Each second countable abelian group is a subgroup of a second countable divisible group

EA CH SECOND COUNT ABLE ABELIAN GR OUP IS A S UBGR OUP OF A SECOND COUNT ABLE DIVISIBLE G R OUP T ARAS BANAKH AND LIUBOMYR ZDOMSKY ˘ I Abstract. It is shown that each pseudonorm | · | H defined o n a subgroup H of an ab elian group G can be extended to a pseudonorm | · | G on G such that the de ns ities of the ps e udo metrizable top olo g ical gro ups ( H , | · | H ) a nd ( G, | · | G ) co incide. W e derive from this that any Hausdo rff ω -b ounded group top ology on H ca n b e extended to a Hausdorff ω -b ounded gr o up top o logy on G . In its turn this result implies that each separable metrizable ab e lian group H is a subgro up of a separa ble metrizable divisible group G . This r e sult essentially relies on the Axio m of Choic e and is not true under the Axio m of Determina cy (which contradicts to the Axiom o f Choice but implies the Countable Axiom of Choice). This paper w as motiv ated by the follo wing question hav ing its origin in functional analysis (see [PZ], [BRZ]): Is it true that every metrizable s ep ar a ble ab elian top olo gic al gr o up with no torsion is a sub gr oup of a metrizable s e p a r ab l e divisible ab elian gr oup with no torsion? F r om now on al l gr oups c onside r e d in the p ap er ar e c ommutative . W e recall that a group G is divisible (r esp. has n o torsio n ) if for any elemen t a ∈ G and a p ositive in teger n the equation nx = a has a solution x ∈ G (resp. do es not hav e t w o distinct solutions in G ). According to the Baer Theorem [F, 21.1] each divisible group G is inje ctive in the sense that eac h homomorphism h : B → G defined on a subgroup B of a gro up A can b e extended to a homomorphism ¯ h : A → G . A classical result of the theory of infinite ab elian groups [F, 24.1] asserts t ha t eac h group (with no torsion) is a subgroup of a divisible gro up (with no torsion). This result a llo ws us to r educe the ab o v e question to the following one: Can every sep ar able gr oup top olo gy on a sub gr oup H of a gr o up G b e extende d to a sep ar abl e gr oup top olo gy on G ? Note that without the separabilit y requiremen t this problem is trivial: just announce H to b e an op en subgroup of G and ta k e the neigh b orho o d base at the or ig in o f H for a neigh b orho o d base at the origin in the group G . Ho w ev er if the quotien t g roup G/H is uncoun table suc h an extension leads to an unseparable to p ology on G . So, another less direct approa c h should b e dev elop ed. A classical result in t he theory of top ological g roups asserts that eac h gr o up top ology is generated by a family of con tinuous pseudonorms, see [Tk, § 2]. This observ ation allo ws us to reduce the problem of extending group top o logies to the problem of extending pseudonorms. As usual, under a (c o n tinuous) pseudonorm of a (top ological) group G w e understand a (con tin uous) non-negative function | · | : G → [0 , ∞ ) suc h that | 0 | = 0 and | x − y | ≤ | x | + | y | for all x, y ∈ G . A pseudonorm | · | is a n o rm provid ed | x | = 0 implies x = 0 . E ac h pse udonorm | · | on a group G generates a group top ology on G whose neighborho o d base at the origin consists of the ε -balls B |·| ( ε ) = { x ∈ G : | x | < ε } , 1991 Mathema tics Subje ct Classific ation. 03E25, 03E 60, 20K3 5, 20K45 , 22A05, 54A25, 5 4A35, 54H05, 54H11. Key wor ds and phr ases. a be lia n gr oup, divisible group, pseudo no rm, second countable g roup, b ounded- ness index, extension of top olog ie s, Polish group, analytic spac e, Axio m of Choic e , Axio m of Determinacy . 1 2 T ARAS BANAKH AND LIUBOMYR ZDOMSKY ˘ I ε > 0. The group G endow ed with this top ology turns into a top ological gr o up denoted b y ( G, | · | ). Giv en a top ological space X b y d ( X ) we denote it s density (that is the smallest size of a dense subset of X ), by w ( X ) its weight (that is the smallest size of a base of the top ology of X ) and by χ ( X ) its cha r acter (i.e., a smallest cardinal τ such that any p oin t x ∈ X p ossesses a neighborho o d base B of size |B | ≤ τ ). It is kno wn that d ( X ) = w ( X ) for any (pseudo)metrizable top olog ical space. No w w e are able to formulate the main result of this pap er. Theorem 1. Any pseudonorm | · | H define d on a s ub gr oup H of an ab elian gr oup G c an b e extende d to a pseudo n orm | · | G on G so that d ( H , | · | H ) = d ( G, | · | G ) . Because of its tec hnical c haracter w e p ostp one the pro of of this theorem till the end of the pap er. No w w e consider some its corollaries. According t o [Tk, 4.1], w ( G ) = χ ( G ) · ib ( G ) fo r an y top ological gro up G where ib ( G ) stands for the b ounde dness index o f G , equal to the smallest cardinal τ suc h that for a n y neigh b orho o d U of the origin of G there is a subset F ⊂ G with G = F · U and | F | ≤ τ , see [Tk, § 3]. T op olog ical groups G with ib ( G ) ≤ ℵ 0 are called ω -b ounde d , see [Gu] or [Tk]. It is known that a metrizable top ological group is ω -b ounded if and only if it is separable. Unlik e to separable groups, the class of ω -b ounded groups is closed under man y op erations, in pa r ticular taking subgroups and T yc ho nov pro ducts, see [Tk] or [Gu]. T aking in to accoun t that | X | ≤ 2 w ( X ) for any Hausdorff top olo gical space X [En, 1.5.1] and w ( G ) = χ ( G ) · i b ( G ) for an y top ological group G , w e get | G | ≤ 2 χ ( G ) for an y Hausdorff ω - b ounded top olog ical gr oup G . This inequalit y can b e rewritten as χ ( G ) ≥ log | G | , where log κ = min { τ : k ≤ 2 τ } for a cardinal κ . Theorem 2. A ny Hausdorff gr oup top olo gy τ H define d on a sub gr oup H of an ab eli a n gr o up G c an b e extende d to a Hausdorff gr oup top olo gy τ G on G so that ib ( G, τ G ) = ib ( H, τ H ) , χ ( G, τ G ) = max { χ ( H , τ H ) , log | G | } and w ( G, τ G ) = max { w ( H , τ H ) , log | G | } . In an ob vious wa y Theorem 2 implies Corollary 1. Any sep ar able metrizable top olo gy define d on a sub gr oup H of an ab elian gr o up G with | G | ≤ c c an b e extende d to a sep a r ab l e metrizab l e top o l o g y on G . Here c stands for the size of con tinuum. The next our corolla r y fo llows fro m Theorem 2 and Theorem 24.1 of [F] asserting that eac h ab elian group H (with no torsion) is a subgroup of a divisible gr o up G (with no torsion) suc h that | G | = | H | . Corollary 2. Any Hausdorff top o lo gic al ab elian gr oup H (w ith no torsion) is a sub gr oup of a Hausdorff ab elian divisib le gr oup G (w ith no torsion) such that w ( G ) = w ( H ) , χ ( G ) = χ ( H ) and ib ( G ) = ib ( H ) . The follo wing particular case of the ab ov e corollary giv es a p o sitiv e answ er to the question stated at the b eginning of the pap er. Corollary 3. Each sep ar able metrizab le a b e lian gr o up H (with no torsion) is a sub gr oup of a sep ar a b le metrizable di v i s ible gr oup G (having no torsion). In fact, the construction o f suc h a divisible gr oup G ⊃ H hardly uses Axiom of Choice (see Remark 1). As a result the group G has a complex descriptiv e structure. W e shall sho w tha t in general the group G is not a nalytic. Let us recall that a t o p ological space is analytic if it is a metrizable con tinuous image of a P olish space. As usual, under a Polish sp ac e w e understand a top o lo gical space homeomorphic to a separable complete metric space. A top olo g ical g roup is Polis h ( analytic ) if its underlying top ological space is P o lish (analytic). ON SU BGROUPS OF SECOND COUNT ABLE DIVIS IBLE ABELIAN GROUPS 3 The w ell-known Op en Mapping Principle for Banach spaces generalizes to top o logical groups a s follo ws: Any c ontinuous gr oup hom o morphism fr om an analytic gr oup onto a Polish gr oup is op en . The pro of of this Op en Mapping Principle fo llo ws from Theorem 9.10 [Ke] asserting that an y homomorphism h : G → H from a P olish group G into a ω - b ounded group H is con tinuous prov ided h has the Baire Prop ert y a nd Theorem 29.5 of [Ke] asserting that a nalytic subspaces of Polish spaces ha ve Baire Prop ert y . W e remind that a subset A of a top ological space X has the Bair e pr op erty if A con tains a G δ -subset G of X suc h that A \ G is meager in X . F o r a group H with no torsion and a p ositiv e in teger n let nH = { ny : y ∈ H } ⊂ H and 1 n : nH → H b e the map assigning to eac h elemen t x ∈ nH a unique y ∈ H suc h that ny = x . Prop osition 1. I f a Polish gr oup H i s a sub gr oup of a divi s i b le analytic gr oup G with no torsion, then for every p ositive inte ger n the ma p 1 n : nH → H is c o n tinuous. Pr o of. The subgroup H , b eing complete, is closed in G . Then the subgroup 1 n H = { g ∈ G : ng ∈ H } , b eing the preimage of H under the con tinuous map n : G → G , n : x 7→ nx , is a closed subset of G and thus is analytic. Since the group G is divisible and has no torsion, the map n : 1 n H → H , n : x → nx , is a bijectiv e contin uous group homomorphism from the analytic group 1 n H on to the P olish group H . Applying the Op en Mapping Principle for top olo gical gro ups w e conclude that this map is a top ological isomorphism and hence the map 1 n : H → 1 n H is con tin uous. Since nH ⊂ H , the map 1 n : nH → H is con tinuous to o.  Finally w e give an example of a P o lish group without torsion admitting no em b edding in to a divisible analytic gr o up without torsion. Example 1. Ther e is a Polish gr oup H without torsion such that the ma p 1 2 : 2 H → H is disc ontinuous. This gr oup H c ann o t b e a sub gr oup of a divisible analytic gr oup with no torsion. Pr o of. F or ev ery k ∈ N let H k b e a copy of the gr oup R of reals and let e k = 1 ∈ H k . Endow the g roup H k with t he norm | x | k = p (cos( π z ) − 1) 2 + sin 2 ( π z ) + (2 − ( k + 1) x ) 2 (whic h is generated b y the usual Euclidean distance under a suitable winding of H k = R a r ound a cylinder in R 3 ). It is easy to v erify that | e k | k > 2 while | 2 e k | k = 2 − k . On the direct sum ⊕ k ∈ N H k consider t he norm | ( x k ) k ∈ N | = P i ∈ N | x k | k and let H b e the completion of ⊕ k ∈ N H k with resp ect to this no r m. Then H is a P olish group. W e claim that H has no tor sion. Consider the iden tit y inclusion i : ⊕ k ∈ N H k → Q k ∈ N H k from the direct sum in to the direct pro duct endow ed with the T yc honov top ology . Observ e that this direct pro duct is a complete group. T o show that the group H has no torsion, it suffices t o v erify that the extension ¯ i : H → Q k ∈ N H k of the homomorphism i o n to the completion H is injectiv e. It will b e con v enien t to think of elemen ts of the groups ⊕ k ∈ N H k and Q k ∈ N H k as func- tions f : N → S k ∈ N H k . Assuming that the homomorphism ¯ i is not injectiv e, w e could find an elemen t f ∞ ∈ H suc h that f ∞ 6 = 0 but ¯ i ( f ∞ ) = 0. Fix an y ε > 0 with ε < | f ∞ | . C ho ose a sequence ( f n ) n ∈ N ∈ ⊕ k ∈ N H k con ve rging to f ∞ in H . W e can assume that | f n | > ε for eve ry n ∈ N . By the contin uit y of the map ¯ i , w e conclude that the sequence { i ( f n ) } n ∈ N con ve rges t o zero in Q k ∈ N H k (this means that the function sequence ( f n ) is p o in twis e con ve rgen t to zero). Since the sequence ( f n ) is Cauc hy in H , there is m ∈ N such that | f m − f j | < ε 2 for any j ≥ m . Without loss of generality , we can assume that f m ( k ) = 0 for all k > m . Since for ev ery k lim j →∞ f j ( k ) = 0, w e can find j > m so large that | f j ( k ) | k < ε 2 m 4 T ARAS BANAKH AND LIUBOMYR ZDOMSKY ˘ I for all k ≤ m . Then | f m − f j | = P ∞ k =1 | f m ( k ) − f j ( k ) | k ≥ P m k =1 | f m ( k ) − f j ( k ) | k ≥ P m k =1 | f m ( k ) | k − P m k =1 | f j ( k ) | k ≥ | f m | − P m k =1 ε 2 m > ε − ε 2 = ε 2 , whic h contradicts to | f m − f j | < ε 2 . Therefore, the homomorphism ¯ i is injectiv e and the group H has no torsion. Since | e k | = | e k | k > 2 and | 2 e k | = | 2 e k | k = 2 − k for all k , w e see that the sequence (2 e k ) con ve rges to zero in H while ( e k ) do es not. This means that the map 1 2 : 2 H → H is discon tinuous .  . Remark 1. Corollary 3 can not b e pro v en without the full Axiom of Choice and is not true under t he Axiom of Determinacy . This axiom con tra dicts the Axiom of Choice but implies its weak er form, the Coun table Axiom of Choice, see [JW, § 9.2 and § 9.3]. It is kno wn that under the Axiom o f Determinacy , an y subset of a P olish space has the Baire Prop ert y , see [K e , 8.35]. This fact and Theorem 9.10 of [Ke] implies that under Axiom of Determinacy the O p en Mapping Principle for top ological gro ups holds in the follo wing more stro ng form: an y contin uous homomorphism h : H → G fr o m a ω - b ounded gro up H on to a P olish group G is o p en. Using this stronger fo rm o f the Op en Mapping Principle and rep eating the pro o f of Prop osition 1 w e see that under the Axiom of D eterminacy this prop o sition holds without the analycit y assumption on the group G . Th us w e come to a rather unexp ected conclusion: U nder the Axiom of Determina cy the gr oup H fr om Example 1 c an n ot b e emb e dde d into a metrizable sep a r abl e divisib l e gr oup with no torsion, in s p ite of the fact that algebr aic al ly, H is a sub gr oup of the c ountable pr o duct R ω of l i n es . This shows that Coro llary 3 is not true under the Axiom of Determinacy . 1. Proof of Theorem 1 In t he pro of of Theorem 1 w e shall need one combinatorial lemma. A collection A of subsets of a set X is called k -uniform where k ∈ N if | A | = k for eac h A ∈ A ; A is disjoint if it consists of pairwise disjoin t sets. Lemma 1. Supp os e k ∈ N and A , B ar e two disjoint k -uniform finite c ol le ctions of subsets of a n infinite set X . Then ther e is a subset I ⊂ X such that | I ∩ C | = 1 for e ach C ∈ A ∪ B . Pr o of. It is easy to construct k - uniform disjoint finite collections C , D of subsets o f X suc h that A ⊂ C , B ⊂ D , |C | = | D | , and ∪C = ∪D . Let n = |C | = |D | and write C = { C 1 , . . . , C n } , D = { D 1 , . . . , D n } . Consider the matrix [ a ij ] n i,j =1 where a ij = 1 k | C i ∩ D j | and observ e that it is double sto chastic , that is P n i =1 a ij = 1 = P n j =1 a ij for all i, j ∈ { 1 , . . . , n } . According to t he Birkhoff Theorem (see [Bi], [Ga, p.556], or [A, 8.4 0 ]) eac h double sto c hastic matrix is a con ve x com bination o f p ermutating matric es , that is matrices of the form [ δ i,σ ( j ) ] n i,j =1 where σ is a p erm utation o f the set { 1 , . . . , n } and [ δ ij ] n i,j =1 is the iden tity matrix. This result implies the existenc e of a p erm utation σ of the set { 1 , . . . , n } suc h that a i,σ ( i ) > 0 for all i . This means that the in t ersection C i ∩ D σ ( i ) is not empty and th us con tains some p oin t x i . Let I = { x 1 , . . . , x n } and observ e that | C ∩ I | = 1 for an y elemen t C ∈ C ∪ D ⊃ A ∪ B .  Theorem 1 will b e prov ed b y induction whose inductiv e step is based of the fo llo wing Lemma 2. L et H b e a sub gr oup of a gr oup G such that pG ⊂ H for some prime numb er p . Then any pseudonorm | · | H on H c an b e e xtende d to a p s e udonorm | · | G on G so that d ( G, | · | G ) = d ( H , | · | H ) . ON SU BGROUPS OF SECOND COUNT ABLE DIVIS IBLE ABELIAN GROUPS 5 Pr o of. The quotien t group G/H has prime exp onent p a nd th us has a basis whic h can b e written as { g α + H : α < µ } for some ordinal µ , see [F, 16.4]. It will b e con ven ien t to complete this basis b y zero letting g µ = 0. It follows that an y elemen t o f G can b e represen ted in the fo rm x = u + P i ∈ ω g α i where u ∈ H and the set { i ∈ ω : α i 6 = µ } is finite. F or an y elemen t x ∈ G let Rep( x ) = { ( u, ( α i ) i ∈ ω ) ∈ H × [0 , µ ] ω : x = u + P i ∈ ω g α i } . Observ e that for any x ∈ H and ( u, ( α i ) i ∈ ω ) ∈ Rep( x ) the num b er p divides the cardinalit y of the set { i ∈ ω : α i = α } for eac h or dina l α < µ . Let | · | H b e a pseudonorm on H . D efine a function ρ : G × G → [0 , ∞ ) letting ρ ( x, y ) = inf {| u − v | H + X i ∈ ω | pg α i − pg β i | H : ( u, ( α i ) i ∈ ω ) ∈ Rep( x ) , ( v , ( β i ) i ∈ ω ) ∈ Rep( y ) } for x, y ∈ G . It is easy to see that ρ is an in v aria nt pseudometric o n G . Let us show that d ( G, ρ ) ≤ d ( H , | · | H ). L et D b e a dense subset of ( H , | · | H ) with | D | = d ( H , | · | H ) and I ⊂ [0 , µ ] b e a subset of size | I | ≤ d ( H , | · | H ) suc h that I ∋ µ and the set { pg α : α ∈ I } is dense in the subspace { pg α : α < µ } of ( H , | · | H ). Then the set E = { x ∈ G : Rep( x ) ∩ ( D × I ω ) 6 = ∅} is a dense subset of ( G, ρ ) with | E | ≤ d ( H , | · | H ). This prov es the inequality d ( G, ρ ) ≤ d ( H , | · | H ). It remains to sho w that ρ ( x, y ) = | x − y | H for an y x, y ∈ H . Fix a rbitrary ( u, ( α i ) i ∈ ω ) ∈ Rep( x ), ( v , ( β i ) i ∈ ω ) ∈ Rep( y ). F or eve ry α < µ let A ( α ) = { i ∈ ω : α i = α } a nd B ( α ) = { i ∈ ω : β i = α } . Since x, y ∈ H , the n um b er p divides the cardinalit ies of the sets A ( α ), B ( α ) fo r all o r dina ls α < µ . Applying Lemma 1, find a subset I ⊂ ω suc h that | C ∩ I | = 1 p | C | for ev ery nonempt y subset C ∈ { A ( α ) , B ( α ) : α < µ } . Then x = u + P i ∈ ω g α i = u + X α<µ | A ( α ) | · g α = u + X α<µ p | I ∩ A ( α ) | g α = = u + X α<µ X i ∈ I ∩ A ( α ) pg α i = u + X i ∈ I pg α i . By analo gy , y = v + P i ∈ I pg β i . Consequen tly , | u − v | H + P i ∈ ω | pg α i − pg β i | H ≥ | u − v | H + P i ∈ I | pg α i − pg β i | H ≥ ≥ | ( u + P i ∈ I pg α i ) − ( v + P i ∈ I pg β i ) | H = | x − y | H P assing to the infim um, we get ρ ( x, y ) ≥ | x − y | H . The pro of of the in ve rse inequalit y is straigh t f orw ar d, hence ρ ( x, y ) = | x − y | H . Letting | x | G = ρ ( x, 0) for x ∈ G w e define a pseudonorm on G extending the pseudonorm | · | H so that d ( G, | · | G ) = d ( H , | · | H ).  Lemma 3. L et H b e a sub gr oup of a gr oup G such that the quotient gr oup G/ H is p e rio dic. Then any pseudonorm | · | H on H c an b e extende d to a pseudonorm | · | G on G so that d ( G, | · | G ) = d ( H , | · | H ) . Pr o of. L et ( p i ) ∞ i =1 b e a sequence of prime num b ers suc h that for ev ery prime n umber p the set { i ∈ N : p i = p } is infinite. Let H 0 = H and for i ≥ 0 let H i +1 = 1 p i +1 H i = { x ∈ G : p i +1 x ∈ H i } . Because of the p erio dicit y of the quotient group G/H w e get G = S ∞ i =1 H i . Let | · | H b e an y pseudonorm on H and D 0 b e a dense subset of the top ological group ( H , | · | H ) with | D 0 | = d ( H , | · | H ). Let | · | 0 = | · | H . Using the previous lemma, by induction for eve ry i ≥ 1 find a pseudonorm | · | i on the group H i and a dense subset D i of the top ological group ( H i , | · | i ) such that | x | i = | x | i − 1 for each x ∈ H i − 1 and | D i | = | D i − 1 | . Completing the inductiv e construction, define a pseudonorm | · | G on the group G letting | x | G = | x | i where x ∈ H i . It is clear that | · | G extends | · | H and D = S ∞ i =1 D i is 6 T ARAS BANAKH AND LIUBOMYR ZDOMSKY ˘ I a dense set in the top ological group ( G, | · | G ) with | D | = | D 0 | = d ( H , | · | H ). This yields d ( G, | · | G ) ≤ d ( H , | · | H ).  Finally we are able to complete the Pr o of of The or em 1. Let H b e a subgroup of a group G a nd | · | H b e a pseudonorm on H . According to [F, 24.1] the gro up H is a subgroup o f a divisible group E . Moreov er, according to Lemma 24 .3 [F] w e can a ssume that the quotien t group E /H is p erio dic. Applying the previous lemma, extend the pseudonorm | · | H to a pseudonorm | · | E on E so that d ( E , | · | E ) = d ( H , | · | H ). According to Ba er Theorem [F, 21.1], eac h divisible group is injectiv e. Consequen tly , t here is a group homomorphism h : G → E extending the iden tity map H → H ⊂ E . D efine a pseudonorm | · | G on G letting | x | G = | h ( x ) | E for x ∈ G and observ e that | · | G extend | · | H and d ( H , | · | H ) ≤ d ( G , | · | G ) ≤ d ( E , | · | E ) = d ( H , | · | H ).  2. Proof of Theorem 2 Let H b e a subgroup of a group G and τ H b e a Hausdorff group top ology on H . First we define a Hausdorff group top olog y τ G/H on the quotient g r o up G/H suc h that ib ( G/H, τ G/H ) ≤ ω and χ ( G/ H , τ G/H ) ≤ log | G/ H | . By [F , 24 .1] G/H is a subgroup of a divisible gro up E with | E | = | G/H | . Applying Theorem 2 3.1 [F ] (on the structure of divisible groups), w e can sho w that the gro up E is isomorphic to a subgroup of t he p ow er T κ of the circle T = R / Z where κ = log | G/H | ≤ log | G | . Observ e that T κ endo we d with the natural T ychono v pro duct top olo gy is a compact top ological group with ib ( T κ ) = ω and χ ( T κ ) = κ ≤ log | G | . Consequen t ly , the gro up G/H , b eing isomorphic to a subgroup of T κ , carries a Hausdorff group top ology τ G/H suc h that ib ( G/H, τ G/H ) ≤ ω and χ ( G/H , τ G/H ) ≤ κ ≤ log | G | . Fix a neighborho o d base B of size |B | = χ ( H , τ H ) at t he origin o f the top olo g ical group ( H , τ H ). Applying [Tk, 2.3], for ev ery U ∈ B fix a contin uous pseudonorm | · | U on H suc h that { x ∈ H : | x | U < 1 } ⊂ U . By Theorem 1, the pseudonorm | · | U can b e extended to a pseudonorm k · k U on G suc h that d ( G, k · k U ) = d ( H , | · | U ). T he contin uit y of the iden tity map ( H , τ H ) → ( H , | · | U ) implies that ib ( H , | · | H ) ≤ ib ( H , τ H ), see [Tk, 3.2]. Since the densit y and the b oundedness index coincide for (pseudo)metrizable top olog ical g roups [Tk, § 3], w e conclude that ib ( G, k · k U ) = d ( G, k · k U ) = d ( H , | · | U ) = ib ( H, | · | U ) ≤ ib ( H , τ H ). Let τ G b e the smallest top ology o n G making contin uous the quotien t homomorphism ( G, τ G ) → ( G/H , τ G/H ) and the iden tity map ( G, τ ) → ( G, k · k U ) for all U ∈ B . It is easy to see that τ G is a Hausdorff group top olo gy on G inducing the top olog y τ H on the subgroup H . Observ e that the top ological group ( G, τ G ) can b e iden t ified with a subgroup of the pro duct G/H × Q U ∈B ( G, k · k U ) of top olo g ical groups whose b o undedness indices do not exceed ib ( H , τ H ) and characters do not exceed log | G | . According to [Tk, 3.2], the b oundedness index of such a pro duct do es dot exceed ib ( H, τ H ) while its c haracter do es not exceed χ ( G/H ) · |B | ≤ log | G | · χ ( H , τ H ). Consequen tly , ib ( G, τ G ) ≤ ib ( H , τ H ), χ ( G, τ G ) ≤ χ ( H , τ H ) · log | G | , and w ( G, τ G ) = χ ( G , τ G ) · ib ( G, τ G ) ≤ log | G | · χ ( H , τ H ) · ib ( H , τ H ) = w ( H , τ H ) · log | G | . Ac kno wledgemen t . The author s express t heir sincere thanks to Sasha Rav sky and Igor Protaso v for v aluable and stim ulating discussions (concerning sto c hastic matrices). ON SU BGROUPS OF SECOND COUNT ABLE DIVIS IBLE ABELIAN GROUPS 7 Reference s [A] M. Aigner. Combinatorial Theory . – Springer - V erlag, 19 79. [BRZ] T. B anakh, A. Plichk o , A. Zagoro dnyuk. Automatic contin uity of p o lynomial o p er ators b etw een top ological ab e lian gr oups (in prepara tion). [Bi] G. Birk ho ff, T res obser v aciones sobre el alg ebra lineal // Rev. Univ. Nac T ucuma´ an. ser. 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E-mail addr ess : tb anakh@ frank o.lviv.ua Dep ar tment of Ma thema tics, Iv an Franko L viv Na tional University, Universytetska 1, L viv, 79000, Ukr aina