Equicontinuity and balanced topological groups

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Topology and its Applications 135 (2004) 63–71

www.elsevier.com/locate/topo

Equicontinuity and balanced topological groups

J.P. Troallic

UMR CNRS 6085, Faculté des Sciences et Techniques, Université du Havre, 25, rue Philippe LeboF-76600 Le Havre, France

Received 5 July 2002; received in revised form 30 April 2003

Abstract

It was proved by V.G. Pestov in 1988 that a locally compact groupG is balanced if and onlyif any countable subset ofG is thin in G. This unexpected result was obtained by using a nelementary transfinite induction involving properties of infinite ordinals. In the present workresult is reconsidered in a more general context by using an approach which is comparable, ito Pestov’s, but uses a notably simplified technique. LetX be a topological space,Y a uniform spaceandH a set of continuous mappings ofX intoY . First, new conditions concerningX are given underwhich H is equicontinuous provided its countable subsets are. Next,X andY are supposed equato a topological group equipped with its right uniform structure, and the setH taken into accounis the group of all its inner automorphisms. We then obtain theorems such as the followingincludes, as a special case, Pestov’s result: LetG be a topological group; let us suppose thatspaceG is strongly functionally generated by the set of all its subspaces of countableo-tightness;thenG is balanced if and only if any right uniformly discrete countable subset ofG is thin inG. Asan application, it is proved that ifG satisfies the above hypothesis and is non-Archimedean, thGis balanced if and only ifG is strongly functionally balanced. 2003 Elsevier B.V. All rights reserved.

MSC:primary 54E15, 22A05; secondary 54D99

Keywords:Countableo-tightness; Countable cellularity; Uniform space; Criteria for equicontinuity; Topologgroup; Balanced topological group; Strongly functionally balanced topological group; Thin subset; Rightuniformly discrete subset; Non-Archimedean topological group

E-mail address:[email protected] (J.P. Troallic).

0166-8641/$ – see front matter 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0166-8641(03)00135-4

64 J.P. Troallic / Topology and its Applications 135 (2004) 63–71

1. Introduction

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A subsetA of a topological groupG is said to be thin inG if for every neighborhoodV of the identitye in G, the set

⋂a∈A a−1V a is a neighborhood ofe in G. If G is thin in

itself, or equivalently, if the left and right uniform structures onG are equal, thenG is saidto be balanced.

In 1976, it was asked by Itzkowitz [7] whether or not the equality of left and runiform structures on a locally compact groupG could be decided by countable subsof G. In 1988, the question was answered in the affirmative by Pestov [13]:G is balancedif and only if any countable subsetA of G is thin in G. This surprising result waobtained by using a transfinite induction involving some non-elementary propertinfinite ordinals. Recall that shortly after, and independently, another interesting pmaking use of Lie groups and of an ordinary induction argument, was given by Itzket al. [8], Itzkowitz [9].

In the present work, the above problem is reconsidered in a more general contusing an approach which is comparable, in spirit, to Pestov’s, but uses a notably simtechnique.

Let X be a topological space,Y a uniform space andH a set of continuous mappingof X into Y . In Section 2, new conditions concerningX are given under whichH isequicontinuous provided its countable subsets are. Recall that in [18] a similar resuproved forX belonging to the class of quasi-k-spaces.

In Section 3, the spacesX andY of the second section are particularised. More precisX = Y =G,G being a topological group equipped with its right (or left) uniform structuOn the other hand, the setH taken into account is the group of all inner automorphisof G. We then obtain theorems such as the following which includes, as a speciaPestov’s result: LetG be a topological group; let us suppose that the spaceG is stronglyfunctionally generated by the set of all its subspaces of countableo-tightness; thenG isbalanced if and only if any countable subsetA ofG is thin inG. Besides extending Pestovresult, it also gives a positive answer to Pestov’s problem [13] as to whether a topolgroup of countable tightness is balanced if all its countable subsets are thin.

In fact, in the above statement, it suffices to consider countable subsets ofG which areuniformly discrete relative to the right uniform structure. In Section 4, this improvemeused to prove that ifG satisfies again the above hypothesis and is non-ArchimedeanG is balanced if and only if every right uniformly continuous real-valued function onG isleft uniformly continuous.

All topological spaces considered in this paper are assumed to be Hausdorff. Noand terminology are as in [2,3].

2. Tests for equicontinuity

For all topological spacesS andT , the set of all continuous mappings ofS into T isdenoted byC(S,T ). LetX be a topological space, letY be a uniform space and letH bea subset ofC(X,Y ). In this section, it is shown that if the spaceX is strongly functionallygenerated by the collection of all its subspaces of countableo-tightness (respectively o

J.P. Troallic / Topology and its Applications 135 (2004) 63–71 65

countable tightness, respectively of countable cellularity), then the setH is equicontinuousll thatf

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provided its countable subsets are. (Cf. Definitions 2.1 and 2.6 below.) Let us recain [18], similar criteria were obtained, by other methods, forX belonging to the class oq-spaces or to the class of quasi-k-spaces.

Definition 2.1.

(1) LetX be a topological space. According to Tkacenko [15], the spaceX is of countableo-tightness(notation:ot (X)� ω) if for every x ∈ X and every collectionA of opensubsets ofX such thatx ∈ ⋃

A, there exists a countable subcollectionB of A suchthatx ∈ ⋃

B. Recall that the spaceX is of countabletightness(notation:t (X)� ω) iffor everyA⊂X and everyx ∈ A, there is a countable subsetB of A such thatx ∈ B(cf., for example, [3]). Obviously, ifX is of countable tightness, thenX is of countableo-tightness.

(2) Let A be a collection of subspaces of a topological spaceX. The spaceX is saidto bestrongly functionally generatedby the collectionA if the following conditionis satisfied: for every discontinuous functionf of X into R (or equivalently, into anyTychonoff spaceY ), there existsA ∈ A such that the restrictionf |A is discontinuousIf X is strongly functionally generated by the collection of all its compact subspX is said to be akR-space.

To obtain the countable criterion for equicontinuity Theorem 2.3 below, the followelementary lemma will be extremely useful.

Lemma 2.2. LetX be a topological space,Y a uniform space andH a set of mappingof X into Y . Let A be a collection of subspaces ofX. Let us suppose thatX is stronglyfunctionally generated byA. ThenH is equicontinuous if and only if for everyA ∈ A, thesetH |A of restrictions toA of mappings ofH is equicontinuous.

Theorem 2.3. LetX be a topological space,Y a uniform space andH a subset ofC(X,Y ).Let us suppose that the spaceX is strongly functionally generated by the set of allsubspaces of countableo-tightness. Then the following statements are equivalent:

(1) H is equicontinuous.(2) Each countable subset ofH is equicontinuous.

Proof. It is obvious that (1) implies (2) (without any hypothesis on the spaceX); let usshow that (2) implies (1).

(i) Suppose first that the spaceX is of countableo-tightness. Letx0 be a point ofX. Assume thatH is not equicontinuous atx0; then there is an entourageV of Y suchthat

⋂h∈H h−1(V [h(x0)]) is not a neighborhood ofx0 in X, or equivalently, such tha

x0 ∈ ⋃h∈HUh, whereUh denotesX \ h−1(V [h(x0)]) (h ∈ H). We chooseV closed in

X ×X, which is possible since the closed entourages ofX in X ×X form a fundamentasystem of entourages ofX; sinceh is continuous, the setUh is then an open subsetX (h ∈ H). The spaceX being of countableo-tightness, there exists a countable sub


L of H such thatx0 ∈ ⋃h∈LUh. Sincex0 ∈ ⋃

h∈LUh, the set⋂h∈L h−1(V [h(x0)]) is

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not a neighborhood ofx0 in X and consequently, the countable subsetL of H is notequicontinuous atx0 . Thus, condition (2) is not satisfied.

(ii) Suppose now thatX is strongly functionally generated by the setA of all itssubspaces of countableo-tightness and that condition (2) holds. By (i), the setH | A ofrestrictions toA of mappings ofH is equicontinuous for everyA ∈A. This implies thatHis equicontinuous by Lemma 2.2.✷

Since any space of countable tightness is of countableo-tightness, it is immediatthat any spaceX strongly functionally generated by{A ⊂ X | t (A) � ω} is stronglyfunctionally generated by{A ⊂ X | ot (A) � ω}. Consequently, Theorem 2.3 admits tfollowing corollary.

Corollary 2.4. LetX be a topological space,Y a uniform space andH a subset ofC(X,Y ).Let us suppose thatX is strongly functionally generated by the set of all its subspacecountable tightness. Then the following statements are equivalent:


Remark 2.5. It is not difficult to verify that the conclusion of the above statemCorollary 2.4 remains valid if condition (2) is replaced by the following condition: For ecountable subsetD of X and each countable subsetL of H, the setL |D of restrictions toD of mappings ofL is equicontinuous. Let us point out that Theorem 2.3 does not asuch an improvement.

Definition 2.6. Let X be a topological space. Recall that the spaceX is of countablecellularity (notation:c(X) � ω), if any collection of pairwise disjoint open sets inX iscountable (cf., for example, [3]).

As observed in [15], if the spaceX is of countable cellularity, then theo-tightness ofXis countable. It follows from the following useful lemma which is proved for the sakcompleteness.

Lemma 2.7. LetX be a topological space of countable cellularity and letA be a collectionof open subsets ofX. Then there is a countable subcollectionB ofA such that

⋃A⊂ ⋃

B.

Proof. Let Λ be the set of all collectionsC of pairwise disjoint open sets inX such thateachC ∈ C is contained in some element ofA. OrderΛ by inclusion; thenΛ is inductive,and according to Zorn’s lemma there is a maximal collectionM in Λ. Sincec(X) � ω,M is countable; moreover by the maximality ofM,

⋃A ⊂ ⋃

M. For eachM ∈ M, letAM ∈ A such thatM ⊂ AM ; thenB = {AM |M ∈ M} is a countable subcollection ofAsuch that

⋃A⊂ ⋃

B. ✷


Since any space of countable cellularity is of countableo-tightness, it is immediate

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that any spaceX strongly functionally generated by{A ⊂ X | c(A) � ω} is stronglyfunctionally generated by{A ⊂ X | ot (A) � ω}. Consequently, Theorem 2.3 above aadmits the following important particulary case.

Corollary 2.8. LetX be a topological space,Y a uniform space andH a subset ofC(X,Y ).Let us suppose thatX is strongly functionally generated by the set of all its subspacecountable cellularity. Then the following statements are equivalent:


3. Balanced topological groups

Recall first some definitions.

Definition 3.1. LetG be a topological group.

(1) Following [1], we say thatG is a balancedtopological group if the left and righuniform structures onG coincide.

(2) A subsetA ofG is thin inG if for every neighborhoodV of the identitye inG, the set⋂a∈A a−1V a is a neighborhood ofe in G [17]. It is well known, and easy to prove

thatG is balanced if and only ifG is thin in itself [6].(3) A subsetA of G is said to beright uniformly discreteif it is uniformly discrete

with respect to the right uniform structure. Obviously,A is right uniformly discreteif and only if there exists a neighborhoodV of the identitye in G such that for alla, b ∈ A with a = b, the relationV a ∩ V b = ∅ holds. A similar definition holds foleft uniformly discretesubsets ofG.

If G is a locally compact topological group, then by Pestov’s theorem [13],G isbalanced if and only if any countable subsetA of G is thin inG. In the present sectionto obtain improvements of this theorem, we use the results of Section 2. Theorebelow contains, as a special case, Pestov’s theorem; to prove that, the main argumeis the following well-known property established in [16] by Tkacenko:Everyσ -compacttopological group has countable cellularity.

In fact, in Pestov’s statement, it suffices to consider countable subsetsA of G whichare uniformly discrete relative to the right uniform structure. It was proved in [8,9].condition is also taken into account in our context.

Theorem 3.2. LetG be a topological group. Let us suppose that the spaceG is stronglyfunctionally generated by the collection of all its subspaces of countableo-tightness. Thenthe following statements are equivalent:


(1) G is a balanced topological group.

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(2) Each countable subset ofG is thin inG.(3) Each right uniformly discrete countable subset ofG is thin inG.

Proof. For everyg ∈ G, let ιg :G→ G be the inner automorphism defined byι(x) =gxg−1 (x ∈ G). Let us equipG with its right (or left) uniform structure; then it is cleathat a subsetA of G is thin in G if and only if the subset{ιg | g ∈ A} of C(G,G) isequicontinuous; moreover, as recalled in 3.1,G is balanced if and only ifG is thin in itself.It follows consequently from Theorem 2.3 that conditions (1) and (2) are equivalent.

It is obvious that (2) implies (3). LetA be a countable subset ofG and letV be aneighborhood of the identitye in G. Suppose that condition (3) holds; let us show thatset

⋂a∈A a−1V a is a neighborhood ofe inG, which establishes (2). LetW be a symmetric

neighborhood ofe in G such thatW5 ⊂ V , and letE be the set of all subsetsE of A suchthat for all a, b ∈ E with a = b the relationWa ∩Wb = ∅ holds. OrderE by inclusion;thenE is inductive, and according to Zorn’s lemma there exists a maximal elementM in E .SinceM is a right uniformly discrete countable subset ofG,M is thin inG by hypothesisconsequently,U = ⋂

m∈M m−1Wm is a neighborhood ofe in G. Let us show thatU iscontained in

⋂a∈A a−1V a, which will complete the proof.

Let a ∈ A. If a ∈M thenU ⊂ a−1Wa, and sinceW ⊂ W5 ⊂ V , the inclusionU ⊂a−1V a holds. If a ∈ A \M then the maximality ofM in E implies thatWa ∩WM = ∅;let x, y ∈W andm ∈M such thatxa = ym; then

aUa−1 = x−1y(mUm−1)y−1x ⊂W5 ⊂ V,

which implies againU ⊂ a−1V a. ✷Remark 3.3. Zorn’s lemma is used in the proof of Theorem 3.2 in comparable condito those where it is used in [11] to establish that any locally connected topologicalin which every left uniformly discrete subset is left neutral is balanced.

Since every topological spaceX strongly functionally generated by the collecti{A⊂ X | t (A)� ω} (respectively{A⊂X | c(A)� ω}) is strongly functionally generateby the collection{A⊂X | ot (A)� ω}, the following corollary of Theorem 3.2 holds.

Corollary 3.4. LetG be a topological group. Let us suppose that at least one of thetwo conditions is satisfied:

(a) The spaceG is strongly functionally generated by{A⊂G | t (A)� ω}.(b) The spaceG is strongly functionally generated by{A⊂G | c(A)� ω}.

Then the following statements are equivalent:

(1) G is a balanced topological group.(2) Each countable subset ofG is thin inG.(3) Each right uniformly discrete countable subset ofG is thin inG.


In order to obtain Corollary 3.6, let us state a lemma. This lemma is an easy consequence

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Lemma 3.5. LetG be a topological group that is akR-space. Then the spaceG is stronglyfunctionally generated by the set of all its subspaces of countable cellularity.

Proof. Let K be the family of all compact subspaces ofG, and letA be the collectionof all subspacesA of X such thatc(A) � ω. For everyK ∈ K there existsA ∈ A suchthatK ⊂ A. To see that, it suffices to chooseA equal to the subgroup ofG generated byK since by Tkacenko’s theorem [16], every compactly generated topological groupcountable cellularity. Consequently,G is strongly functionally generated byA since, byhypothesis,G is strongly functionally generated byK. ✷

The following corollary is just a consequence of Corollary 3.4 and Lemma 3.5. Severy locally compact topological space is akR-space, it contains Pestov’s result recalat the beginning of this section as a particular case.

Corollary 3.6. Let G be a topological group that is akR-space. Then the followinstatements are equivalent:

(1) G is a balanced topological group.(2) Each countable subset ofG is thin inG.(3) Each right uniformly discrete countable subset ofG is thin inG.

Remark 3.7. Recall that in [18] a similar result to Corollary 3.6 was proved forGbelonging to the class of quasi-k-spaces. Previously, the particular case of CorollarywhenG is almost metrizable was established by Protasov in [14]. The methods ofused in [14], [18], and here are very different.

4. Strongly functionally balanced topological groups

Definition 4.1. Following [10,14], we say that a topological groupG is strongly func-tionally balancedif every right uniformly continuous real-valued function onG is leftuniformly continuous (or equivalently, if every left uniformly continuous real-valfunction onG is right uniformly continuous).

If a topological groupG is balanced, then it is obviously strongly functionally balancIt is still an open question, formulated in [9] by Itzkowitz, whether or not the convis true. Nevertheless, it is known that the answer is positive ifG is locally compact[12,8,9,4], almost metrizable [14], a quasi-k-space [18], locally connected [11]. Morecently, Hernández [5] has established that the answer is also positive forω-boundednon-Archimedean topological groups. In this section, as an application of Theoremwe obtain an analogous result for non-Archimedean topological groups belongingclass considered in Theorem 3.2.


Definition 4.2. A topological groupG is said to benon-Archimedeanif the open subgroups

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of G form a fundamental system of neighborhoods of the identitye.

Theorem 4.3. Let G be a non-Archimedean topological group. Let us suppose thaspaceG is strongly functionally generated by the set of all its subspaces of couno-tightness. Then the following statements are equivalent:

(1) G is a balanced topological group.(2) G is a strongly functionally balanced topological group.

Proof. It is clear that (1) implies (2) (without any hypothesis on the topological groupG);let us show that (2) implies (1). Let us consider a right uniformly discrete countable sA of G; by Theorem 3.2, it suffices to establish thatA is thin in G. If A is finite, Ais obviously thin inG. Let us supposeA infinite. SinceA is right uniformly discretethere is a neighborhoodV of the identitye in G such thatV a ∩ V b = ∅ for all distincta, b ∈A. LetL be an open subgroup ofG such thatL⊂ V ; let us show that

⋂a∈A a−1La

is a neighborhood ofe in G; the set of all open subgroups ofG contained inV beinga fundamental system of neighborhoods ofe in G, it will follow that A is thin in G.If a, b ∈ A and if a = b, then the right cosetsLa andLb of L in G are disjoint. Letφ :A→ N

∗ a bijection,N∗ being the set of all integers> 0. We definef :G→ R inthe following way: letx ∈ G; if there isa ∈ A such thatx ∈ La, thenf (x) = φ(a); ifx ∈G\LA, thenf (x)= 0. For everyx, y ∈G such thatxy−1 belongs to the neighborhooL of e inG, the equalityf (x)= f (y) holds; consequentlyf is right uniformly continuousBy hypothesisf is left uniformly continuous, hence there exists a symmetric neighborU of e in G such that|f (x)− f (y)|< 1 for all x, y ∈G such thatx−1y ∈ U . To achievethe proof, it suffices to verify thatU ⊂ ⋂

a∈A a−1La. Let u ∈ U anda ∈ A; the equality(au)−1a = u−1 shows that(au)−1a ∈ U which implies that|f (au)−f (a)|< 1. It followsfrom this inequality and from the definition off thatau ∈ La. HenceU ⊂ a−1La for alla ∈A. ✷

Obviously, in the same way that Theorem 3.2 admits Corollary 3.4 as a corollartwo following particular cases of Theorem 4.3 hold.

Corollary 4.4. LetG be a non-Archimedean topological group. Let us suppose that atone of the next two conditions is satisfied:

(a) The spaceG is strongly functionally generated by{A⊂X | t (A)� ω}.(b) The spaceG is strongly functionally generated by{A⊂X | c(A)� ω}.

Then the following statements are equivalent:

(1) G is a balanced topological group.(2) G is a strongly functionally balanced topological group.


Remark 4.5. Following again [10,14], we say that a topological groupG is functionally

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balanced if each boundedright uniformly continuous real-valued function onG isleft uniformly continuous (or equivalently, if eachboundedleft uniformly continuousreal-valued function onG is right uniformly continuous). Obviously, ifG is stronglyfunctionally balanced, thenG is functionally balanced; it is a question raised by Itzkowin [10] whether or not the converse holds. In fact, we do not know if the above stateTheorem 4.3 (or even, its corollary) remains true when the condition “strongly functiobalanced” is replaced by the condition “functionally balanced”.

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Equicontinuity and balanced topological groups