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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 29 - 43

Some properties of o-bounded and strictlyo-bounded groupsC. Hern�andez, D. Robbie, M. Tkachenko�Abstract. We continue the study of (strictly) o-boundedtopological groups initiated by the �rst listed author and solvetwo problems posed earlier. It is shown here that the product ofa Comfort-like topological group by a (strictly) o-bounded groupis (strictly) o-bounded. Some non-trivial examples of strictly o-bounded free topological groups are given. We also show thato-boundedness is not productive, and strict o-boundedness can-not be characterized by means of second countable continuoushomomorphic images.2000 AMS Classi�cation: Primary 54H11, 22A05; Secondary 22D05, 54C50Keywords: o-bounded group, strictly o-bounded group, @0-bounded group,OF-undetermined group, Comfort-like group1. IntroductionThe class of �-compact topological groups has many nice properties. Forexample, every �-compact group is countably cellular [11] and perfectly �-normal [13, 15]. The subgroups of �-compact groups inherit these properties,but clearly need not be �-compact. The notions of o-boundedness and stricto-boundedness introduced by O. Okunev and M. Tkachenko respectively, wereconsidered in [9]. The idea was to �nd a wider class of topological groups asclose to the class of �-compact groups as possible which is additionally closedunder taking subgroups. Let us recall the corresponding de�nitions.A topological group G is called o-bounded if for every sequence fUn : n 2 Ngof open neighborhoods of the neutral element in G, there exists a sequencefFn : n 2 Ng of �nite subsets of G such that G = Sn2N Fn � Un. It is clearthat all �-compact groups as well as their subgroups are o-bounded. In a sense,o-bounded groups have to be small: the group R! fails to be o-bounded [9,�The research is partially supported by Consejo Nacional de Ciencias y Tecnolog��a (CONA-CyT) of Mexico, grant no. 400200-5-28411E, and grant for Special Studies Program (Long)The University of Melbourne (Science Faculty).



30 C. Hern�andez, D. Robbie, M. TkachenkoExample 2.6]. The class of o-bounded groups has good categorical properties:all subgroups and all continuous homomorphic images of an o-bounded groupare o-bounded [9]. It was not known, however, whether this class was �nitelyproductive [9, Problem 5.2]. We show in Example 2.12 that there exists asecond countable o-bounded group G whose square is not o-bounded. Actually,the group G �rst appeared in [9, Example 6.1] in order to distinguish the classesof o-bounded and strictly o-bounded groups. However, the properties of thisgroup were not completely exhausted there. As is shown in [4], the group G isadditionally analytic, that is, G is a continuous image of a separable completemetric space.To de�ne strictly o-bounded groups, we need to describe the OF-game (see[9] or [14]). Suppose that G is a topological group and that two players, say Iand II, play the following game. Player I chooses an open neighborhood U1 ofthe identity in G, and player II responds choosing a �nite subset F1 of G. Inthe second turn, player I chooses another neighborhood U2 of the identity inG and player II chooses a �nite subset F2 of G. The game continues this wayuntil we have the sequences fUn : n 2 Ng and fFn : n 2 Ng. Player II winsif G = S1n=1 Fn � Un. Otherwise, player I wins. The group G is called strictlyo-bounded if player II has a winning strategy in the OF-game on G. It is easy tosee that �-compact groups are strictly o-bounded and every strictly o-boundedgroup is o-bounded. As we mentioned above, o-bounded groups need not bestrictly o-bounded. In addition, there are lots of strictly o-bounded groupsthat are neither �-compact nor isomorphic to subgroups of �-compact groups[9, Example 3.1]. However, an o-bounded continuous homomorphic image of aWeil-complete group is �-bounded, hence strictly o-bounded [3]. All this makesthe problem of studying the properties of these two classes of topological groupsfairly interesting.The class of o-bounded groups is not productive in view of Example 2.12.However, we have no examples of strictly o-bounded groups G and H such thatthe product G � H is not strictly o-bounded (see Problem 4.1). On the otherhand, it was known that a product of an o-bounded group by a �-compact groupwas o-bounded [9, Theorem 5.3], and a similar result for strictly o-boundedgroups was recently proved by Jian He (see Theorem 2.7) who in fact hasproved the result with `�-bounded' instead of `�-compact' and by a methodthat extends the o-bounded result as well. It turns out that there are manytopological groups G (far from being �-compact) with the property that theproduct G �H is (strictly) o-bounded for every (strictly) o-bounded group H.Let G be a �-product of countable discrete groups endowed with the @0-boxtopology. We shall call any subgroup of such a group G a Comfort-like group.(It was W. Comfort who proved that every �-product of countable discretespaces with the @0-box topology inherited from the whole product is Lindel�of,see [5]). We prove in Section 2 that multiplication by a Comfort-like groupG does not destroy (strict) o-boundedness: the product G � H is (strictly) o-bounded for every (strictly) o-bounded group H. It is also shown that thefree topological group F(X) is strictly o-bounded whenever X is the one-point



Some properties of o-bounded and strictly o-bounded groups 31Lindel�o�cation of any uncountable discrete space (Theorem 2.8). In fact, theproduct F(X) � H is strictly o-bounded for every strictly o-bounded group H(see Theorem 2.11).It is clear that every o-bounded group is @0-bounded in the sense of [6],that is, it can be covered by countably many translates of any neighborhoodof the identity. By Theorem 4.1 of [9], if G is @0-bounded and all secondcountable continuous homomorphic images of G are o-bounded, then G itselfis o-bounded. In Section 3 we use � to construct an o-bounded group G whosesecond countable continuous homomorphic images are countable (hence strictlyo-bounded), but G itself is not strictly o-bounded. Therefore, the class ofstrictly o-bounded groups is considerably more complicated than that of o-bounded groups. In other words, strict o-boundedness is not re
ected in theclass of second countable groups.The group G in Theorem 3.1 has another interesting feature. Let us call atopological group H OF-undetermined if neither player I nor player II has awinning strategy in the OF-game in H. It was an open problem whether thereexist OF-undetermined groups. It turns out that the group G in Example3.1 is OF-undetermined. We do not know, however, if such a group can beconstructed in ZFC. Another problem is considered by T. Banakh in [4]: Doesthere exist a metrizable OF-undetermined group? He shows that such groupsexist under Martin's Axiom and have necessarily to be second countable.1.1. Notation and terminology. We denote by N the positive integers, by Zthe additive group of integers, and by R the group of reals. A topological groupG is called @0-bounded [6] if countably many translates of every neighborhoodof the identity in G cover the group G. By a result of [6], G is @0-bounded ifand only if it is topologically isomorphic to a subgroup of a direct product ofsecond countable topological groups. This class of groups is closed under takingdirect products, subgroups and continuous homomorphic images.We say that H is a P-group if the intersection of any countable family of opensets in H is open. Every topological group H admits a �ner group topologythat makes it a P-group: a base of such a topology consists of all G�-subsets ofH.If X is a subset of a group G, we use hXi to denote the subgroup of Ggenerated by X. Finally, the families of all non-empty �nite and countablesubsets of a set A will be denoted by [A]<! and [A]�!, respectively.2. Productive properties of o-bounded groupsHere we introduce the class of Comfort-like topological groups and show thatthe product G�H is (strictly) o-bounded whenever G is a Comfort-like groupand H is (stricly) o-bounded. We start with a simple but useful lemma.Lemma 2.1. Suppose that G, H and K are groups that ': G ! H and : G ! K are homomorphisms such that ker � ker'. Then there existsa homomorphism f : K ! H such that ' = f �  . If in addition, G, H and Kare topological groups, ' and  are continuous, and for each neighborhood U



32 C. Hern�andez, D. Robbie, M. Tkachenkoof the identity eH in H there exists a neighborhood V of the identity eK in Ksuch that  �1(V ) � '�1(U), then f is continuous.Proof. The algebraic part of the lemma is well known. Let us verify the conti-nuity of f in the second part of the lemma. Suppose that U is a neighborhoodof eH in H. By our assumption, there exists a neighborhood V of eK in K suchthat W =  �1(V ) � '�1(U). Then f(V ) = '(W) � U, that is, f is continuousat the identity of K. Therefore, f is continuous. �Our second auxiliary result concerns continuous homomorphic images of @0-bounded P-groups.Lemma 2.2. Let ': G ! H be a continuous homomorphism of an @0-boundedP-group G to a topological group H of countable pseudocharacter. Then theimage '(G) is countable.Proof. Let fUn : n 2 Ng be a countable pseudobase at the identity eH of H.Since G is a P-group, the kernel N = ker' = Tn2N '�1(Un) is a normal opensubgroup of G. Let  : G ! G=N be the quotient homomorphism. By Lemma2.1, there exists a homomorphism f : G=N ! H such that ' = f �  . SinceG is @0-bounded, the quotient group G=N is countable, and hence j'(G)j =jf( (G))j � jG=Nj � !. �Let � = Qi2I Gi be the direct product of topological groups Gi and let e bethe identity of �. For every x 2 �, put supp(x) = fi 2 I : x(i) 6= eig, where eiis the identity of Gi, i 2 I. Then we de�ne�� = fx 2 � : jsupp(x)j < !g:It is clear that �� is a subgroup of �. This subgroup is called the �-productof the groups Gi, i 2 I. Suppose that � carries the @0-box topology T! thestandard base of which consists of the sets ��1J (V ), where J is a countablesubset of I, �J : � ! �J = Qj2J Gj is the projection, and V = Qj2J Vj is aproduct of open subsets Vj � Gj, j 2 J. Then � with the topology T! becomesa topological group. Note that if all groups Gi are discrete, then every G�-setin (�;T!) is open. In the special case when the groups Gi are countable anddiscrete, we shall call �� (as well as every subgroup of ��) a Comfort-like group.Therefore, every Comfort-like group is a P-group. In particular, such a groupis zero-dimensional.Let us show that Comfort-like groups form a subclass of @0-bounded P-groups. It is helpful to note that by Theorem 2.4 of [9], every @0-boundedP-group is o-bounded (in precise terms, the result in [9] was formulated forLindel�of P-groups, but its proof remains valid for @0-bounded groups as well).Corollary 2.3. Every Comfort-like group G is @0-bounded. Therefore, G is an@0-bounded P-group, hence o-bounded.Proof. Every Comfort-like group G is a P-group. We show that the group G is@0-bounded. Since a subgroup of an @0-bounded group is also @0-bounded [6],we can assume that G = ��, where � = Qi2I Gi is the product of countablediscrete groups Gi, and � is endowed with the @0-box topology. Let U =



Some properties of o-bounded and strictly o-bounded groups 33G\��1J (x) be a non-empty basic open set in G, where J 2 [I]�!, �J : � ! �Jis the projection and x 2 �J. Since �J(G) is countable, there exists a countablesubset K of G such that �J(K) = �J(G). One easily veri�es that G = K � U.This proves that G is @0-bounded.To �nish the proof, note that the class of @0-bounded P-groups is closedwith respect to taking arbitrary subgroups, and every group in this class iso-bounded by Theorem 2.4 of [9]. �Now we present one of the main results of this section.Theorem 2.4. Let G be an @0-bounded P-group and H be an o-bounded topo-logical group. Then G � H is o-bounded.Proof. Let us show that if ': G � H ! K is a continuous epimorphism,where K is a second countable group, then K is o-bounded. Suppose thatW1, W2, : : : is a neighborhood basis at the identity eK of K. For eachi 2 N, we take a neighborhood Ui � Vi of (eG;eH), where Ui and Vi areneighborhoods of the identities eG and eH of G and H respectively, such thatUi � Vi � '�1(Wi). Since G and H are @0-bounded, there exist continuoushomomorphisms fi : G ! Gi and hi : H ! Hi, where Gi and Hi are secondcountable groups, and neighborhoods U 0i and V 0i of the identities eGi and eHiof Gi and Hi respectively, such that f�1i (U 0i) � Ui and h�1i (V 0i ) � Vi (see [14,Lemma 3.7]). Let f = �i2Nfi : G ! Qi2N Gi and h = �i2Nhi : H ! Qi2N Hibe the diagonal products of the families ffi : i 2 Ng and fhi : i 2 Ng, re-spectively. The groups G0 = f(G) and H0 = h(H) are second countable andby Theorem 2.3 of [9], are o-bounded as continuous homomorphic images ofo-bounded groups G and H, respectively. In addition, G0 = f(G) is countableby Lemma 2.2, and so, �-compact. It then follows from [9, Theorem 5.3] thatG0�H0 is o-bounded. Observe that ker(f �h) � ker' and then, for each neigh-borhood W of the identity in K, there exists a neighborhood V of the identity(eG0;eH0) in G0�H0 such that (f�h)�1(V ) � '�1(W). So, by Lemma 2.1, thereexists a continuous homomorphism  : G0 �H0 ! K such that ' = (f �h)� .Applying again Theorem 2.3 of [9], we infer that K is o-bounded. Therefore,all continuous homomorphic images of G � H are o-bounded, so [9, Theorem4.1] implies that the group G � H is o-bounded. �The above theorem and Corollary 2.3 together imply the following.Corollary 2.5. If G is a Comfort-like group, then the product G � H is o-bounded for every o-bounded group H.It is shown in [9, Example 3.1] that every �-product of countable discretegroups that carries the @0-box topology is strictly o-bounded. Since subgroupsof a strictly o-bounded group inherit this property [9, Theorem 2.1], everyComfort-like group is strictly o-bounded. Here we strengthen this result byconsidering the product of a Comfort-like group by a strictly o-bounded group.Theorem 2.6. If G is a Comfort-like group and H is a strictly o-boundedgroup, then G � H is strictly o-bounded.



34 C. Hern�andez, D. Robbie, M. TkachenkoProof. By Theorem 2.1 of [9], a subgroup of a strictly o-bounded group isstrictly o-bounded. Therefore, it su�ces to consider the case G = �� � �,where the product � = Q�<� G� of countable discrete groups G� is equippedwith the @0-box topology. For every A 2 [�]�!, let �A : � ! �A = Q�2A G�be the projection. Denote by eA the identity of �A. Then the family U = fUA :A 2 [�]�!g is a base at the identity e of G, where UA = G \ ��1A (eA). Oneeasily veri�es the following:(1) every UA is a normal subgroup of G;(2) the sets UA are clopen in G;(3) jG=UAj � @0 for each A 2 [�]�!.It was proved in [9, Example 3.1] that G is strictly o-bounded. We need torefer to aspects of that proof, so the needed parts are reproduced here withappropriate adaptation for our present proof. Suppose that player I chooses inthe turn i the neighborhood Wi of the identity in G�H. We can assume withoutloss of generality that each Wi has the form Wi = Ui � Vi where Ui and Vi areneighborhoods of the identity in G and H respectively. In addition, we canassume that Ui = UAi, where Ai 2 [�]�!, and that A1 � A2 � ��� � Ai � ��� .First consider the group G. If x 2 G, put supp(x) = f� < � : x� 6= e�g,where e� is the identity of G�. Clearly, supp(x) is a �nite subset of � for eachx 2 G.Since each UA has a countable number of cosets in G we may do as follows:For every A 2 [�]�!, we de�ne a countable set BA = fxA1 ;xA2 ; : : :g choosingelements xAi in every coset of UA in G in such a way that supp(xAi ) � A foreach i 2 N.Choose x 2 G. Then of course x 2 xAi UA for some xAi 2 BA. We note thatas any element u 2 UA has u� = e�, for � 2 A, therefore x� = (xAi )� for � 2 A,and moreover, supp(xAi ) � supp(x). We further note that if A;B 2 [�]�! andA � B then UB � UA, and if x 2 xBj UB then xBj UB � xAi UA, so supp(xAi ) �supp(xBj ) � supp(x).With reference to the Ui = UAi above, and writing xij for xAij we see thatfor each n, we have that x 2 xnjnUn � ��� � x2j2U2 � x1j1U1 and supp(x1j1) �supp(x2j2) � ��� � supp(xnjn) � ��� � supp(x).As all of these sets are �nite we must have some n = n(x) such thatsupp(xn+kjn+k) = supp(xnjn), for each k = 1;2; : : : . Since also by our remarksabove x must agree with each one of the xnjn at each one of their coordinates ofsupport, therefore xn+kjn+k = xnjn for all k � 1.Thus we have our main point as follows: x 2 xn(x)jn(x)Un for each n � n(x).For each n, let En = fxij : i;j � ng. Then certainly x 2 En � Un for eachn � n0 = maxfn(x);jn(x)g. To prove that G is strictly o-bounded we only needone n � n0, but to prove our present theorem we need the full set of n � n0 aswe will now see.Since H is strictly o-bounded, player II has a winning strategy in the OF-game on H. So, we are able to construct �nite non-empty subsets Fi;j of H as



Some properties of o-bounded and strictly o-bounded groups 35set out in the following scheme:V1 V2 V3 : : : Vp : : : Vj : : :F1;1 F2;1 F3;1 : : : Fp;1 : : : Fj;1 : : :F2;2 F3;2 : : : Fp;2 : : : Fj;2 : : :... ...Fp;p : : : Fj;p : : :...Fj;j : : :such that for each p � 1, H = 1[q=p Fq;p � Vq:If we put Fi = Sij=1 Fi;j, clearly then H = S1i=p Fi � Vi for each p � 1. Now weshall prove that G � H = S1i=1(Ei � Fi) � (Ui � Vi). Let (x;y) 2 G � H. Thenx 2 En0 �Un0 where n0 = maxfn(x);jn(x)g. Now, H = S1i=n0 Fi�Vi. So y 2 Fj�Vj,for some j � n0. But then x 2 Ej � Uj. Hence (x;y) 2 (Ej � Uj) � (Fj � Vj) =(Ej � Fj) � (Uj � Vj), as required. �It is clear that every �-compact group is strictly o-bounded. By [9, Theorem2.1], subgroups of �-compact groups inherit this property. The following theo-rem proved by Jian He strengthens this result and complements Theorem 2.6.We present its proof with his kind permission.Theorem 2.7. If G is a strictly o-bounded group and H is a subgroup of a�-compact group, then G � H is strictly o-bounded.Proof. Since H is a subgroup of a �-compact group it is �-bounded, that is, H =S1i=1 Xi, where the sets Xi are precompact in H and may be taken such thatXi is included in Xj whenever i � j. Therefore, for each sequence fVi : i 2 Ngof neighborhoods of the identity, eH, of H, there exists a sequence fQi : i 2 Ngof �nite non-empty subsets of H such that for each i, Xi � Qi � Vi. Now wesuppose that at turn i, player I chooses a neighborhood Ui � Vi of the identityin G�H. Since player II has a winning strategy in the OF-game on G, we areable to construct �nite non-empty subsets Fi;j of G as set out in the followingscheme: U1 U2 U3 : : : Up : : : Uq : : :F1;1 F1;2 F1;3 : : : F1;p : : : F1;q : : :F2;2 F2;3 : : : F2;p : : : F2;q : : :... ...Fp;p : : : Fp;q : : :...Fq;q : : :



36 C. Hern�andez, D. Robbie, M. Tkachenkosuch that for each p � 1, G = 1[q=pFp;q � Uq:For every i � 1, letFi = F1;i [ F2;i [ �� � [ Fi;i and Ki = Fi � Qi:We claim that G � H = S1i=1 Ki � (Ui � Vi). That is because if (x;y) 2 G � H,then there exists p � 1 such that y 2 Xp. But then there exists q � p such thatx 2 Fp;q � Uq. Now as Qq � Vq � Xq � Xp, then y 2 Qq � Vq. Since Fp;q � Fq, wehave (x;y) 2 Kq � (Uq � Vq);and so G � H is strictly o-bounded. �Another interesting problem is to characterize the spaces X such that thefree topological group F(X) is strictly o-bounded. Here we �nd a special classof spaces X with this property.Let D be a discrete space of uncountable cardinality. Denote by D� =D [ fx�g the space obtained by adjoining to D the point x� not in D, whosetopology consists of all subsets of D and all subsets of D� with countablecomplements. Such a space D� is known as the one-point Lindel�o�cation of D.Theorem 2.8. The free topological group F(D�) is strictly o-bounded.To prove the above theorem we need two auxiliary results.Lemma 2.9. The family 
 = fUK : K 2 [D]�!g, where UK is the normalsubgroup of F(D�) generated by D� n K, is a base at the identity of F(D�).Proof. Let K 2 [D]�! and K� = K [ fx�g. De�ne the natural retractionr : D� ! K� by the formula r(x) = x if x 2 K� and r(x) = x� if x =2 K�. Now,consider r̂: F(D�) ! F(K�), the extension of r to a continuous homomorphism.It is easy to see that ker r̂ = UK. Since K� is a discrete space, the groupF(K�) is discrete, so UK is a normal open subgroup of F(D�). Let us provethat T
 = feg, where e is the identity of F(D�). Note that an element x =a"11 a"22 : : :a"nn 2 F(D�), with ai 2 D� and "i = �1, belongs to UK if and onlyif the word representing x becomes equal to e when all elements ai 2 D� n K�are replaced by x�. Hence if we choose K = fa1; : : : ;ang, then x 6= e impliesx =2 UK.Clearly, D� is a Lindel�of P-space, and hence all �nite powers of D� are alsoLindel�of [10]. Therefore, the group F(D�) is Lindel�of (see also [11]). Supposethat U is an open neighborhood of the neutral element e. Since the groupF(D�) is Lindel�of and feg = T
 � U, we can �nd a countable subfamily � of
 such that T� � U. Finally, observe that the intersection of any countablesubfamily of 
 is in 
. So, 
 is a base at e for F(D�). �Lemma 2.10. Let UK be the normal subgroup of F(D�) generated by D� n K,where K 2 [D]�!. Then F(D�) = hKi � UK.



Some properties of o-bounded and strictly o-bounded groups 37Proof. Let g be an element of F(D�). Then g = x1 � � �xn, where each xi is inD�[(D�)�1. We use mathematical induction on n to prove that g is in hKi�UK.If n = 1, then g = x1 is either in K [K�1 � hKi or in (D� nK)[(D� nK)�1 �UK. In either case, g 2 hKi�UK. We suppose that n > 1 and x1 � � �xn�1 = f �u,where f 2 hKi and u 2 UK.If xn 2 D�nK, then it is clear that uxn 2 UK, hence g = f�uxn 2 hKi�Un. Onthe other hand, if xn is in K, then for UK to be a normal subgroup, u0 = x�1n uxnis in UK. Therefore, g = fxn � u0 2 hKi � UK. This �nishes the proof. �Proof of Theorem 2.8. For every K 2 [D]�!, let fgKn : n 2 Ng be an enu-meration of the group hKi. Without loss of generality, we may suppose thatplayer I chooses open sets of the form UK (Lemma 2.9 applies here). If playerI chooses UK1, player II chooses F1 = fgK11 g. In general, if player I choosesUKn, then player II chooses Fn = fgKij : 1 � i;j � ng. We can also as-sume that K1 � K2 � ��� � Kn � ��� . Let K = S1i=1 Ki. Observe thathKi = S1n=1hKni = S1n=1 Fn. Finally, since UK � UKn for each n, Lemma 2.10implies thatF(D�) = hKi � UK = � 1[n=1 Fn� � UK = 1[n=1(Fn � UK) � 1[n=1 Fn � UKn:Then, F(D�) is strictly o-bounded. �In fact, the above theorem admits a stronger form.Theorem 2.11. The product F(D�)�H is strictly o-bounded for each strictlyo-bounded group H.Proof. We can modify slightly the proof of Theorem 2.6 and obtain the proofof our theorem. Indeed, the sets UK are now the normal subgroups of F(D�)generated by D� n K, where K 2 [D]�!. These sets were used in the proof ofTheorem 2.8 and, as before, form a base for the identity that has the followingproperties:(1) each UK is a normal subgroup of F(D�);(2) the subsets UK are clopen in F(D�);(3) jF(D�)=UKj � @0 for each K 2 [D]�!.We may suppose that player I chooses neighborhoods of the form Ui � Vi,where Ui and Vi are neighborhoods of the identity e of F(D�) and eH of Hrespectively, and Ui = UKi, where Ki is in [D]�!, i 2 N.As in the proof of Theorem 2.8, we choose an enumeration fgKn : n 2 Ng ofhKi, and put En = fxKij : i;j � ng. At this point, the proof of the theoremcontinues in the same way as the proof of Theorem 2.6. �The following example shows that the class of o-bounded groups is not �nitelymultiplicative. This answers the corresponding problem posed in [9] in thenegative. It turns out that the o-bounded group G from [9, Example 8] suits.Example 2.12. There exists a second countable o-bounded topological groupG such that G � G is not o-bounded.



38 C. Hern�andez, D. Robbie, M. TkachenkoFor every x 2 R!, de�ne suppx = fn 2 N : x(n) 6= 0g. Let fnk(x) : k 2 !gbe the enumeration of suppx in the increasing order. Denote by X the set ofall x 2 R! such that limk!1 x(nk)nk+1(x) = 0:Consider the subgroup G of R! generated by X, i.e., G = hXi. In what followswe use the additive notation for the group operation in R! .We already know that G is o-bounded. We shall prove that G2 is not o-bounded describing a sequence fUn : n 2 Ng of open neighborhoods of theidentity e 2 G for which no sequence of �nite subsets fEn : n 2 Ng in Gwill make G2 = S1n=1[(En � En) + (Un � Un)]. For every n 2 N, let Un =G \ Q1j=1 Vn;j, where Vn;j = (�1;1) for 0 � j � n and Vn;j = R if j > n. Now,when considering En +Un the only coordinates of the elements En that matterare 0;1; : : : ;n since Un is unrestricted on ! nn coordinates. So, we may as wellonly consider En where the elements have 0 at each of the !nn places. Moreover,we can assume that En � En+1. Let An = maxfjz(i)j : z 2 En; 0 � i � ng.Observe that A0 < A1 < � � � . We shall prove that G2 6= S1n=1[(En � En) +(Un � Un)] for any �nite subsets En � G. That is, there exists at least onepair of elements x, y 2 G such that (x;y) =2 S1n=1[(En � En) + (Un � Un)]. Weconstruct x and y as follows. Choose n0 = 0, n1 = 1 and set x(0) = x0 > An1.We now choose any n2 such that x0=n2 < 1=2. Now, for all i, 0 < i < n2, we putx(i) = 0. Let y(n1) = yn1 > An2. Then we choose n3 2 ! so that yn1=n3 < 1=3.We set y(j) = 0 if 0 � j < n1 or n1 < j < n3. We continue in this way tode�ne numbers fnk : k 2 !g. We put x(nk) = xnk > Ank+1 for k even and suchthat x(nk)=nk+2 < 1=(k + 2). The other values for x(j) so far unde�ned forj < nk+2 are set as 0. Similarly, if k is odd, then de�ne y(nk) = ynk > Ank+1and nk+2 is de�ned so that y(nk)=nk+2 < 1=(k + 2). It is clear that x,y 2 G.We claim that (x;y) =2 S1n=1[(En � En) + (Un � Un)]. Indeed, suppose thatn 2 N and that nk � n < nk+1. If k is even, then xnk > Ank+1 � An, sox =2 En + Un. If k is odd, then ynk > Ank+1 � An, so y =2 En + Un. Hence(x;y) =2 S1n=0[(En�En)+(Un�Un)]. This shows that G2 is not o-bounded. �3. An example of an OF-undetermined groupBy Theorem 4.1 of [9], an @0-bounded group G is o-bounded if and onlyif all second countable continuous homomorphic images of G are o-bounded.Here we show that strictly o-bounded groups cannot be characterized this way,thus answering [9, Problem 4.2] in the negative. In addition, the group G weconstruct below will be OF-undetermined, that is, neither player I nor playerII has a winning strategy in the OF-game on G.Theorem 3.1. Under �, there exists a topological group G with the followingproperties:(a) every countable intersection of open sets in G is open;(b) the image f(G) is countable for every continuous homomorphism f :G ! H to a second countable topological group H; in particular, G iso-bounded;



Some properties of o-bounded and strictly o-bounded groups 39(c) G is OF-undetermined, hence not strictly o-bounded.Proof. We shall construct G as a subgroup of the group Z!1 endowed withthe @0-box topology, where the group Z has the discrete topology. This willguarantee (a). For every � < !1, let �� : Z!1 ! Z� be the projection andK� be the kernel of ��. Then K� is an open subgroup of Z!1, and we putN� = G \ K�. Clearly, the family fN� : � < !1g forms a decreasing base atthe neutral element of G. The subgroup G of Z!1 will also satisfy the followingstrong condition:(B) jGj = @1, but ��(G) is countable for each � < !1.Let us show that (B) implies (b). Suppose that f : G ! H is a continuoushomomorphism to a second countable topological group H. Choose a countablebase fUn : n 2 Ng at the neutral element of H. For every n 2 N, thereexists an ordinal �n < !1 such that N�n � f�1(Un). Let � be a countableordinal satisfying �n < � for each n 2 N. Then N� � kerf, so by Lemma2.1 there exists a homomorphism g : ��(G) ! H such that f = g � ��. Sincethe group ��(G) is countable by (B), we have jf(G)j � j��(G)j � !. Clearly,every countable group is o-bounded, so Theorem 4.1 of [9] implies that G iso-bounded.The di�cult part of our construction is to guarantee (c). This requires somepreliminary work. For a point x 2 Z!1, put supp(x) = f� < !1 : x(�) 6= 0gand consider the subgroup � of Z!1 de�ned by� = fx 2 Z!1 : jsupp(x)j � !g:It is clear that j�j = c = @1. Actually, our group G will be constructedas a subgroup of �. Since fN� : � < !1g is a base at the neutral elementof G, we can assume without loss of generality that player I always makes hischoice from this family, and this choice, say N�, is de�ned by the correspondingordinal �. Therefore, every possible winning strategy for player II is a function : Seq ! [G]<!, where Seq is the family of all �nite sequences (�0;�1; : : : ;�n)with �0 < �1 < � � � < �n < !1 and [G]<! is the family of all non-empty �nitesubsets of G.Denote by Lim the set of all in�nite limit ordinals in !1. For every � 2Lim, denote by Seq(�) the family of all �nite sequences (�0;�1; : : : ;�n), where�0 < �1 < � � � < �n < �. Using � and Lemma 2 of [Fed], we can �nd a familyf � : � 2 Limg satisfying the following conditions:(i)  � : Seq(�) ! [Z�]<! is a function for each � 2 Lim;(ii) for every function  : Seq ! [�]<! satisfying j�
( (Seq))j � ! foreach 
 < !1, there exists � 2 Lim such that  � = �� �  jSeq(�), i.e., �(�0; : : : ;�n) = ��( (�0; : : : ;�n)) for any sequence �0 < � � � < �n < �.If � < � < !1, denote by ��� the projection of Z� to Z�. Now, we willconstruct a sequence fG� : � < !1g satisfying the following conditions for each� < !1:(1) G� is a countable subgroup of Z�;(2) ���(G�) = G� if � < �;



40 C. Hern�andez, D. Robbie, M. Tkachenko(3) G�+1 = G� � Z;(4) if � 2 Lim, then S�<�(G� � f0��g) � G�, where 0�� is the neutralelement of Z�n�;(5) if � 2 Lim, then  � is not a winning strategy for player II in G�.Put G0 = Z. Suppose that for some � < !1, we have de�ned a sequencefG� : � < �g satisfying (1){(5). If � is non-limit, say � = � + 1, then we putG� = G� � Z � Z� � Z = Z�. Let us consider, therefore, the case � 2 Lim.Set H = S�<�(G� � f0��g). Clearly, the subgroup H of Z� is countable. Fixa sequence f�n : n 2 Ng � � such that limn2N �n = �. For every n 2 N, putFn =  �(�0; : : : ;�n). We claim that there exists a point x 2 Z�nSn2N(Un+Fn)such that ���n(x) 2 G�n for each n 2 N, where Un = ��(K�n) � Z�. Indeed,choose a point x0 2 G�0 n ���0(F0). By induction, with the help of (2) and (3),de�ne a sequence fxn : n 2 Ng such that xn 2 G�n n���n(Fn) and ��n+1�n (xn+1) =xn for each n 2 N. This is possible because (2) and (3) together imply thatfor every �;
 with 
 < � < � and every z 2 G
, there exist in�nitely manyy 2 G� with ��
 (y) = z. Let x 2 Z� be a point satisfying ���n(x) = xn for eachn 2 N. It is easy to see then that x =2 Sn2N(Un + Fn). Put G� = H � hxi.Then G� n Sn2N(Un + Fn) 6= ?, i.e.,  � is not a winning strategy for player IIin G�. Since ���(H) = G� and ���(x) 2 G� for each � < �, we conclude that���(G�) = G� for all � < �. Clearly, the group G� is countable, so that thesequence fG
 : 
 � �g satis�es (1){(5). This �nishes our construction.Consider the subgroup G = S�<!1(G� � f0(�)g) of Z!1, where 0(�) is theneutral element of Z!1n� for every � < !1. Then G � �, jGj = @1 and��(G) = G� for each � < !1, i.e., G satis�es (B). Let us verify that G isOF-undetermined.First, we show that player II has no winning strategy. Let  : Seq ! [G]<! bea function. Then j��( (Seq))j � j[G�]<!j � ! for each � < !1, so (ii) impliesthat there is � 2 Lim such that  � = �� �  jSeq(�). However,  � fails to bea winning strategy for player II in G�, and hence one can �nd an increasingsequence �0 < � � � < �n < � � � < � and a point x 2 G� n Sn2N(Un + Fn), whereFn =  �(�0; : : : ;�n) and Un = ��(K�n) for each n 2 N. Choose an elementy 2 G with ��(y) = x. Since Fn = ��( (�0; : : : ;�n)) for every integer n, weconclude that y =2 Sn2N(K�n +  (�0; : : : ;�n)), i.e.,  is not a winning strategyfor player II in G.Let us show that player I has no winning strategy. Since player I alwayschooses elements of the base fN� : � < !1g, every possible winning strategyfor him is a function �: F ! !1, where F is the family of all �nite sequences(F0; : : : ;Fn) with F0; : : : ;Fn 2 [G]<!. The equality � = �(F0; : : : ;Fn) meansthat player I chooses the neighborhood N� at the step n + 1. Suppose that�: F ! !1 is a function, and �0 is the player I's choice at the �rst step. SinceG � �, for every x 2 G there exists � < !1 such that supp(x) � �. Therefore,for every �nite subset F of G, we can de�nes(F) = minf� < !1 : supp(x) � � for each x 2 Fg:



Some properties of o-bounded and strictly o-bounded groups 41Given an ordinal � < !1, put�(�)= supf�(F0; : : : ;Fn) : (n 2 N) ^ (8i � n)[Fi 2 [G]<! ^ s(Fi) � �]g:De�ne a strictly increasing sequence f�n : n 2 Ng � !1 such that �(�n) � �n+1for each n 2 N and put � = supn2N �n. Let G� = fxn : n 2 Ng. For everyn 2 N, Hn = G�n �f0(n)g is a subgroup of G, where 0(n) is the neutral elementof Z!1n�n (see (4)). It is easy to de�ne a sequence fFn : n 2 Ng of �nite subsetsof G satisfying the following conditions for all n 2 N:(6) Fn � Hn;(7) ��n(xi) 2 ��n(Fn) for i = 0; : : : ;n.We claim that if x 2 G and ��(x) = xi, then x 2 N�n + Fn for each n � i.Indeed, by (7), we have ��n(xi) 2 ��n(Fn), so xj�n = xij�n = yj�n for somey 2 Fn. This implies that x 2 K�n + y � K�n + Fn, and hence x 2 N�n + Fn.This proves the claim. From (6) it follows that 
n+1 = �(F0; : : : ;Fn) � �n+1for each n 2 N, so we conclude thatG � 1[n=1(N�n + Fn) � 1[n=1(N
n + Fn):Therefore, � cannot be a winning strategy for player I, and hence G is OF-un-determined. It remains to note that an OF-undetermined group is not strictlyo-bounded. This completes the proof. �4. Open problemsThe class of o-bounded groups is not productive by Example 2.12. Thismotivates the followingProblem 4.1. Does there exist strictly o-bounded groups G and H such thatthe product G � H is not strictly o-bounded?It is known that the product G � H is o-bounded whenever G is o-boundedand H is either a �-compact or Comfort-like group (see [9, Theorem 5.3] andour Corollary 2.5). We do not know whether the multiplication by a strictlyo-bounded group can destroy o-boundedness:Problem 4.2. Is it true that a product of an o-bounded group by a strictlyo-bounded group is o-bounded?Theorem 2.8 suggests the following problem:Problem 4.3. Characterize the spaces X such that the free (Abelian) topologicalgroup F(X) (A(X)) is o-bounded or strictly o-bounded.In fact, the above problem splits up into four distinct subproblems.By [9, Theorem 4.1], an @0-bounded group G is o-bounded i� all secondcountable continuous homomorphic images of G are o-bounded. We do notknow whether \@0-bounded" can be omitted here:Problem 4.4. Suppose that all second countable continuous homomorphic im-ages of an (Abelian) topological group G are o-bounded. Is then G o-bounded?



42 C. Hern�andez, D. Robbie, M. TkachenkoIn Theorem 3.1 we constructed an example of an OF-undetermined topologi-cal group G. Our group G is very far from being metrizable: it is a non-discreteP-group. The following question is considered by T. Banakh in [4]:Problem 4.5. Does there exist an OF-undetermined metrizable group?Since every OF-undetermined group is o-bounded and o-bounded groups are@0-bounded, such a group has to be second countable. It is also shown in [4]that the answer to the above question is \yes" under Martin's Axiom.Acknowledgements. The second author wishes to thank the Department ofMathematics at UAM-Iztapalapa, Mexico, for the opportunity to work therefor two months in 1999 and also to thank the Department of Mathematics atUJI-Riu Sec, Castell�on de la Plana, for a similar opportunity for one month in1999. References[1] A. V. Arhangel'ski��, Cardinal invariants of topological groups: Embeddings and condensa-tions, Dokl. AN. SSSR 247 (1979), 779{782.[2] A. V. Arhangel'ski�� and V. I. Ponomarev, Fundamentals of General Topology, D. ReidelPublishing Company (1984).[3] T. Banakh, Locally minimal topological groups and their embeddings into products of o-bounded groups, Comment. Math. Univ. Carolin., to appear.[4] T. Banakh, On index of total boundedness of (strictly) o-bounded groups, Preprint.[5] W. W. Comfort, Compactness-like properties for generalized weak topological sums, Paci�cJ. Math. 60 (1975), 31{37.[6] I. Guran, On topological groups close to being Lindel�of, Soviet Math. Dokl. 23 (1981),173{175.[7] R. Engelking, General Topology, Heldermann Verlag, (1989).[8] V. V. Fedorchuk, On the power of hereditarily separable compact spaces, Dokl. AN SSSR222 (1975), 302{305 (in Russian).[9] C. Hern�andez, Topological groups close to being �-compact, Topology Appl. 102 (2000),101{111.[10] N. Noble, A note on z-closed projection, Proc. Amer. Math. Soc. 23 (1969), 73{76.[11] M. Tkachenko, The Souslin property in free topological groups over compact spaces, Math.Notes 34 (1983), 790{793. Russian original in: Mat. Zametki 34 (1983), 601{607.[12] M. Tkachenko, On the completeness of free Abelian topological groups, Soviet Math. Dokl.27 (1983), 341{345. Russian original in: Dokl. AN SSSR 269 (1983), 299{303.[13] M. Tkachenko, On boundedness and pseudocompactness in topological groups, Math.Notes 41 (1987), 229{231. Russian original in: Mat. Zametki 41 (1987), 400{405.[14] M. Tkachenko, Introduction to topological groups, Topology Appl. 86 (1998), 179{231.[15] V. Uspenskij, On continuous images of Lindel�of topological groups, Soviet Math. Dokl.32 (1985), 802{806. Received March 2000
C. Hern�andez and M. Tkachenko



Some properties of o-bounded and strictly o-bounded groups 43Departamento de Matem�aticasUniversidad Aut�onoma MetropolitanaAv. Michoac�an y Pur��sima s/nIztapalapa, C.P. 0934, M�exicoE-mail address: chg@xanum.uam.mxE-mail address: mich@xanum.uam.mx
D. RobbieDepartment of Mathematics and StatisticsThe University of MelbourneParkville, Victoria, 3052, AustraliaE-mail address: D.Robbie@ms.unimelb.edu.au