Bounded sets in spaces and topological groups

Topology and its Applications 101 (2000) 21–43

Bounded sets in spaces and topological groups

Salvador Hernándeza,1, Manuel Sanchisa,∗, Michael Tkacenkob,2,3

a Departament de Matemàtiques, Universitat Jaume I, Campus de Penyeta Roja s/n, 12071, Castelló, Spainb Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. Michoacan y la Purísima,

Iztapalapa, A.P. 55-532, C.P. 09340, D.F., Mexico

Received 13 May 1997; received in revised form 6 April 1998

Abstract

We investigateC-compact and relatively pseudocompact subsets of Tychonoff spaces with aspecial emphasis given to subsets of topological groups. It is shown that a relatively pseudocompactsubset of a spaceX isC-compact inX, but not vice versa. If, however,X is a topological group, thenthese properties coincide. A product of twoC-compact (relatively pseudocompact) subsetsA of XandB of Y need not beC-compact (relatively pseudocompact) inX×Y , but if one of the factorsX, Yis a topological group, then bothC-compactness and relative pseudocompactness are preserved. Weprove under the same assumption that, withA andB being bounded subsets ofX andY , the closureof A×B in υ(X×Y) is naturally homeomorphic to clυXA× clυYB, whereυ stands for the Hewittrealcompactification. One of our main technical tools is the notion of anR-factorizable group. Weshow that anR-factorizable subgroupH of an arbitrary groupG is z-embedded inG. This fact isapplied to prove that the group operations of anR-factorizable groupG can always be extended tothe realcompactificationυG of G, thus giving toυG the topological group structure. We also provethat aC-compact subsetA of a topological groupG is relatively pseudocompact in the subspaceB =A ·A−1 ·A of G. 2000 Elsevier Science B.V. All rights reserved.

Keywords:Bounded; Relatively pseudocompact;C-compact;z-embedded;R-factorizable group;Realcompactification; Distribution law

AMS classification: 54H11; 54D60; 54C30; 54D35; 22A05

∗ Corresponding author. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] The work was done during the third author’s visit to the Universitat Jaume I, Spain, in June–July, 1996. Hethanks for the hospitality and financial support from the Departament de Matemàtiques de la Universitat Jaume I.The first and second listed authors were partially supported by Generalitat Valenciana, Grant GV-2223-94 andFundació Caixa Castello, Grant P1B95-18.

0166-8641/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0166-8641(98)00098-4

22 S. Hernández et al. / Topology and its Applications 101 (2000) 21–43

1. Introduction

One of the most interesting problems in the theory of topological groups is to findcertain topological properties stable with respect to the product operation in the class oftopological groups which fail to be productive in the class of (Tychonoff) spaces. Thefirst (and the unique, as far as the authors know) “absolute” topological property meetingthese requirements was discovered by Comfort and Ross [10]: any Cartesian product ofpseudocompacttopological groups ispseudocompact, but a well-known example of Novak[30] shows that this is not the case when the factors are Tychonoff spaces. The Novak’sexample gives even more: there exist two countably compact Tychonoff spacesX andYsuch that the productX × Y is not pseudocompact. Surprisingly, it is a very difficult taskto construct two countably compact topological groups whose product is not countablycompact. All known examples of such products require extra axioms like the ContinuumHypothesis CH or Martin’s Axiom MA (see [11,24,37]). The problem whether there existtwo such groups in ZFC is still open (see Question 2 of [1] and Question 1 A.2 of [9]).

On the other hand, the list of non-productive properties for topological groups isquite long, it includes normality, the Lindelöf property, cellularity, countable tightness,sequentiality, the Fréchet–Urysohn property, etc. (see [27,28,36]).

Our aim is to study certain “relative” properties of subsets of Tychonoff spaces andHausdorff topological groups. We deal withC-compactnessandrelative pseudocompact-nessand establish some kind of stability of these properties with respect to the product op-eration and taking certain subgroups. A subsetA⊆X is calledC-compactin X if any con-tinuous functionf :X→R takesA to a compact subset ofR (see [8,17,18,25,26]). So, bydefinition, aC-compact subsetA of a spaceX is boundedin X, which means that all con-tinuous real-valued functions defined onX are bounded onA. It is clear that a pseudocom-pact space is bounded andC-compact in itself. Neither boundedness norC-compactnessis a productive property: pseudocompact spacesX andY with a non-pseudocompact prod-uctX × Y give us a counterexample. It turns out that the situation changes completely ifthe “enveloping” spaces in question have the topological group structure (see Theorem 2.2of [32]):

Theorem 1.1. Suppose that for everyi ∈ I , Ai is a bounded subset of a topological groupGi . Then the product

∏i∈I Ai is bounded in

∏i∈I Gi .

This result generalizes the theorem of Comfort and Ross [10] on products ofpseudocompact groups and, in a sense, is a relativization of that theorem. Furthermore,the above result admits the following generalization in the case of two factors:

Theorem 1.2. Let A be a bounded subset of a topological groupG and B a boundedsubset of a spaceY . ThenA×B is bounded inG× Y .

In fact, Theorem 1.2 was implicitly proved in [32] (it suffices to combine Corollary 2.5,Lemmas 2.8 and 2.10 of [32]). Our start point is the following result which also impliesthe Comfort–Ross theorem (see Corollary 3 of [25]).

S. Hernández et al. / Topology and its Applications 101 (2000) 21–43 23

Theorem 1.3. Let {Gi : i ∈ I } be a family of topological groups and for everyi ∈ I , letAibe aC-compact subset ofGi . Then the set

∏i∈I Ai isC-compact in

∏i∈I Gi .

Thus, boundedness andC-compactness are (relatively) productive properties in the classof topological groups. It is worth mentioning that the property of the factorsGα of beingtopological groups in Theorem 1.3 can not be replaced by compactness even in the case oftwo factors: there existC-compact subsetsA andB of the Cech–Stone compactificationβω of the discrete spaceω such thatA × B is notC-compact inβω × βω (see (4) ofExample 2.4). This still leaves a hope that if one of the factorsX or Y is a topologicalgroup, then the productA× B of C-compact subsetsA⊆ X andB ⊆ Y is C-compact inX× Y . To solve this central problem of the article, we use the following notion introducedby Arhangel’skiı and Genedi in [2] under a slightly different name.

Definition 1.4. A subsetA of a spaceX is calledrelatively pseudocompactinX, or brieflyr-pseudocompactinX, if any infinite family of open sets inXmeetingA has a cluster pointin A.

It is easy to see that everyr-pseudocompact subset ofX is C-compact inX, but notvice versa. The property of beingr-pseudocompact subset is not productive because of theNovak’s example, but everyC-compact subset of a topological group isr-pseudocompactin this group (see Corollary 3.11), so thatr-pseudocompactness is productive in the class oftopological groups by Theorem 1.3. We also show that ifA is anr-pseudocompact subsetof a topological groupG andB is anr-pseudocompact subset of a spaceY , thenA×B is r-pseudocompact inG× Y (see Theorem 3.4). In other words, anyr-pseudocompact subsetof a topological groupG is stronglyr-pseudocompactin G (see Definition 2.3 below).

Let us again turn to the problem of productivity ofC-compactness in the case whenone of two factors is a topological group. We can reformulate the problem asking whethereveryC-compact subset of a topological group isstronglyC-compactin this group. Thisquestion will seem even more natural if Theorem 1.2 is reformulated as follows: Everybounded subset of a topological groupG is strongly boundedin G (see Definition 2.3below).

One of the main results of the article is Theorem 4.8 which states that everyC-compactsubsetA of a topological groupG is stronglyC-compact inG or, equivalently, ifB isan arbitraryC-compact subset of a spaceY , thenA × B is C-compact inG × Y . Theproof of Theorem 4.8 is not straightforward and requires certain combination of differentmethods. Our proof is based on the relative distribution law for the closure of a productof two bounded subsets (see Theorem 4.7) which in turn depends on Theorem 1.2 andinvolves the Frolík technique from [15].

The article is organized as follows. Section 2 contains a detailed discussion ofrelationships between boundedness,r-pseudocompactness andC-compactness. It alsoincludes a number of examples clarifying the things.

In Section 3, we start a thorough study of bounded andC-compact subsets of topologicalgroups. The technique we use there depends on the notions of az-set and anR-factorizable


topological group. The first one is well known and refers to a subset of a spaceX which isof the formf−1(0) for some continuous functionf :X→R. As forR-factorizable groups,see Definition 3.1 or consult the articles [33,34] which contain the main results concerningthis notion. We only recall here that the class ofR-factorizable groups contains all Lindelöfgroups and arbitrary subgroups ofσ -compact groups, as well as Cartesian products ofσ -compact groups and dense subgroups of these products. One of the new results provedin Section 3 is Theorem 3.2: anR-factorizable subgroupH of an arbitrary topologicalgroupG is z-embedded inG, that is, for everyz-setF in H there exists az-setΦ in Gsuch thatΦ ∩ H = F . We apply this result to show that the Hewitt realcompactificationof anyR-factorizable topological groupH is again a topological group containingH asa subgroup (see Theorem 3.3). Another application of Theorem 3.2 is the following result(see Proposition 3.1): ifA is aC-compact subset of a topological groupG, thenA is C-compact in the subgroupH = 〈A〉 of G generated byA. We also show that the groupHisR-factorizable and perfectlyκ-normal (see Lemma 3.6), and henceH is z-embedded inG. Given aC-compact subsetA of an arbitrary topological groupG, Proposition 3.1 andLemma 3.6 sometimes enable us to replace the groupG by its subgroupH = 〈A〉 whichhas better properties.

Further, by Corollary 3.11, everyC-compact subsetA of a topological groupG is r-pseudocompact in the subspaceB = A ·A−1 ·A of G. This useful fact can be applied inSection 4 to give an alternative proof of Theorem 4.8.

In the sequel we consider only Tychonoff spaces. All topological groups are assumed tobe Hausdorff. The realcompactification of a spaceX is denoted byυX, andµX is used forthe Dieudonné-completion ofX. The Raıkov completion of a topological groupG (that is,the completion ofG with respect to its bilateral group uniformity) is denoted byG and issimply referred to as the group completion ofG. The family of all continuous real-valuedfunctions on a spaceX is C(X), andC∗(X) stands for the subfamily ofC(X) consistingof bounded functions. Iff :X→ R is a function andU ⊆X, we define the oscillation off atU by

osc(f,U)= sup{|f (y)− f (x)|: x, y ∈ U}.

If X is a space andx ∈ X, thenNX(x) or simplyN (x) stands for the family of openneighborhoods ofx in X. We say that a subsetY of X is Gδ-dense inX if every non-emptyGδ-set inX intersectsY . Finally, X is called anOz-space[4,6] if the closure ofevery open set inX is az-set. Such spaces are also known asperfectlyκ-normal[31].

2. Preliminary facts and results. Some examples

Here we discuss the relations between the notions of boundedness,r-pseudocompactnessandC-compactness in spaces and topological groups. It is well known that a subsetA of aspaceX is C-compact inX if and only if A is bounded in every cozero set ofX contain-ing A (see Proposition 2.7 of [6]). This fact implies that a subsetA of an Oz-spaceX isC-compact inX if and only if for every neighborhoodU of A in X, A is bounded inU .The following result characterizes this kind of sets for an arbitrary spaceX:


Proposition 2.1. The following assertions are equivalent for a subsetA of a spaceX:(1) A is r-pseudocompact inX;(2) A is r-pseudocompact in every neighborhoodU ofA in X;(3) A isC-compact in every neighborhoodU ofA in X;(4) A is bounded in every neighborhoodU ofA in X.

Proof. The implications(1)⇒ (2), (2)⇒ (3) and(3)⇒ (4) are clear, so we only needto show(4)⇒ (1). Consider a sequence{Un}n∈ω of open subsets ofX meetingA. LetKbe the set of cluster points of the sequence{Un}n∈ω in X. By (4), A is bounded inX, andhenceK is a non-empty closed set. Suppose thatA∩K = ∅ and consider the sequence ofopen sets{Vn}n∈ω whereVn =Un \K for everyn ∈ ω. Then{Vn}n∈ω has no cluster pointsin U =X \K. SinceA⊆ U , the latter contradicts(4). 2

Clearly, the wordneighborhoodin Proposition 2.1 can be replaced byopen neighbor-hood. It is well known that a subsetA of anOz-spaceX is C-compact inX iff for everyneighborhoodU of A in X, A is bounded inU (see [6, Corollary 2.11]). So, we have

Corollary 2.2. AnyC-compact subset of an Oz-spaceX is r-pseudocompact inX.

Since r-pseudocompactness,C-compactness and boundedness are not stable withrespect to the product operation, it seems interesting to separate and study the followingstronger versions of these notions.

Definition 2.3. A subsetA of a spaceX is calledstrongly r-pseudocompact(stronglyC-compact, strongly bounded) in X if for every spaceY and everyr-pseudocompact (C-compact, bounded) subsetB of Y , A× B is r-pseudocompact (C-compact, bounded) inX× Y .

To clarify relations between different kinds of bounded sets we present several examples.First, the following implications are clear:

pseudocompact⇒ r-pseudocompact⇒ C-compact⇒ bounded.

Neither of them can be inverted as (1), (2) and (3) of the example below show.

Example 2.4.(1) LetX be a pseudocompact space which is not countably compact. ThenX contains

an infinite closed discrete subsetA, and henceA is notC-compact inX [26, p. 460].However, all subsets of the pseudocompact spaceX are bounded inX (see also [25,Example 4]).

(2) Consider the Alexandroff compactificationαD of an infinite discrete spaceD.According to [17, Corollary 3.8],D is C-compact inαD. It is clear, however, thatD is notr-pseudocompact inαD.

(3) If A is a maximal almost disjoint family of infinite subsets ofω, then the Mrówka–Isbell spaceΨ (A)= ω∪A is pseudocompact [20, 5 I (5)]. Since every point inω is


isolated inΨ (A), A is r-pseudocompact inΨ (A). However,A is a closed discretesubspace ofΨ (A) [20, 5 I (3)], and henceA is not pseudocompact.

(4) Let P be a dense pseudocompact subspace ofβω such thatP × P is not pseu-docompact (see [20, Example 9.15]). ThenP is r-pseudocompact andC-compactin itself, butP × P is neither bounded inP × P norC-compact inβω× βω.

(5) ConsiderP as in (4). ThenP is strongly bounded and stronglyC-compact inβω by [7, Corollary 4] or [18, Corollary 5.7]. However,P is not stronglyr-pseudocompact inβω. Indeed, the sequence{{n} × {n}}n∈ω of open subsets ofβω × βω has no cluster points inP × P . Note thatβω is a compactOz-space[4, Theorem 5.8].

(6) Fix p ∈ ω∗ and consider the spaceX = βω \ {p}. Since every infinite closed subsetof βω has the cardinality 2c, X is strongly bounded in itself (see [7, Proposition 1]or [14, Theorem 3.5.1]). Proposition 1 of [7] and Theorem 2.5 imply thatX isstrongly r-pseudocompact in itself. In addition, according to [18, Theorem 5.5],X is stronglyC-compact in itself. We will show thatX is notp-bounded (in itself).Suppose that there existsq ∈ ω∗ with

q ∈⋂F∈p

clXF.

So, if we considerq as a free ultrafilter inω, F ∈ q for everyF ∈ p [20, 6.5(c)].Therefore,p ⊆ q and hencep = q . We have proved that the sequence of opensubsets{n}n∈ω has nop-limit points inX.

The following theorem characterizes stronglyr-pseudocompact sets in a similar way tothe one used in [32] for strongly bounded subsets. We omit its proof which goes like inProposition 1 of [7].

Theorem 2.5. LetA be a subset of a spaceX. The following conditions are equivalent:(1) A is stronglyr-pseudocompact inX;(2) every infinite family of pairwise disjoint open subsets ofX meetingA contains an

infinite subfamily{Un}n∈ω such that

A∩⋂F∈F

clX

(⋃n∈F

Un

)6= ∅

for each filterF of infinite subsets ofω;(3) for every pseudocompact spaceY , A× Y is r-pseudocompact inX× Y .

In the sequelβω is identified with the set of ultrafilters onω, and the set of all freeultrafilters onω is denoted byω∗.

Definition 2.6. Let {Sn}n∈ω be a sequence of non-empty subsets of a spaceX, and supposethatp ∈ ω∗. We callx ∈X is ap-limit point of the sequence{Sn}n∈ω if {n ∈ ω: Sn ∩ V 6=∅} ∈ p for every neighborhoodV of x in X.


The above notion was introduced by Ginsburg and Saks in [21] as a generalization ofthe Bernstein’s concept of ap-limit of a sequence of points [3]. The following concept wasintroduced and investigated in [16].

Definition 2.7. Let p ∈ ω∗. A subsetA of a spaceX is calledp-bounded inX if everysequence{Un}n∈ω of open subsets ofX with A∩Un 6= ∅ for eachn ∈ ω has ap-limit pointin X.

It is clear that for everyp ∈ ω∗, p-bounded sets are bounded. However, there existbounded sets which are notp-bounded for anyp ∈ ω∗ (see [21] for details). The followingresult can be deduced from Lemmas 2.10 and 2.8 of [32] after a slight modification of thelatter (see also the proof of Theorem 4.3 in [18]).

Lemma 2.8. LetV be an open set in a topological groupG. The following conditions areequivalent for a subsetA⊆ V :

(1) A is bounded inV ;(2) A is p-bounded inV for eachp ∈ ω∗;(3) A is p-bounded inV for somep ∈ ω∗.

Corollary 2.9. The following conditions are equivalent for a subsetA of a topologicalgroup:

(1) A is r-pseudocompact inG;(2) for any p ∈ ω∗, every sequence{Un}n∈ω of open subsets ofG meetingA has a

p-limit in A;(3) for somep ∈ ω∗, every sequence{Un}n∈ω of open subsets ofG meetingA has a

p-limit point inA.

Proof. The implications(2)⇒ (3) and(3)⇒ (1) are obvious. Therefore, we only needto prove(1)⇒ (2). Letp ∈ ω∗ be arbitrary. Suppose that there exists a sequence{Un}n∈ωof open subsets ofX with Un ∩ A 6= ∅ for all n ∈ ω which has nop-limit points in A.Let K be the set of allp-limit points of {Un}n∈ω in G. It is clear thatK is closed anddisjoint fromA. Therefore,A is notp-bounded in the open setU = G \ K. Apply (1)and the equivalence of (1) and (4) of Proposition 2.1 and the equivalence of (1) and (2) ofLemma 2.8 to obtain a contradiction.2Theorem 2.10.Every r-pseudocompact subset of a topological groupG is stronglyr-pseudocompact inG.

Proof. Let A be anr-pseudocompact subset ofG. Consider a sequence{Un}n∈ω of opensubsets ofG meetingA and a free filterF of infinite subsets ofω. Let p ∈ ω∗ be anultrafilter containingF . According to Theorem 2.5, it suffices to prove that there existsx ∈ A with x ∈ clG(

⋃n∈F Un) for everyF ∈ p. By Corollary 2.9, the sequence{Un}n∈ω

has ap-limit point x ∈A. Therefore,{n ∈ ω: V ∩Un 6= ∅}∩F 6= ∅ for every neighborhoodV of x and eachF ∈ p. This completes the proof.2


In the rest of this section we shall study several properties ofC-compact subsets relatedto metrizability. Let us start with two lemmas.

Lemma 2.11. LetA be aC-compact subset of a spaceX. Suppose thatf is a continuousmapping ofX to a spaceY of countable pseudocharacter. Thenf (A) is closed inY .

Proof. Suppose thatf (A) is not closed inY and choosey ∈ Y \ clY f (A). Accordingto [20, 3.11(b)], the singleton{y} is a zero-set ofY which does not meetf (A). Letg > 0 bea continuous real-valued function onY satisfyingg−1(0)= {y}. Theng is positive onf (A)and infz∈f (A) g(z)= 0. Sincef (A) isC-compact inY , we obtain a contradiction.2

A spaceX is called submetrizable if there exists a continuous one-to-one function fromX onto a metrizable space. It is easy to see that everyC-compact subset of a submetrizablespace is compact. The following lemma generalizes this fact.

Lemma 2.12. Let A be aC-compact subset of a spaceX and let f be a continuousmapping ofX to a submetrizable spaceM. Thenf (Z ∩ A) is a compact subset ofMfor every zero-setZ in X.

Proof. By assumption,Z = g−1(0) whereg is a continuous function fromX into the unitintervalI = [0,1]. The diagonal producth = f 4 g of f andg is a continuous functionfromX to the submetrizable spaceM×I . So, Lemma 2.11 implies thath(A) is a closedC-compact subset ofM × I . Therefore,h(A) is compact. LetpM andpI be the projectionsof M × I to M andI , respectively. It is clear that the setsK andpM(K) are compact,whereK = p−1

I (0)∩ h(A). SincepI ◦ h= g andpM ◦ h= f , we have

pM(K)= pM(p−1I (0)∩ h(A))= f (A∩ g−1(0)

)= f (A∩Z),and the proof is complete.2Corollary 2.13. LetA be aC-compact subset of a spaceX. ThenA∩Z isC-compact inX for each zero-setZ ofX.

We can now obtain a metrization theorem forC-compact sets which is close to Lem-ma 2.1 of [18].

Theorem 2.14.Letf be a continuous mapping of a spaceX to a submetrizable spaceMsuch that the restrictionf |A of f to aC-compact subsetA ofX is one-to-one. ThenA isa metrizable compact space.

Proof. Denote byB the family of all cozero sets inX. ThenBA = {B ∩ A: B ∈ B}is a base for the induced topology inA. Let K = f (A). Since f |A is one-to-one,f (A∩B)=K \f (A\B) for eachB ∈ B. According to Lemma 2.12,f (A\B) is closed inK. Therefore,f (A∩B) is open inK for eachB ∈ B, and hencef |A is a homeomorphism


of A ontoK. SinceK is compact by Lemma 2.12, the result immediately follows from thefact that every submetrizable compact space is metrizable.2

Let us say that a familyγ of subsets ofX T0-separatesthe points of a subsetA⊆X iffor any distinctx, y ∈A there existsZ ∈ γ such that eitherx ∈Z 63 y or y ∈ Z 63 x.

Corollary 2.15. LetA be aC-compact subset of a spaceX and letγ be a countable familyof zero-sets inX whichT0-separates the points ofA. ThenA is a metrizable compact space.

Proof. For everyZ ∈ γ , there exists a continuous functionfZ fromX to the unit intervalI = [0,1] such thatZ = f−1

Z (0). The diagonal productf = 4Z∈γ fZ is a continuousfunction fromX to Iℵ0 and the restrictionf |A is one-to-one. The result now follows fromTheorem 2.14. 2

We shall apply the above results to obtain a metrization criterion for locally pseudocom-pact groups. The following result generalizes Theorem 2.5 of [18] proved for the Abeliancase.

Theorem 2.16.The following assertions are equivalent for a locally pseudocompactgroupG:

(1) there exist a neighborhoodV of the identity inG and a countable familyγ of zero-sets ofV whichT0-separates the points ofV ;

(2) there exist a neighborhoodV of the identity inG and a countable familyγ of zero-sets ofG whichT0-separates the points ofV ;

(3) G is metrizable.

Proof. (1)⇒ (2) Let V andγ be as in(1). There exists an open neighborhoodW of theidentity satisfying clGW ⊆ V and such that clGW is pseudocompact. Consider the family

µ= {Z ∩ clGW : Z ∈ γ }.Thenµ is a family of zero-sets in clGW whichT0-separates points of clGW . Since clGWisC-embedded inG [25, Corollary 9], we conclude that there exists a countable family ofzero-sets ofG whichT0-separates the points of clGW .(2)⇒ (3) LetV andγ be as in(2). Choose an open neighborhoodW of the identity such

thatA= clGW is pseudocompact and lies inV . For everyZ ∈ γ , there exists a continuousfunctionfZ :G→ [0,1] such thatZ = f−1

Z (0). The diagonal productf = 4Z∈γ fZ is acontinuous mapping ofG to the metrizable space[0,1]ω andf |A is one-to-one because ofthe choice ofγ . Therefore,A is metrizable by Theorem 2.14. SinceeG ∈W ⊆ clGW = AandW is open inG, the groupG has a countable base at the identityeG. The result nowfollows from the Birkhoff–Kakutani metrization theorem.(3)⇒ (1) SinceG is metrizable and locally pseudocompact, it is locally compact.

The conclusion immediately follows from the fact that a compact metrizable space hasa countable base which consists of cozero sets.2


3. R-factorizable groups andC-compact sets

We start with the definition ofR-factorizable groups [33,34].

Definition 3.1. A topological groupG is calledR-factorizable if for every continuousfunction f :G→ R there exist a continuous homomorphismπ of G onto a secondcountable topological groupH and a continuous functionh :H →R such thatf = h ◦ π .

Recall that a subsetS of a spaceX is z-embeddedin X if for every zero-setZ in S, thereis a zero-setZ∗ in X with Z∗ ∩ S =Z.

Theorem 3.2. EveryR-factorizable subgroupH of a topological groupG is z-embeddedin G.

Proof. Let F be a zero-set inH , and consider a continuous real-valued functionf onHsuch thatF = f−1(0). SinceH isR-factorizable, we can find a continuous homomorphismπ :H → P onto a second countable groupP and a continuous functiong :P → R suchthat f = g ◦ π . Denote byL the kernel ofπ . Let {On: n ∈ ω} be a countable base atthe identity ofP . We can define by induction a sequence{Un: n ∈ ω} of open symmetricneighborhoods of the identitye in G satisfying the following conditions for eachn ∈ ω:

(i) U2n+1⊆Un;

(ii) Un ∩H ⊆ π−1(On).

It is clear thatK =⋂n∈ω Un is a closed subgroup ofG andK ∩ H ⊆ L. Let ϕ be thecanonic mapping ofG onto the left coset spaceG/K. SinceK ∩H is a subgroup ofL,there exists a functionψ :ϕ(H)→ P satisfyingψ ◦ ϕ|H = π .

Open setsUn satisfye ∈ Un =U−1n andU2

n+1⊆Un, so Theorem 8.1.10 of [13]) impliesthat there exists a continuous left invariant pseudometricd onG such that

(∗) {x ∈G: d(x, e) < 1/2n

}⊆Un ⊆ {x ∈G: d(x, e)6 2/2n}.

One easily verifies thatd(x, y)= 0 iff x−1y ∈K. The latter enables us to define a metric% onG/K such thatd(x, y)= %(ϕ(x),ϕ(y)) for all x, y ∈G.

For everyx ∈ G, y ∈ G/K and ε > 0, defineBε(x) = {x ′ ∈ G: d(x ′, x) < ε} andCε(y)= {y ′ ∈G/K: %(y ′, y) < ε}. By the definition of%, we have

(a) ϕ(Bε(x)

)= Cε(ϕ(x)) for eachx ∈G.In other words, the images underϕ of open balls inG are open in the metric space(G/K,%). One can easily verify that the ballsBε(x) satisfy

(b) Bε(x)= ϕ−1ϕ(Bε(x)

)for all x ∈G andε > 0.

Let t% be the topology onG/K generated by%. Note thatt% is coarser that the quotienttopology onG/K. We claim that the homomorphismψ of ϕ(H) to P remains continuousif ϕ(H) is considered as a subspace of(G/K, t%), and this is the key point of the proof.Indeed, let a pointy ∈ ϕ(H) and an open setO ⊆ P with z = ψ(y) ∈ O be arbitrary.


There existsn ∈ ω such thatzOn ⊆ O . Choosex ∈ H with ϕ(x) = y; thenπ(x) = z.Note that the setU = B1/2n(e) is contained inUn by (∗) and the imageϕ(xU) is an openneighborhood ofy in (G/K, t%) by (a), so (b), (ii) and the equalityH ∩ x U = x (H ∩U)together imply that

ψ(ϕ(x U)∩ ϕ(H))=ψ(ϕ(H ∩ x U))= π(H ∩ x U)

= y π(H ∩U)⊆ y π(H ∩Un)⊆ y On ⊆O.This proves the continuity ofψ on the subspaceϕ(H) of (G/K, t%).

It remains to find a zero-setF0 in G such thatF0 ∩H = F . DenoteΦ = g−1(0)⊆ P .Fromf = g ◦ π it follows thatπ−1(Φ) = F . It is clear thatΦ∗ = ψ−1(Φ) is a zero-setin ϕ(H). Being a subspace of the metric space(G/K,%), ϕ(H) is z-embedded inG/K.Therefore, there exists a zero-setF ∗ in (G/K,%) such thatF ∗ ∩ϕ(H)=Φ∗. Let us verifythatF = F0 ∩ H whereF0 = ϕ−1(F ∗) is a zero-set inG. Sinceψ ◦ ϕ|H = π , we haveϕ−1(ψ−1(Φ))∩H = π−1(Φ), that is,ϕ−1(Φ∗)∩H = F . Consequently,

F0 ∩H = ϕ−1(F ∗)∩H = ϕ−1(F ∗ ∩ ϕ(H))∩H = ϕ−1(Φ∗) ∩H = F.This completes the proof.2

In 1985, V. Pestov and M. Tkacenko posed the problem whether the DieudonnécompletionµG of a topological groupG is a topological group (see [38, Problem III.28]).In other words, the problem is to extend continuously the group operations fromG toµG.Uspenskiı [39] gave the positive answer for the special case when a groupG has countableo-tightness. In particular, the result is valid for all separable groups and for groups ofcountable cellularity. According to [34, Corollary 4.10], every topological groupG whosetopology is defined by compact sets has countable o-tightness, and hence both extensionsµG andυG are topological groups. Theorem 3.2 enables us to resolve the problem for theHewitt realcompactification in the class ofR-factorizable groups.

Recall that a topological groupG is said to beℵ0-bounded [23] if for everyneighborhoodU of the identity inG, there exists a countable subsetK of G such thatK · U =G. It is known that everyR-factorizable group isℵ0-bounded (see the commentafter Definition 1.12 of [33]).

We say thatS isGδ-densein X if every non-emptyGδ-set inX meetsS. The definitionsof aGδ-closedset and theGδ-closureare self-explanatory.

Theorem 3.3. If G is anR-factorizable topological group, then the group operations ofG can be extended continuously over the realcompactificationυG ofG, thus makingυGa topological group.

Proof. Let G be the completion ofG, and denote byH the Gδ-closure ofG in G.A routine verification shows thatH is a subgroup ofG. We shall prove thatH = υG.SinceG is R-factorizable, it isℵ0-bounded. So,G is alsoℵ0-bounded as the completionof an ℵ0-bounded group [23, Proposition 3], and henceG can be embedded as atopological subgroup to a Cartesian productΠ of second countable topological groups [23,Corollary 1]. Applying the fact thatG is complete, we conclude thatG can be identified


with a closed subgroup ofΠ . Thus,G is realcompact. By Theorem 3.2,G is z-embeddedin G, and hence Lemma 1.1(b) of [5] implies thatυG coincides with theGδ-closure ofGin G, that is,H = υG. 2R-factorizable groups need not have countable o-tightness. The simplest counterexam-

ple is the free Abelian groupA(X) on the one-point LindelöficationX of an uncountablediscrete space. Another one is the weak sum of uncountably many copies of the discretegroupZ2 endowed with theℵ0-box topology (see Example 2.1 of [34]). Both topologicalgroups are Lindelöf and henceR-factorizable by [33, Assertion 1.1]. Therefore, Theo-rem 3.3 actually extends our knowledge on topological groups whose realcompactificationremains a topological group.

On the other hand, it is not known whether separable topological groups areR-fac-torizable. A similar problem for topological groups of countable cellularity also remainsopen [34, Problem 3.1].

The following result refines Theorem 3.3.

Theorem 3.4. If G is anR-factorizable group, so isυG.

Proof. Let f be a continuous real-valued function onυG. SinceG is R-factorizable, wecan find a continuous homomorphismπ of G to a second countable topological groupK and a continuous real-valued functionϕ on K such thatf |G = ϕ ◦ π . Consider thecontinuous homomorphismπ of the completionG of G to the completionK of Ksatisfyingπ |G = π . Theorem 3.3 implies thatG⊆ υG⊆ G. SinceG is Gδ-dense inυGandK is first countable space, we haveπ(υG)=K. Let us definep = π |υG andg = ϕ◦p.Both functionsf andg are continuous onυG andg|G = f |G = ϕ ◦π . SinceG is dense inυG andK is Hausdorff, we conclude thatf = g, that is,f = ϕ ◦ p. Therefore, the groupυG isR-factorizable. 2Problem 3.5. Is the completionG of anR-factorizable groupG R-factorizable?

Our aim now is to establish several interesting properties ofC-compact subsets oftopological groups. Let us show thatR-factorizable groups naturally correspond tobounded subsets of topological groups.

Lemma 3.6. If A is a bounded subset of a topological groupG, then the subgroup〈A〉 ofG generated byA is anR-factorizable group which is also an Oz-space.

Proof. Let G be the completion ofG. SinceA is bounded inG, the setY = clGA iscompact [12, 3.1]. The subgroup ofG generated byY is σ -compact and contains〈A〉.Therefore, Corollary 1.13 of [33] implies that〈A〉 is R-factorizable. Since everyσ -compact group is anOz-space and the property of being anOz-space is inherited by densesubsets [4, Proposition 5.3], we conclude that〈A〉 is anOz-space. 2Proposition 3.7. Let A be aC-compact subset of a topological groupG. ThenA is aC-compact subset of the subgroup〈A〉 ofG generated byA.


Proof. By Lemma 3.6, the group〈A〉 is R-factorizable. Therefore, Theorem 3.2 impliesthat 〈A〉 is z-embedded inG. SinceA is bounded inG, from [6, Proposition 2.7(3)] itfollows thatA is bounded in〈A〉. By [6, Proposition 2.7(5)], it suffices to prove thatAis completely separated from every zero-setZ of 〈A〉 such thatZ ∩ A = ∅. Let Z be azero-set in〈A〉 disjoint fromA. Since〈A〉 is z-embedded inG, there exists a zero-setZ∗of G such thatZ∗ ∩ 〈A〉 =Z. Consequently,Z∗ ∩A= ∅. SinceA isC-compact inG, wecan find a continuous real-valued functionf onG with Z∗ ⊆ f−1(0) andA⊆ f−1(1) [6,Proposition 2.7(5)]. Therefore,Z andA are completely separated byg = f |〈A〉. 2

The following example shows that we can not replaceC-compactby boundedinProposition 3.7. Given an ordinalα, [0, α] stands for the space of ordinal numbersα + 1= {σ : σ 6 α} endowed with the order topology. As usual, we denote byω andω1, respectively the first infinite and the first uncountable ordinals.

Example 3.8. Let Y be the space obtained by deleting the corner point{(ω,ω1)} of theproduct space[0,ω] × [0,ω1]. It is easy to check that the countable closed discrete setH = ω× {ω1} is bounded inY . LetG be the free Abelian topological group on the spaceY [29]. SinceH is bounded inY , it is also bounded inG. Consider the subgroup〈H 〉generated byH in G. By Assertion D of Section 4 in [22],〈H 〉 is closed inG. It is clearthat〈H 〉 is normal as a topological space because it is countable. SinceH is closed in〈H 〉,it is C-embedded in〈H 〉. Therefore,H is a discreteC-embedded subset of〈H 〉, and henceit is not bounded in〈H 〉.

In what follows we will improve Proposition 3.7 and present one more property ofC-compact subsets of topological groups. First, we need a lemma.

Lemma 3.9. Let A be a Gδ-dense subset of a countably compact spaceX and let{(an, γn): n ∈ ω} be a sequence such thatan ∈ X andγn is an open cover ofX for eachn ∈ ω. Then there exists a pointb ∈A such that for any open coverµ ofX one can find aninfinite subsetP ofω such that for eachn ∈ P :

(∗) there areU ∈ γn andV ∈µ such thatan ∈ V, b ∈U andU ∩ V 6= ∅.In addition, ifX is compact andp ∈ ω∗, one can chooseP ⊆ ω with P ∈ p.

Proof. Suppose thatX is countably compact. Leta ∈X be a cluster point of the sequence{an: n ∈ ω}. For everyn ∈ ω, choose an elementUn ∈ γn with a ∈ Un and putF =⋂n∈ω Un. Thena ∈ F 6= ∅. SinceF is aGδ-set inX andA isGδ-dense inX, there exists

a pointb ∈ F ∩A.Letµ be an open cover ofX. We fixV ∈µ with a ∈ V and defineP = {n ∈ ω: an ∈ V }.

The setP is infinite and we claim that everyn ∈ P satisfies(∗). Indeed, ifn ∈ P , thena ∈ Un ∩ V 6= ∅, an ∈ V ∈µ andUn ∈ γn.

The case whenX is compact can be considered similarly taking the pointa as ap-limitof the sequence{an: n ∈ ω}. 2


Theorem 3.10.Let p ∈ ω∗. If A is a C-compact subset of a topological groupG, thenevery sequence{On}n∈ω of open sets inA ·A−1 ·A meetingA has ap-limit in A.

Proof. Let G be the completion of the groupG. ConsiderX = clGA. ThenX is a compactspace and by Lemma 1 of [25],A is Gδ-dense inX. Consider a family{On}n∈ω of opensets inB =A ·A−1 ·A meetingA. For everyn ∈ ω, pick a pointan ∈A∩On and chooseopen symmetric neighborhoodsUn andVn of the identity inG such that(anUn)∩B ⊆OnandV 2

n ⊆ Un. It is clear thatγn = {(xVn) ∩ X: x ∈ X} is an open cover ofX. ApplyLemma 3.9 to the sequence{(an, γn): n ∈ ω} and choose a corresponding pointb ∈A. Weclaim thatb is ap-limit of the family {On: n ∈ ω}.

Indeed, letO be a neighborhood ofb in B. We can find open symmetric neighborhoodsU andV of the identity inG such that(Ub)∩B ⊆O andV 2⊆U . Consider the open coverµ = {(V x) ∩ X: x ∈ X} of X. By the choice of the pointb ∈ A, there exists an infinitesubsetP of ω with P ∈ p such that everyn ∈ P satisfies(∗) of Lemma 3.9. For everyn ∈ P , choose elementsK ∈ γn andL ∈ µ such thatb ∈K, an ∈L andK ∩L 6= ∅. Pick apointx ∈A∩K∩L. We shall show thatanx−1b ∈On∩O 6= ∅. Indeed, fromx, b ∈K ∈ γnit follows thatx, b ∈ yVn for somey ∈X, and hencex−1b ∈ V−1

n y−1yVn = V 2n ⊆ Un. So,

we haveanx−1b ∈ (anUn) ∩ B ⊆ On. On the other hand, a similar argument shows thatanx−1b ∈ B ∩ (Ub)⊆ O , which implies thatOn ∩O 6= ∅. Since the latter conclusion is

valid for everyn ∈ P andP ∈ p, we conclude thatb is ap-limit point of the sequence{On: n ∈ ω}. 2

Making use of Theorems 3.10, 2.10 and the characterization of strongly bounded subsetsgiven in [32] (see also [7, Proposition 1]), we obtain the following corollary.

Corollary 3.11. AnyC-compact subsetA of a topological groupG is strongly boundedand stronglyr-pseudocompact inA ·A−1 ·A and hence inG.

Since the product of a family ofC-compact subsets of topological groups isC-compactin the product group (Theorem 1.3), we have the following.

Corollary 3.12. For everyi ∈ I , letAi be aC-compact subset of a topological groupGi .Then

∏i∈I Ai is stronglyr-pseudocompact in

∏i∈I Gi .

Following Frolík [14], we say that a spaceP belongs to the classP if P × Y ispseudocompact for every pseudocompact spaceY . Making use of Corollary 3.11, wegeneralize Corollary 2.14 of [32] as follows.

Corollary 3.13. EveryC-compact subgroup of a topological group is pseudocompact andbelongs to the classP .

In [2], r-pseudocompact subsets were introduced with a slightly different name anddistinct (but equivalent) definition. It is proved in [2] under CH that the discrete space ofcardinalityℵ1 can be embedded as anr-pseudocompact subset into a regular space and the


problem is posed whether one can drop CH in that assertion. We answer this question in theaffirmative. Recall that a spaceX is said to beweakly pseudocompactif it is Gδ-dense insome compactification ofX [17]. The following result gives a new characterization of thiskind of spaces and we apply it to embed uncountable discrete sets into Abelian topologicalgroups asr-pseudocompact subsets (see Corollary 3.15 below).

Proposition 3.14. A spaceX is weakly pseudocompact if and only if it can be embeddedinto a topological group as anr-pseudocompact subset.

Proof. Suppose thatX is weakly pseudocompact and letK be a compactification ofXsuch thatX is Gδ-dense inK. It is easy to see thatX is C-compact inK (see also [6,Proposition 2.7(2)]). Consider the free topological groupF(K) on K. SinceX is C-compact inK, it is alsoC-compact inF(K). It follows from Corollary 3.11 thatX isr-pseudocompact inF(X).

Conversely, ifX is an r-pseudocompact subset of a topological groupG, thenXis C-compact inG by Proposition 2.1. According to Corollary 3 of [25],X is Gδ-dense in the compact space clGX whereG is the completion ofG. Thus,X is weaklypseudocompact.2

According to Corollaries 3.8 and 3.9 of [17], a locally compact space is weaklypseudocompact if and only if it is not Lindelöf. This result and Proposition 3.14 implythe following answer to the problem posed in [2].

Corollary 3.15. A discrete space of cardinalityλ> ℵ0 can be embedded in a topologicalgroup as anr-pseudocompact subset if and only ifλ is uncountable.

4. Bounded rectangular subsets of products

In this section, we prove two results about the relative distribution of the closure operatorin the realcompactification of a product of two spaces when one factor is a topologicalgroup (see Theorem 4.7 and Corollary 4.9). Theorem 4.7 is applied to prove the main resultof the article: ifA is aC-compact subset of a topological groupG andB is aC-compactsubset of a spaceY , thenA×B isC-compact inG× Y (Theorem 4.8).

We combine different techniques here. When dealing with a bounded subset of a productof two topological spaces, one of the main tools is a refinement of methods used by Frolíkin [14]. On the other hand, if one factor in the product is a topological group, we applythe methods developed in Sections 2 and 3 to prove the relative distribution law (seeTheorem 4.7). Along this line, Lemma 4.1 below provides a clear argument that explainsthe good behaviour of pseudocompactness and boundedness in the class of topologicalgroups.

Lemma 4.1. Let G be a topological group,{xn: n ∈ ω} a sequence of points inG andlet F be a filter onω. If a ∈ µG anda ∈ clµG{xn: n ∈K} for eachK ∈ F , then for any


sequence{Un: n ∈ ω} of open sets inG such that xn ∈ Un for all n ∈ ω, there exists anelementb ∈G satisfying

(∗) b ∈ clG(⋃{Un: n ∈K}

)for everyK ∈F .

Proof. Denote byG the completion of the groupG. If {Un: n ∈ ω} is a sequence ofopen sets inG with xn ∈ Un, for everyn ∈ ω choose a symmetric subsetOn ∈ NG(e)such that(xn On)∩G⊆Un. Letϕ :µG→ G be a continuous extension of the embeddingi :G ↪→ G. We defineWn = ϕ−1(ϕ(a)On) ∈ NµG(a) for everyn ∈ ω. SinceG is Gδ-dense inµG, we can pick a point, sayb, belonging toG ∩ (⋂n∈ω Wn). We claim thatbsatisfies(∗).

Indeed, letO be a symmetric neighbourhood of the identity inG and letK ∈F , we haveto prove thatO b intersects the set

⋃{Un: n ∈ K}. DefineU = ϕ−1(O ϕ(a)) ∈NµG(a).By the assumption of the lemma, there isn ∈ K such thatxn ∈ U . Sinceϕ|G = idG, itfollows thatxn ∈O ϕ(a) and, therefore,ϕ(a) ∈O xn.

On the other hand,b ∈ G ∩ Wn or, equivalently,b ∈ ϕ(a)On, which implies thatϕ(a) ∈ bOn. Thus,(O xn) ∩ (bOn) 6= ∅. SinceG is dense inG, there is a pointx ∈(O xn)∩ (bOn)∩G. It is easy to verify then that

xn x−1b ∈ (Ob)∩ (xnOn)∩G 6= ∅.

Since(xnOn) ∩G⊆Un andn ∈K, this proves thatb satisfies the condition(∗). 2Corollary 4.2. Lemma4.1remains valid forυG if one additionally assumes that eitherGhas a nonmeasurable cardinality or the set{xn: n ∈ ω} is bounded inG.

Proof. If G has a nonmeasurable cardinality,υG andµG coincide. On the other hand,if P = {xn: n ∈ ω} is bounded inG, thenP is totally bounded with respect to both thebilateral group uniformity and the finest uniformity onG. Thus,P is relatively compact inµG. Since there exists a natural embeddingi :µG→ υG such thati|G = idG, we concludethat the closures ofP in µG andυG coincide. 2

Two lemmas below generalize similar results of Frolík’s [14]. Although our proofs ofLemmas 4.3 and 4.4 exploit the same idea as in [14], for the reader’s convenience weinclude main details of the corresponding reasoning (see also [8,19]).

Lemma 4.3. Let X and Y be Tychonoff spaces and letf ∈ C∗(X × Y ). If L ⊆ A ⊆ X,B ⊆ Y andA×B is bounded inX× Y , then the functionsFL andGL defined by

FL(y)= inf{f (x, y): x ∈ L}, GL(y)= sup

{f (x, y): x ∈ L}

are both continuous onB.

Proof. It suffices to prove the continuity ofFL. SinceFL is the infimum of a family ofcontinuous functions, it is clear that this function is upper-semicontinuous.


Suppose thatFL is not lower-semicontinuous. Then there exist a pointy0 ∈ B andε > 0such that for everyV ∈N (y0) one can findyV ∈ V ∩B with FL(yV ) < FL(y0)− 3ε. Wedefine by induction the sequences of points and open sets

{xn: n ∈ ω} ⊆ L, {yn: n ∈ ω} ⊆ B,{Un: n ∈ ω}, {Vn: n ∈ ω}, {V ′n: n ∈ ω}

which satisfy the following conditions for eachn ∈ ω:(1) Un ∈N (xn), Vn ∈N (yn), V ′n ∈N (y0);(2) osc(f,Un × Vn) < ε and osc(f,Un × V ′n) < ε;(3) f (xn, yn) < FL(y0)− 3ε;(4) clY (Vn+1 ∪ V ′n+1)⊆ V ′n.

Suppose thatxn, yn,Un,Vn and V ′n have already been defined. By the choice ofy0

and ε, there isyn+1 ∈ V ′n such thatFL(yn+1) < FL(y0) − 3ε. There exists a pointxn+1 ∈ L satisfyingf (xn+1, yn+1) < FL(y0) − 3ε. By the continuity off we can findUn+1 ∈N (xn+1), Vn+1 ∈N (yn+1) andV ′n+1 ∈N (y0) such that clY (Vn+1 ∪ V ′n+1)⊆ V ′n,osc(f,Un+1 × Vn+1) < ε and osc(f,Un+1 × V ′n+1) < ε. SinceA × B is bounded inX × Y , the sequence{Un × Vn: n ∈ ω} has a cluster point(u, v) ∈ G × Y . From thisand the continuity off at (u, v) it follows thatf (u, v)6 FL(y0)− 2ε. On the other hand,since clYVn+1 ⊆ V ′n ⊆ clY V ′n ⊆ V ′n−1 for eachn ∈ ω, we conclude that(u, v) is also acluster point of the sequence{Un × V ′n: n ∈ ω}. Again, the continuity off implies thatf (u, v)> FL(y0)− ε. This contradiction completes the proof.2

From now on, for a subsetA of a topological spaceX, we shall denote byA∗ the closureof A in υY .

Lemma 4.4. Under the conditions of Lemma4.3, the functionf admits a continuousextensionf ∗ over(A∗ ×B)∪ (A×B∗).

Proof. By the symmetry argument, it suffices to extendf overA∗ × B. In its turn, thisonly requires to prove that ifp ∈ A∗ \ A and y0 ∈ B, thenf continuously extends to(A×B)∪ {(p, y0)}.

Take a pointy ∈ Y and consider the functionfy ∈ C∗(X) defined byfy(x)= f (x, y)for all x ∈ X. Denote byf υy the continuous extension offy over υX. We now definef ∗(p, y0)= f υy0

(p). Let us verify thatf ∗ is continuous at(p, y0). By the continuity off υy0,

there isU ∈N (p) such that|f ∗(x, y0)−f (p,y0)|< ε for all x ∈ U ∩A. TakeL=U ∩Aand defineFL andGL as in Lemma 4.3. Since the mappingsFL andGL are continuous aty0, there isV ∈N (y0) such that

FL(y0)− ε < FL(y) < FL(y0)+ ε,GL(y0)− ε <GL(y) <GL(y0)+ ε

for all y ∈ V ∩B. If now (x, y) ∈ (U ∩A)× (V ∩B), then


f ∗(p, y0)− 2ε 6 FL(y0)− ε < FL(y)6 f (x, y)6GL(y)<GL(y0)+ ε 6 f ∗(p, y0)+ 2ε,

and hencef ∗ is continuous at the point(p, y0). 2Lemma 4.5. LetA be a bounded subset of a topological groupG and letB be a boundedsubset of a spaceY . If f ∈ C∗(G× Y ), L⊆A andf ∗ is a continuous extension off over(A∗ ×B)∪ (A×B∗) defined in Lemma4.4,then the functions

F ∗L(q)= inf{f ∗(x, q): x ∈ L} and G∗L(q)= sup

{f ∗(x, q): x ∈L}

are continuous onB∗.

Proof. It suffices to prove the continuity of the functionF ∗L onB∗. Suppose thatF ∗L is notlower-semicontinuous at a pointq ∈ B∗ \ B. Then there exists aε > 0 such that for everyV ∈NυY (q), there isyV ∈ V ∩B with F ∗L(yV ) < F ∗L(q)−4ε. As in Lemma 4.3, we definethe sequences{xn: n ∈ ω} ⊆ L, {yn: n ∈ ω} ⊆ B, {f υn : n ∈ ω}, {U ′n: n ∈ ω}, {Vn: n ∈ ω}and {V ′n: n ∈ ω} which satisfy the following properties for eachn ∈ ω:

(1) U ′n ∈NυG(xn), Vn ∈NυY (yn) andV ′n ∈NυY (q);(2) f υn ∈ C∗(υY );(3) osc(f, (U ′n × Vn)∩ (G× Y )) < ε and osc(f υn ,V

′n) < ε;

(4) f (xn, yn) < F ∗L(q)− 4ε;(5) clY (Vn+1 ∪ V ′n+1)⊆ V ′n.Indeed, suppose thatxn, yn, f υn ,U

′n,Vn andV ′n have already been defined. SinceF ∗L is

not lower-semicontinuous atq , there isyn+1 ∈ V ′n ∩ B such thatF ∗L(yn+1) < F∗L(q)− 4ε.

Therefore,f (xn+1, yn+1) < F∗L(q)− 4ε for some pointxn+1 ∈ L. By the continuity off

on G × Y , there areU ′n+1 ∈ NυG(xn+1) andVn+1 ∈ NυY (yn+1) such that clυYVn+1 ⊆V ′n and osc(f, (U ′n+1 × Vn+1) ∩ (G × Y )) < ε. Define a functionfn+1 ∈ C∗(Y ) byfn+1(y) = f (xn+1, y) for all y ∈ Y , and letf υn+1 be its continuous extension overυY .Sincef υn+1 is continuous atq , there existsV ′n+1 ∈NυY (q) such that clυY V ′n+1 ⊆ V ′n andosc(f υn+1,V

′n+1) < ε. This completes the stepn+ 1 of our construction.

SinceB is a bounded subset ofY andyn ∈ Vn ∩ B 6= ∅ for eachn ∈ ω, the sequence{Vn∩Y : n ∈ ω} of open sets inY has a cluster pointy ∈ Y . Letfy be a continuous functiononG defined byfy(x)= f (x, y) for all x ∈G. Again, the continuity offy implies thatfor everyn ∈ ω, there isU ′′n ∈NυG(xn) such thatosc(fy,U ′′n ) < ε. DefineUn =U ′n ∩U ′′n .

ForV ∈NυY (y), defineK(V )= {n ∈ ω: V ∩ Vn 6= ∅}. Clearly,{K(V ): V ∈NυY (y)}is a base of a filterF on ω. SinceA∗ is a compact subset ofυG, there is a pointa ∈⋂K∈F clA∗{xn: n ∈K}. By Lemma 4.1, there exists a pointb ∈G such that

b ∈ clG(⋃{Un: n ∈K}

)for eachK ∈F .

The definition of the pointsy ∈ Y andb ∈ G implies that(b, y) is a cluster point of thesequence{Un × Vn: n ∈ ω}. Hence, by (3), we havef (b, y)6 F ∗L(q)− 3ε.

As in Lemma 4.3, it is clear thaty ∈ V ′n for all n ∈ ω. Again, by (3),f (xn, y)= f υn (y)>F ∗L(q) − ε. SinceUn ⊆ U ′′n and osc(fy,U ′′n ) < ε, we havef (x, y) > F ∗L(q) − 2ε for


all x ∈ Un. The fact thaty is a cluster point of the sequence{Un: n ∈ ω} implies thatf (b, y)> F ∗L(q)− 2ε. This contradiction completes the proof.2Lemma 4.6. LetA be a bounded subset of a topological groupG and letB be a boundedsubset of a spaceY . Then every functionf ∈ C∗(G× Y ) admits a continuous extensionoverA∗ ×B∗.

Proof. First, we shall prove thatf admits a continuous extension overβA × B∗. Weproceed as in the proof of Lemma 4.4.

By Theorem 1.2, the setA× B is bounded inG× Y . Making use of Lemma 4.4, weextendf to a continuous functionf ∗ overA× B∗. Now, it suffices to prove that ifp ∈βA \A andq ∈B∗, thenf ∗ admits a continuous extensionf β over(A×B∗)∪ {(p, q)}.

Pick a pointq ∈ B∗ and consider the functionfq ∈ C∗(A) defined byfq(x)= f ∗(x, q)for all x ∈ A. Denote byf βq a continuous extension offq over βA. We now define

f β(p,q)= f βq (p). Let us verify thatf β is continuous at(p, q). By the continuity off βq ,there isU ∈N (p) such that|f β(x, q)− f ∗(p, q)|< ε for all x ∈ U ∩A. PutL=U ∩Aand defineF ∗L andG∗L as in the Lemma 4.5. Since the functionsF ∗L andG∗L are bothcontinuous atq , there isV ∈N (q) such that

F ∗L(q)− ε < F ∗L(y) < F ∗L(q)+ ε,G∗L(q)− ε <G∗L(y) <G∗L(q)+ ε

for eachy ∈ V ∩B∗. If (x, y) ∈ (U ∩A)× (V ∩B∗), then

f β(p,q)− 2ε6 F ∗L(q)− ε < F ∗L(y)6 f (x, y)6G∗L(y)<G∗L(q)+ ε 6 f β(p,q)+ 2ε,

which implies the continuity off β at (p, q).So, we have proved thatf extends to a continuous functionf β ∈ C(βA × B∗).

Let η :B∗ → C∗u(A) be the evaluation mapping defined byη(q)(x) = f β(x, q) for allq ∈ B∗ andx ∈A, whereC∗u(A) stands forC∗(A) endowed with the topology of uniformconvergence. From the continuity off β it follows that η is continuous onB∗. Notethat η(B) ⊆ C(G)|A, i.e., every function inη(B) is the restriction toA of a continuousfunction inC(G), and hence inC(υG). SinceA∗ ⊆ υG, every function inη(B) is, infact, a restriction toA of a continuous function inC(A∗). Let r :Cu(A∗)→ C∗u(A) bethe restriction mapping defined byr(g) = g|A for everyg ∈ Cu(A∗). It is easy to seethatr is a homeomorphic embedding andr(Cu(A∗)) is a closed subspace ofC∗u(A) [20].Summarizing, we have

η(B)⊆ r(Cu(A∗))⊆ C∗u(A).Sinceη(B) is dense inη(B∗) andr(Cu(A∗)) is closed inC∗u(A), we conclude thatη(B∗)⊆r(Cu(A

∗)). Sincer is a homeomorphic embedding, the mappingϕ = r−1 ◦ η :B∗ →Cu(A

∗) is continuous.It remains to define a continuous functionf∗ onA∗ ×B∗ by f∗(x, y)= ϕ(y)(x) for all

x ∈A∗ andy ∈ B∗. It is clear thatf∗|A×B = f . 2


The following two theorems are the main results of this section. They clarify the reasonfor the productivity of boundedness andC-compactness in topological groups and showthat everyC-compact subset of a topological groupG is stronglyC-compact inG.

Theorem 4.7. LetA andB be bounded subsets of a topological groupG and a spaceY ,respectively. Then the following relative distribution law is valid:

clυ(G×Y )(A×B)= (A×B)∗ ∼=A∗ ×B∗ = clυGA× clυY B.

Proof. Letψ :υ(G×Y )→ υG×υY be a continuous extension of the identity embeddingofG×Y intoυG×υY . Denote byϕ the restriction ofψ to (A×B)∗. Both the domain andimage ofϕ are compact spaces, and henceϕ((A×B)∗)=A∗×B∗. If ϕ were not injective,there would exist two distinct pointsp andq in (A× B)∗ such thatϕ(p) = ϕ(q). Thereexists a continuous functionf υ on υ(G× Y ) such thatf υ(p) 6= f υ(q). It is clear thatf = f υ |G×Y does not admit a continuous extension overA∗ × B∗, and this contradictsLemma 4.6. Thus,ϕ is continuous bijection between compact spaces and hence is ahomeomorphism. This completes the proof.2

Recall that the productA × B of C-compact subsetsA andB of spacesX and Yrespectively need not beC-compact inX × Y (takeA = X andB = Y , whereX andY are pseudocompact spaces constructed by Novak [30]). If, however, one of the factorsX, Y is a topological group, the situation changes completely as the following theoremshows.

Theorem 4.8. If A is aC-compact subset of a topological groupG andB is aC-compactsubset of a spaceY , then the productA×B isC-compact inG× Y .

Proof. Let f be a continuous real-valued function onG× Y . Denote byf υ a continuousextension off overυ(G×Y ). By Theorem 4.7, the closure(A×B)∗ ofA×B in υ(G×Y )is naturally homeomorphic to the productA∗ × B∗ whereA∗ = clυGA andB∗ = clυY B.SinceB is C-compact inY , it easily follows thatB isGδ-dense inB∗ (a similar fact wasproved in Lemma 5.5 of [18] for the closure ofB in µY , but the proof given there worksfor υY as well). Analogously,A is Gδ-dense inA∗. Therefore,A × B is Gδ-dense in(A×B)∗ ∼=A∗ ×B∗. Since(A×B)∗ is compact and the real lineR is first countable, weconclude that

f (A×B)= f υ(A×B)= f υ((A×B)∗)is a compact subset ofR. 2

The following corollary to Theorem 4.7 implies the Comfort–Ross theorem on theproductivity of pseudocompactness in topological groups [10] and Tkacenko’s theoremabout boundedness of a product of bounded subsets of topological groups [32].


Corollary 4.9. LetG=∏i∈I Gi be a Cartesian product of topological groupsGi . If Aiis a bounded subset ofGi andA∗i = clυGiAi for eachi ∈ I , then

clυG

(∏i∈IAi

)∼=∏i∈IA∗i .

Proof. The following simple observation enable us to simplify the proof. LetB be abounded subset of a topological groupK andL the subgroup ofK generated byB. Denoteby ψ :υL→ υK a continuous extension of the identity embeddingL ↪→ K. Then therestriction ofψ to the closureB∗ of B in υL is a homeomorphism ofB∗ onto the closureof B in υK. Indeed, by Lemma 3.6, the groupL isR-factorizable, and hence Theorem 3.3implies thatυL is a topological group containing a dense subgroupL. Therefore,υL isa topological subgroup of the completionL which in its turn is a subgroup ofK . Theconclusion now is immediate.

Applying the above fact, we can assume without loss of generality that for everyi ∈ I ,Ai generates the groupGi . Let f :G→ R be a continuous function. By Theorem 1.1,the setA =∏i∈I Ai is bounded inG. Therefore, Corollary 2.29 of [32] implies that therestriction off toA is uniformly continuous with respect to the bilateral group uniformityV of G. For eachi ∈ I , the Hewitt extensionυGi of Gi is a topological group containingGi as a dense subgroup (Lemma 3.6 and Theorem 3.3), soG is a dense subgroup ofthe Cartesian productGυ = ∏i∈I υGi . It is clear that the restriction of the bilateraluniformityVυ of the groupGυ toG coincides withV . Therefore, the functionf |A extendscontinuously over the closureAυ of A in Gυ . Note thatAυ =∏i∈I A∗i .

Let υG be the Hewitt realcompactification ofG and ϕ :υG → Gυ a continuousextension of the identity embeddingi :G ↪→ Gυ . The argument in the proof ofTheorem 4.7 enables us to conclude that the restriction ofϕ to the closureA∗ of A inυG is a one-to-one continuous mapping of the compact setA∗ ontoAυ , and henceϕ|A∗ isa homeomorphism. This completes the proof.2Added in proof. Theorem 3.3 remains valid for the Dieudonné completionµG of anR-factorizable groupG. The proof of this result makes use of the facts that anR-factorizablegroupG satisfiesc(G)6 c and every metrizable space of cardinality6 c is realcompact.For details, see Theorem 5.23 of [35].

Acknowledgement

The authors thank the referee for his/her helpful suggestions and comments.

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Bounded sets in spaces and topological groups