8, one has that (G×X)/G=Xand (G×βGX)/G=βGX. In sum, we getβX=βGX, which implies that each bounded continuous function f :X→R isG-uniform. However, this is not true.
by continuity of theG-action atxn∈X, there exists an elementgn∈Onsuch thatgnis different from the unity ofGandgnxn∈Un,n=1, 2,.... SinceXis a freeG-space, we see thatgnxn=xn,n≥1.
Now, let fn:X→[0, 1] be a continuous function such that fn(xn)=1, fn(gnxn)=0 and fn(X\Un)= {0}. Define f(x)=∞n=1fn(x),x∈X. Since{U1,U2,...}is disjoint and locally finite, f is a well-defined, continuous, bounded functionX→R. Hence, it should be alsoG-uniform, which yields a neighborhoodQof the unity inGsuch that|f(gx)− f(x)|<1/2 for allx∈Xandg∈Q. We choosen≥1 so large thatOn⊂Q. This implies thatgn∈Q, and hence 1= |f(gnxn)−f(xn)|<1/2, a contradiction. In general, if the acting groupGis not discrete, an actionG×X→X cannot be ex-tended (continuously) to an actionG×βX→βX; the natural rotation-action of the circle group on the planeR2provides a counterexample (see [19, Section 1.5]). However, the following result holds true.
Theorem 2.10. LetGbe a pseudocompact group andXa pseudocompactG-space. ThenX isG-TychonoffandβGX=βX.
Proof. The actionα:G×X→X uniquely extends to a continuous mapϕ:β(G×X)→ βX. ByLemma 2.5(2), the productG×X is pseudocompact, and hence, according to Glicksberg’s theorem [16],β(G×X)=βG×βX. Thus,ϕcan be treated as a continuous map ofβG×βX inβX which extendsα. But remember thatβG is a topological group containingGas a dense subgroup (see, e.g., [6, Theorem 4.1(f)]).
Further, the fact thatαsatisfies the two algebraic conditions of action implies easily that the mapϕ:βG×βX→βX satisfies these conditions as well. Thus,ϕis an action, and henceβX is a βG-space. In particular,βX is aG-space. Consequently,βX is a G -compactification ofX, and henceXis aG-Tychonoffspace. It is also clear thatβXis the maximalG-compactification ofX, that is,βGX=βX, as required.
Remark 2.11. It is worth to mention that there exists a pseudocompact group whose underlying topological space is not ak-space (see, e.g., [12,20]).
Acknowledgments
The author was supported by the Grants U42573-F from CONACYT and IN-105803 from PAPIIT, Universidad Nacional Aut ´onoma de M´exico (UNAM). We are thankful to the referee for useful comments.
References
[1] S. A. Antonjan and Yu. M. Smirnov, Universal objects and bicompact extensions for topological
groups of transformations, Doklady Akademii Nauk SSSR 257 (1981), no. 3, 521–526 (Russian),
English translation: Soviet Mathematics Doklady 23 (1981), no. 2, 279–284.
[2] S. A. Antonyan, Equivariant embeddings andω-bounded groups, Vestnik Moskovskogo
Univer-siteta. Seriya I. Matematika, Mekhanika 49 (1994), no. 1, 16–22, 95 (Russian), English transla-tion: Moscow University Mathematics Bulletin 49 (1994), no. 1, 13–16.
[3] N. Antonyan, Equivariant embeddings and compactifications of freeG-spaces, International
[4] N. Antonyan and S. A. Antonyan, FreeG-spaces and maximal equivariant compactifications,
An-nali di Matematica 184 (2005), no. 3, 407–420.
[5] G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York, 1972. [6] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups,
Pacific Journal of Mathematics 16 (1966), no. 3, 483–496.
[7] J. de Vries, Topological Transformation Groups. 1, Mathematisch Centre Tracts, no. 65, Mathe-matisch Centrum, Amsterdam, 1975.
[8] , Equivariant embeddings of G-spaces, General topology and Its Relations to Modern
Analysis and Algebra, IV (Proceedings of 4th Prague Topological Symposium, Prague, 1976), Part B, Society of Czechoslovak Mathematicians and Physicists, Prague, 1977, pp. 485–493. [9] , On the existence ofG-compactifications, Bulletin de l’Acad´emie Polonaise des Sciences.
S´erie des Sciences Math´ematiques 26 (1978), no. 3, 275–280.
[10] , On theG-compactification of products, Pacific Journal of Mathematics 110 (1984), no. 2,
447–470.
[11] , Elements of Topological Dynamics, Mathematics and Its Applications, vol. 257, Kluwer Academic, Dordrecht, 1993.
[12] D. Dikranjan and D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on
abelian groups, Topology and Its Applications 151 (2005), no. 1–3, 2–54.
[13] J. Dugundji, Topology, Allyn and Bacon, Massachusetts, 1966.
[14] R. Engelking, General Topology, PWN—Polish Scientific, Warsaw, 1977.
[15] Z. Frol´ık, The topological product of two pseudocompact spaces, Czechoslovak Mathematical Jour-nal 10(85) (1960), 339–349.
[16] I. Glicksberg, Stone- ˇCech compactifications of products, Transactions of the American
Mathemat-ical Society 90 (1959), 369–382.
[17] J. L. Kelley, General Topology, D. Van Nostrand, Toronto, 1955.
[18] M. G. Megrelishvili, A TikhonovG-space not admitting a compact HausdorffG-extension orG -linearization, Russian Mathematical Surveys 43 (1988), no. 2, 177–178.
[19] R. S. Palais, The Classification ofG-spaces, Memoirs of the American Mathematical Society, no.
36, American Mathematical Society, Rhode Island, 1960.
[20] M. G. Tkaˇcenko, Compactness type properties in topological groups, Czechoslovak Mathematical Journal 38(113) (1988), no. 2, 324–341.
[21] C. Todd, On the compactification of products, Canadian Mathematical Bulletin 14 (1971), 591– 592.
Natella Antonyan: Departamento de Matem´aticas, Divisi ´on de Inginier´ıa y Arcitectura, Instituto Tecnol ´ogico y de Estudios Superiores de Monterrey, 14380 M´exico,
Distrito Federal, M´exico