Quasi-convex density and determining subgroups of compact abelian groups

Quasi-con v ex densit y and determining s ubgroups of compact ab eli an groups Dikran Dikranjan ∗ and Dmitri Shakhmatov † De dic a te d to W. Wistar Comfo rt on t he o c c asio n of his 75th anniversary Abstract F or an a b elian top olo gical gro up G , let b G denote the dual group of all con tinuous c har - acters endow ed with the compac t op en top olo gy . Given a clos ed subset X of an infinite compact ab elian group G such that w ( X ) < w ( G ), and an op en neighbourho o d U of 0 in T , w e show that |{ χ ∈ b G : χ ( X ) ⊆ U }| = | b G | . (Here, w ( G ) de no tes the weigh t of G .) A subgroup D of G deter mines G if the map r : b G → b D defined b y r ( χ ) = χ ↾ D for χ ∈ b G , is an isomorphism betw een b G and b D . W e pr ov e that w ( G ) = min {| D | : D is a s ubgroup of G that deter mines G } for e very infinite compact ab elian group G . In pa r ticular, an infinite co mpact ab elian group determined b y a coun table subgroup is metrizable. This gives a negative answ er to questions of Co mfort, Hern´ andez, Macario, Raczkowski and T rigos -Arrieta from [5, 6 , 1 3]. As a n application, w e furnish a shor t elementary pro o f of the re s ult from [1 3] that a co mpact ab elian group G is metr iz a ble provided that every dense subgr oup o f G determines G . All top ological groups are assumed to b e Hausd orff, and all top ological spaces are assum ed to b e T yc honoff. As u sual, R denotes the group of real n umb er s (with the usual top ology), Z denotes the grou p of integ er n umb ers, T = R / Z denotes the circle group (w ith the usual top ology), N denotes the set of natural n umbers , P denotes th e set of prime num b ers, ω denotes the first infinite cardinal, and w ( X ) denotes the w eigh t of a space X . If A is a subset of a space X , then A denotes the closure of A in X . 1 Preliminaries and bac kground In this section we giv e necessary definitions and collect fiv e facts th at w ill b e needed later. T h ese facts are either known or part of the folklore. How ev er, to make this manuscript self-con tained, w e pr o vide th eir pro ofs in Section 5 for the reader’s conv enience. F or sp aces X and Y , we denote by C ( X , Y ) the space of all con tin uou s functions from X to Y endo we d with the c omp act op en top ol o gy , that is, the top ology generated b y the family { [ K, U ] : K is a compact su bset of X an d U is an op en subset of Y } ∗ Dipartimento d i Matematica e Informatica, Universit` a di U dine, Via delle Scienze 206, 33100 Udine, Italy; e- mail : dikranja@dimi.uniu d.it ; the first author was partially supp orted by MEC.MTM2 006-02036 and FEDER FUNDS. † Graduate School of S cience and Engineering, Division of Mathematics, Physics and Earth S ciences, Ehime Universit y , Matsuyama 790-8577, Japan; e-mail : dmitri@dpc.ehime-u .ac.jp ; the second author was p artially supp orted by the Grant-in-Aid for S cien tific Research no. 19540092 by the Japan So ciety for th e Promotion of Science (JSPS). 1 as a su bbase, where [ K, U ] = { g ∈ C ( X, Y ) : g ( K ) ⊆ U } . F act 1.1 If X is a c omp act sp ac e and Y is a sp ac e, then w ( C ( X, Y )) ≤ w ( X ) + w ( Y ) + ω . F or a top ological group G , w e denote by b G the Pon try agin-v an Kamp en dual of G , namely the group of all con tinuous characte rs χ : G → T endow ed with the compact op en top ology . Clearly , b G is a closed subgroup of C ( G, T ). In particular, a base of n eigh b orh o o ds of 0 in b G is giv en by the sets W ( K , U ) = { χ ∈ b G : χ ( K ) ⊆ U } = [ K, U ] ∩ b G, (1) where K is a compact subset of G and U is an op en neighbourh o o d of 0 in T . W e iden tify T = R / Z w ith th e real in terv al ( − 1 / 2 , 1 / 2] in the obvi ous w a y , and write T + = { x ∈ T : − 1 / 4 ≤ x ≤ 1 / 4 } . Definition 1.2 L et G b e an ab elian top ological group . (i) F or E ⊆ G and A ⊆ b G , define the p olars E ⊲ = { χ ∈ b G | χ ( E ) ⊆ T + } and A ⊳ = { x ∈ G | χ ( x ) ∈ T + for all χ ∈ A } . (ii) A set E ⊆ G is said to b e quasi-c onvex if E = E ⊲⊳ . (iii) The quasi-c onvex hul l Q G ( E ) of E ⊆ G is the smallest quasi-con v ex set of G con taining E . (iv) F ollo wing [8, 7], we will sa y that E ⊆ G is qc- dense (an abbr eviation for quasi-c onvexly dense ) p ro vided that Q G ( E ) = G , or equiv alen tly , if E ⊲ = { 0 } . Ob viously , E ⊆ E ⊲⊳ . Th erefore, a set E ⊆ G is quasi-con v ex if and only if for ev ery x ∈ G \ E there exists χ ∈ E ⊲ suc h that χ ( x ) 6∈ T + . The n otion of quasi-con v exit y was introd uced by Vilenkin [18] as a n atural counterpart for top ological group s of the fund amental notion of con v exit y from the theory of top ological vec tor spaces (w e refer the r eader to [2 , 1 ] for add itional information). F act 1.3 Supp ose that U i s an op en neighb ourho o d of 0 in T and X is a c omp act subset of a top olo g ic al gr oup G such that W ( X, U ) = { 0 } . Then: (i) ther e exists n ∈ N su c h that the sum K n = ( X ∪ { 0 } ) + ( X ∪ { 0 } ) + . . . + ( X ∪ { 0 } ) of n many c opies of the set X ∪ { 0 } i s qc-dense in G ; (ii) the sub gr oup of G gener ate d by X must b e dense in G . Item (i) of ou r next fact can b e foun d in [7, 8]. F act 1.4 L et f : G → H b e a c ontinuous homomorp hism of top olo gic al ab elian gr oups. Then: (i) f ( Q G ( X )) ⊆ Q H ( f ( X )) for every subset X of G . (ii) If f ( G ) b e dense in H and X is a qc-dense subset of G , then f ( X ) is qc- dense in H . 2 Definition 1.5 F ollo wing [5, 6], we say that a sub group D of an ab elian group G determines G if the r estriction homomorphism r : b G → b D (defin ed by r ( χ ) = χ ↾ D for χ ∈ b G ) is an isomorphism b et w een the top ological groups b G and b D . This notion is relev ant to extending the P ont ryag in-v an K amp en dualit y to n on-lo cally com- pact groups [3, 1]. In d eed, if G is lo cally compact and ab elian, then ev ery subgroup D that determines G m ust b e dens e in G ; in particular, no pr op er lo cally compact su bgroup of G can determine G . (W e note that the original defin ition in [5, 6] assumed upf r on t that D is d ense in G .) When D is d ense in G , the restriction homomorphism r : b G → b D is alwa ys a cont inuous isomorphism. The u ltimate connection b et w een the notions of d etermined s ubgroup and qc-density is es- tablished in the next fact. This fact is a particular case of a more general fact stated withou t pro of (and in equiv alen t terms) in [6, Remark 1.2(a)] and [13, Corollary 2.2]. F act 1.6 F or a sub gr oup D of a c omp act ab elian gr oup G the fol lowing c onditions ar e e quivalent: (i) D determines G ; (ii) ther e exists a c omp act subset of D which is qc-dense in G . Definition 1.7 According to [5, 6], a top ological group G is said to b e determine d if ev ery dense subgroup of G determines G . Chasco [3, Theorem 2] and Außenhofer [1 , Th eorem 4.3] p r o v ed that all metrizable ab elian groups are determined. Comfort, Raczk o wski and T rigos-Arrieta established the follo wing amaz- ing inv erse of this theorem for compact groups: Under the Contin uum Hyp othesis CH, ev ery determined compact ab elian group is metrizable ([5, Corollary 4.9] and [6, Corollary 4.17 ]). Quite recent ly , Hern ´ andez, Macario and T r igos-Arrieta remo v ed the assumption of CH fr om their result [13, Corollary 5.11]. W e note that this theorem b ecomes an immediate consequence of our m ain result, see Corollary 2.6. F act 1.8 [6, Corollary 3.15 ] If f : G → H i s a c ontinuous surje ctive homomorp hism of c omp act ab elian gr oups and G is determine d, then H is determine d as wel l. The follo wing question remained the last principal u n solv ed problem in the theory of compact determined groups: Question 1.9 [5, Question 7.1], [6, Qu estion 7.1], [13, Q u estion 5.12] (a) Is there a compact group G with a countable d ense s ubgroup D su c h that w ( G ) > ω and D determines G ? (b) What if G = T κ ? W e completely resolv e this question in Corollary 2.5. In fact, w e eve n solv e the most general v ersion of this qu estion with ω replaced by an arbitrary cardinal, see Corollary 2.4. 2 Main results Definition 2.1 I f X is a sub s et of a compact ab elian group G , then r G X : b G → C ( X , T ) denotes the “restriction map” defin ed by r G X ( χ ) = χ ↾ X for χ ∈ b G . Observe that C ( X , T ) is a top ological group an d r G X is a contin uous group homomorph ism. W e refer the reader to (1) for the defin ition of W ( X , U ). 3 Theorem 2.2 L et X b e a close d subset of an infinite c omp act ab elian gr oup G su ch that w ( X ) < w ( G ) . Then for ev ery op en neighb ourho o d U of 0 in T one has | W ( X , U ) | = | b G | . Pr o of . Consider fi rst the case when w ( X ) < ω . Then X m ust b e finite. Note that th e set W ( X , U ) is an op en neighborh o o d of 0 in the initial top ology T of b G with resp ect to the family { η x : x ∈ X } of ev aluation c haracters η x : b G → T defined by η x ( π ) = π ( x ) for ev ery π ∈ b G . Since top ologies generated by c haracters are totally b ound ed, fi nitely man y translations of W ( X , U ) co v er the whole group b G . Since b G is in finite, th is yields | W ( X , U ) | = | b G | . F rom now on w e will assum e that w ( X ) ≥ ω . The inequalit y | W ( X , U ) | ≤ | b G | b eing trivial, it suffices to c hec k that | b G | ≤ | W ( X , U ) | . Let r G X b e the m ap from Definition 2.1, and let H = r G X ( b G ). Note that ker r G X ⊆ W ( X , U ), so | k er r G X | ≤ | W ( X, U ) | . If | ke r r G X | = | b G | , we are done. Ass u me no w that | ke r r G X | < | b G | . Since b G is in fi nite, we obtain | b G | = | b G/ ker r G X | = | r G X ( b G ) | = | H | . (2) Let N b e the subgroup of H generate d by the op en subset [ X, U ] ∩ H of H . Then N is a clop en subgroup of H , so the index of N in H cannot exceed w ( H ), w hic h gives | H | = | N | + | H / N | ≤ | N | + w ( H ) ≤ | [ X , U ] ∩ H | + ω + w ( H ) . (3) Since w ( H ) ≤ w ( C ( X, T )) ≤ w ( X ) by F act 1.1, and w ( X ) + ω = w ( X ) b y our assu mption, we obtain from (3) that | H | ≤ | [ X, U ] ∩ H | + w ( X ) . (4) As w ( X ) < w ( G ) = | b G | = | H | by (2), and | H | = | b G | ≥ ω , f rom (4 ) it follo ws that | [ X, U ] ∩ H | = | H | . (5) Finally , note that [ X , U ] ∩ H = r G X ( W ( X, U )), which yields that | [ X, U ] ∩ H | = | r G X ( W ( X, U )) | ≤ | W ( X, U ) | . (6) Com bining (2), (5) and (6), w e obtain the inequalit y | b G | ≤ | W ( X , U ) | .  Corollary 2.3 If a close d subsp ac e X of an infinite c omp act ab elian gr oup G i s qc-dense in G , then w ( X ) = w ( G ) . Pr o of . Let U b e an op en neighbour ho o d of 0 in T s u c h th at U ⊆ T + . Since X is qc-dense in G , w e ha v e W ( X , U ) ⊆ X ⊲ = { 0 } . Now Theorem 2.2 yields w ( X ) ≥ w ( G ). The reverse inequalit y w ( X ) ≤ w ( G ) is trivial.  Our next corollary constitutes a ma jor br eakthrough in the theory of compact determined groups. Corollary 2.4 If a sub gr oup D of an infinite c omp act ab elian gr oup G determines G , th en | D | ≥ w ( G ) . Pr o of . According to F act 1.6, D cont ains a compact subset X th at is qc-dense in G , so | D | ≥ | X | ≥ w ( X ) (see, f or example, [11, Theorem 3.1.21]). Finally , w ( X ) = w ( G ) by Corollary 2.3.  Ev en the particular case of Corollary 2.4 p ro vides a complete answer to Q uestion 1.9: Corollary 2.5 A c omp act ab elian gr oup determine d by a c ountable sub gr oup i s metrizable. Corollary 2.6 [13, Corollary 5.11] Every determine d c omp act ab elian gr oup is metrizable. 4 Pr o of . Assum e that G is a non-metrizable determined compact ab elian group. Then w ( G ) ≥ ω 1 , and so we can find a con tinuous surjectiv e group h omomorphism h : G → K = H ω 1 , where H is either T or Z ( p ) for some prime num b er p (see, f or example, [6, Theorem 5.15 and Discussion 4.14]). As a con tin uous homomorphic image of the determined grou p G , the group K is d etermined by F act 1.8. Since K is separable (see, for example, [11, Theorem 2.3.15]), th ere exists a coun table d ense su bgroup D of K . S in ce K is determined, we conclude th at D m us t determine K . T herefore, K must b e metrizable b y Corollary 2.5, a contradicti on.  Useful p rop erties of determined group s can b e found in [4]. A sup er-se quenc e is a non-emp t y compact Hausd orff sp ace X with at most one non-isolated p oint x ∗ [10]. When X is infin ite, we will call x ∗ the limit of X and sa y that X c onver ges to x ∗ . Observe that a con v ergen t sequen ce is a countably in fi nite su p er-sequen ce. Außenhofer [1] essen tially pr o v ed that ev ery in finite compact metric ab elian group has a qc- dense s equence con v erging to 0. 1 Our next theorem extends th is result to all compact group s b y replacing con vergi ng s equences with su p er-sequences. Theorem 2.7 E very infinite c omp act ab elian gr oup c ontains a qc-dense sup er-se quenc e c on- ver ging to 0 . The pro of of Theorem 2.7 is p ostp oned u n til Section 4. Corollary 2.8 E very infinite c omp act ab elian gr oup G has a (dense) sub gr oup D which deter- mines G such that | D | ≤ w ( G ) . Pr o of . Apply T h eorem 2.7 to fin d a sup er-sequen ce X that is qc-dense in G . Let D b e the subgroup of G generated by X . Clearly , | X | = w ( X ) ≤ w ( G ). Since G is infin ite, w ( G ) m ust b e infinite, and therefore | D | ≤ ω + | X | ≤ w ( G ). Finally , D d etermines G b y F act 1.6.  Our next corol lary pro vides another ma jor adv ance in the theory of compact determined groups: Corollary 2.9 If G is an infinite c omp act ab elian g r oup, then w ( G ) = min {| D | : D is a sub gr oup of G that determines G } . Pr o of . C ombine Corollaries 2.4 and 2.8.  W e ha ve b een kind ly informed by Chasco that our next corollary was ind ep endently pr o v ed b y Bruguera and Tk ac henk o: Corollary 2.10 Ev ery infinite c omp act ab elian gr oup G c ontains a pr op er (dense) sub gr oup D which determines G . Pr o of . Let D b e a subgroup of G as in the conclusion of Corollary 2.8. S ince G is an infinite compact group, w e ha ve | D | ≤ w ( G ) < 2 w ( G ) = | G | . Therefore, D must b e a pr op er subgroup of G .  A subsp ace X of a top ological group G top olo gic al ly gener ates G if G is the sm allest closed subgroup of G that con tains X . Remark 2.11 (i) Item (ii) of F act 1.3 can b e restated as follo ws: A qc-dense subset of a c omp act ab elian gr oup G top olo gic al ly gene r ates G . T herefore, for a subset X of a compact ab elian group G , one has the follo wing implications: X is dense in G − → X is qc-dense in G − → X top ologically generates G. (7) 1 This is an immediate consequence of [1 , Theorem 4.3 or Corollary 4.4]. In fact, a more general statement immediately follo ws from these results: Every dense subgroup D of a compact metric abelian group G contains a sequence conv erging to 0 whic h is qc-dense in G . 5 (ii) Th e first arro w in (7) cannot b e r eversed. In deed, tak e any qc-dense sequence S in T (see Lemma 4.5 for an example of such a sequence). C learly , S is not d ense in T . (iii) The last arrow in (7) cannot b e rev ersed either. Indeed, it follo ws from the resu lts in [10] that T c con tains a con vergi ng sequence (i.e., countably infi nite su p er-sequence) top o- logica lly generating T c . This sequence, how ev er, cannot b e qc-dense in T c b y Corollary 2.3. According to a wel l-known result of Hofmann and Morris [15, 16], ev ery compact group G con tains a sup er -sequ ence top ologically generating G . (See also [17] for a “purely top ological” pro of of th is result based on Mic hael’s selection theorem.) The emphasized text in Remark 2.11(i) allo ws us to conclude that T heorem 2.7 implies the particular case of the theorem of Hofmann and Morr is for compact ab elian groups G . As it wa s d emonstrated in Remark 2.11(iii), a (sup er-)sequence topologically generating a compact ab elian group G need not b e qc-dense in G . Therefore, the conclusion of our Theorem 2.7 is formal ly str onger than that of (the ab elian case of ) the r esult of Hofmann and Morr is. 3 Characterization of qc-dense subsets and determining sub- groups of compact ab elian group s in terms of C ( X , T ) W e refer the reader to (1) and Definition 2.1 for notations u sed in our n ext theorem. Theorem 3.1 F or a c lose d subset X of a c omp act ab e lian gr oup G the fol lowing c ond itions ar e e quivalent: (i) W ( X, U ) = { 0 } for some op en neighb ourho o d U of 0 in T ; (ii) r G X is an isomorphism b etwe en the top olo gic al gr oups b G and H = r G X ( b G ) . Pr o of . (i) → (ii) Let U b e as in (i). Since k er r G X ⊆ W ( X , U ) = { 0 } , w e conclude that r G X is an injection. Sin ce X is compact, { r G X (0) } = r G X ( { 0 } ) = r G X ( W ( X, U )) = H ∩ { g ∈ C ( X, T ) : g ( X ) ⊆ U } is an op en sub set of H . Sin ce H is a sub group of C ( X, T ), w e conclude that H is d iscrete. Therefore, r G X is an op en map on to its image. (ii) → (i) T he assump tion from item (ii) imp lies that H is a discrete su bgroup of C ( X, T ). Hence, we can find n ∈ N , compact sub sets K 0 , . . . , K n of X and op en neighbour ho o ds U 0 , . . . , U n of 0 in T s uc h that H ∩ \ i ≤ n { g ∈ C ( X, T ) : g ( K i ) ⊆ U i } = { r G X (0) } . (8) Define U = T i ≤ n U i . No w equation (8 ) yields W ( X , U ) = { 0 } .  Corollary 3.2 If a close d subset X of a c omp act ab elian gr oup G is qc-dense in G , then r G X is an isomorph ism b etwe en the top olo gic al gr oups b G and r G X ( b G ) . Pr o of . C h o ose an op en neigh b ourho o d U of 0 with U ⊆ T + . Since X is qc-dense in G , w e h av e W ( X , U ) ⊆ X ⊲ = { 0 } , and we can apply Theorem 3.1 to this U .  Corollary 3.3 L et X b e a close d subset of a c omp act ab elian gr oup G such that r G X is an isomorph ism b etwe en the top olo gic al g r oups b G and r G X ( b G ) . Then ther e exists n ∈ N such that the sum K n = ( X ∪ { 0 } ) + ( X ∪ { 0 } ) + . . . + ( X ∪ { 0 } ) of n many c opies of the set X ∪ { 0 } is (c omp act and) qc-dense in G . 6 Pr o of . Ap p ly Theorem 3.1 to find an op en neighbour ho o d U o f 0 in T as in item (i) of this theorem. Then apply F act 1.3(i) to obtain the r equired n ∈ N .  Corollary 3.4 F or a sub gr oup D of a c omp act ab elian gr oup G the fol lowing c onditions ar e e quivalent: (i) D determines G ; (ii) ther e exists a c omp act set X ⊆ D such that r G X is an isomorph ism b etwe en the top olo gic al gr oups b G and r G X ( b G ) . Pr o of . (i) → (ii) Since D determines G , there exists a compact set X ⊆ D w hic h is qc-dense in G (F act 1.6). Then r G X is an isomorphism b et we en the top ological groups b G and r G X ( b G ) (Corollary 3.2). (ii) → (i) Let X b e as in item (ii). Apply Corollary 3.3 to get n ∈ N and K n as in the conclusion of this corolla ry . Clearly , K n is a compact subs et of D . Since K n is qc-dense in G , D determines G b y F act 1.6.  4 Pro of of Theorem 2.7 W e start w ith a p artial inv erse of F act 1.4(ii). Lemma 4.1 Supp ose that f : G → H is a c ontinuous surje ctive homomorphism of c omp act ab elian gr oup s and X is a sub set of G such that X ∩ ker f is q c-dense in ker f . Then X is qc-dense in G if and only if f ( X ) is qc-dense in H . Pr o of . If X is qc-dense in G , then f ( X ) is qc-dens e in H by F act 1.4(ii). Assume that f ( X ) is qc-dense in H . Let χ ∈ X ⊲ . Since Q G ( X ) ⊇ Q G ( X ∩ ker f ) ⊇ Q k er f ( X ∩ ker f ) = k er f , one h as χ ∈ (k er f ) ⊲ . S ince ker f is a sub group and T + con tains no non-trivial sub groups, this yields that χ v anishes on k er f . Thus, χ factorizes as χ = ξ ◦ f , w here ξ ∈ b H . No w χ ∈ X ⊲ ob viously yields ξ ∈ f ( X ) ⊲ . As f ( X ) is qc-dense in H by th e hypothesis, this yields ξ ∈ f ( X ) ⊲ = { 0 } . Hence ξ = 0, and so χ = 0 as well. Therefore, X ⊲ = { 0 } , and thus X is qc-dense in G .  The follo wing d efinition is an adaptation to the ab elian case of [10, Definition 4.5]: Definition 4.2 L et { G i : i ∈ I } b e a family of ab elian top ologica l groups. F or every i ∈ I , let X i b e a su bset of G i . I d en tifying eac h G i with a s ubgroup of the direct pro d uct G = Q i ∈ I G i in the obvious wa y , d efi ne X = S i ∈ I X i ∪ { 0 } , where 0 is the zero elemen t of G . W e will call X the fan of the family { X i : i ∈ I } and w ill denote it by fan i ∈ I ( X i , G i ). The pro of of the follo wing lemma is straigh tforwa rd . Lemma 4.3 L et { G i : i ∈ I } b e a family of ab elian top olo gic al gr oups. F or every i ∈ I , let X i b e a se quenc e c onver ging to 0 in G i . Then fan i ∈ I ( X i , G i ) is a sup er-se quenc e in G = Q i ∈ I G i c onver ging to 0 . Lemma 4.4 L et { G i : i ∈ I } b e a family of ab elian top olo gic al gr oups. F or e ach i ∈ I let X i b e a qc-dense subset of G i . Then X = fan i ∈ I ( X i , G i ) is q c -dense in G = Q i ∈ I G i . 7 Pr o of . Let χ : G → T b e a non-trivial cont inuous c haracter. There exist a non-empt y fi nite subset J of I and a family { χ j ∈ c G j : j ∈ J } suc h that χ ( g ) = P j ∈ J χ j ( g ( j )) for g ∈ G . Since J 6 = ∅ , w e can fix j 0 ∈ J . Since X j 0 is qc-dens e in G j 0 , there exists x ∈ X j 0 ⊆ X suc h that χ j 0 ( x ) 6∈ T + . Finally , note that χ ( x ) = X j ∈ J χ j ( x ( j )) = χ j 0 ( x ( j 0 )) + X j ∈ J \{ j 0 } χ j ( x ( j )) = χ j 0 ( x ) + X j ∈ J \{ j 0 } χ j (0) = χ j 0 ( x ) 6∈ T + . Therefore, χ 6∈ X ⊲ . This giv es X ⊲ = { 0 } , and so X is qc-dense in G .  Our next th r ee lemmas are p articular cases of a general r esult of Außenh ofer quoted in the text preceding Theorem 2.7. Her p ro of relies on Arzela-Ascoli theorem and ind uctiv e constru ction, so the qc-dense sequence she constructs in her pro of is “generic”. T o k eep th is manuscript self- con tained, we provide “constructiv e” examples of “concrete” qc-dens e sequences in the circle group (Lemma 4.5), in the group of p -adic in tegers (Lemm a 4.6) and in th e du al grou p of the rationals equ ipp ed with the discrete top ology (Lemma 4.7). Lemma 4.5 L et T =  1 2 n : n ∈ N  ∪ { 0 } ⊆ R , and let ϕ : R → R / Z b e the natur al quotient map. Then ϕ ( T ) is a c onver ging to 0 se quenc e in T = R / Z that i s qc-dense in T . Pr o of . Let χ ∈ b T b e a non-zero c haracter. Then there exists m ∈ Z \ { 0 } such that χ ( x ) = m x for all x ∈ T . Let n = | m | . Then 1 2 n ∈ T and so x = ϕ  1 2 n  ∈ ϕ ( T ). Moreo v er, χ ( x ) = mx = ϕ  m 2 n  6∈ T + , w hic h sho ws that χ 6∈ ϕ ( T ) ⊲ . Hence ϕ ( T ) ⊲ = { 0 } , and so ϕ ( T ) is qc-dens e in T .  Lemma 4.6 F or every prime numb er p , the gr oup Z p of p -adic inte gers c ontains a qc-dense se quenc e c onver ging to 0 . Pr o of . Recall that the family { p n Z p : n ∈ N } consisting of clop en s u bgroups of Z p forms a basis of neigh b our ho o ds of 0. Therefore, S = { k p n : n ∈ N , 1 ≤ k ≤ p − 1 } ∪ { 0 } is a sequence con ve rging to 0 in Z p . Let u s sho w that S is q c-den s e in Z p . T o this end tak e a non-zero charac ter χ : Z p → T . Since Z p is a zero-dimensional compact group, its image χ ( Z p ) under the con tin uous homomorp hism χ must b e a closed zero-dimensional sub group of T . In particular, χ ( Z p ) 6 = T . Being a p r op er closed subgroup of T , χ ( Z p ) m us t b e finite. It follo ws that k er χ is a clop en subgroup of Z p , and so there exists n ∈ N such th at k er χ = p n Z p . Hence χ ( Z p ) ∼ = Z p /p n Z p ∼ = Z ( p n ). Th erefore, χ (1) = m p n for some m coprime to p . C ho ose k with 1 ≤ k ≤ p − 1 and such that: (a) k m ≡ p − 1 2 ( mod p ), if p > 2; (b) k = 1 if p = 2. Then χ ( k p n − 1 ) = p − 1 2 p 6∈ T + , in case (a). Otherwise, χ (2 n − 1 ) = 1 2 6∈ T + , in case (b). In b oth cases, χ ( k p n − 1 ) 6∈ T + , so χ 6∈ S ⊲ . This pro ves that S ⊲ = { 0 } . Therefore, S is qc-dense in Z p .  Lemma 4.7 L et Q b e the gr oup of r ational numb e rs with the discr ete top olo gy. Then b Q c ontains a qc-dense se quenc e c onver gi ng to 0 . Pr o of . By L emm a 4.6, for ev ery p rime num b er p the group Z p con tains a qc-den s e sequence S p con v erging to 0. By Lemma 4.4, S = fan p ∈ P ( S p , Z p ) is qc-dense in K = Q p ∈ P Z p . In view of Lemma 4.3 , S is a sequence con v erging to 0 in K . Define v = { 1 p } p ∈ P ∈ K , w here eac h 1 p is the ident it y of Z p . Let N = R × K and u = (1 , v ) ∈ N . Then the cyclic subgroup h u i of N is discrete and G = N / h u i is isomorphic to b Q [9, § 2.1], so we w ill iden tify b Q with the quotien t G = N / h u i . Define H = N/ ( Z × K ) 8 and note that H ∼ = T . Let θ : N → G = N / h u i and ψ : N → H b e the (con tin uous) quotient homomorphisms. Since ke r θ = h u i ⊆ Z × K = k er ψ , there exists a (unique) con tin uous surjectiv e group homomorphism f : G → H such that ψ = f ◦ θ . In particular, k er f = θ (k er ψ ) = θ ( Z × K ). Since Z × K = h u i + ( { 0 } × K ), w e hav e k er f = θ ( Z × K ) = θ ( h u i + ( { 0 } × K )) = θ ( h u i ) + θ ( { 0 } × K ) = θ ( { 0 } × K ) . (9) Let T and ϕ b e as in Lemma 4.5. Since T is a s equ ence con v erging to 0 in R , and S is a sequence conv erging to 0 in K , it f ollo ws that X = θ ( T × { 0 } ) ∪ θ ( { 0 } × S ) is a sequence con v erging to 0 in G . Note that ϕ ( T ) = ψ ( T × { 0 } ) = f ( θ ( T × { 0 } )) is qc-dense in f ( G ) = H ∼ = T b y L emm a 4.5. Since θ ( T × { 0 } ) ⊆ X , w e conclude that f ( X ) is qc-dense in H as w ell. S in ce S is qc-dense in K , θ ( { 0 } × S ) is qc-dense in θ ( { 0 } × K ) = ker f b y F act 1.4(ii) and (9). F rom θ ( { 0 } × S ) ⊆ X , w e conclude that X ∩ ker f is qc-dens e in ker f . It no w f ollo ws from Lemma 4.1 that X is qc-dense in G .  The next lemma is probably kno wn , bu t we includ e its pro of f or the reader’s conv enience. Lemma 4.8 Every infinite c omp act ab elian gr oup of weight κ is isomorph ic to a quotient gr oup of the gr oup b Q κ × Q p ∈ P Z κ p . Pr o of . Let H b e an infin ite compact ab elian group su c h that w ( H ) = κ . C learly , κ is infinite and X = b H is a discrete ab elian group of size κ [14, Theorem (24.15) ]. Let Y = X ⊕ L κ ( Q ⊕ Q / Z ). By [12, Theorem 24 .2] there exists a divisible abelian group D conta ining Y s uc h that no pr op er subgroup of D contai nin g Y is divisib le. According to the text immediately follo wing [12, Theorems 24.2], r 0 ( D ) = r 0 ( Y ) and r p ( D ) = r p ( Y ) for every prime p , w here r 0 ( N ) and r p ( N ) denote the free-rank and th e p -rank of an ab elian group N , resp ectiv ely (see, for example, [12, § 16]). Since r 0 ( Y ) = κ and r p ( Y ) = κ for every pr ime p , we conclude that D ∼ = M κ ( Q ⊕ Q / Z ) ∼ = M κ Q ! ⊕ M p ∈ P M κ Z ( p ∞ ) ! (10) b y the structure theorem for divisible ab elian group s (see [12, Th eorem 23.1] ). Consider the compact group G = b D . By (10), G ∼ = b Q κ × Q p ∈ P Z κ p . According to [14, Theorem (24. 5)], H ∼ = b X ∼ = G/X ⊥ , where X ⊥ = { χ ∈ b D : χ ( X ) = { 0 }} . Therefore, H is isomorphic to a qu otient group of b Q κ × Q p ∈ P Z κ p ∼ = G .  Pro of of Theorem 2.7 : Let H b e an infinite compact ab elian group. Define κ = w ( H ) and G = b Q κ × Q p ∈ P Z κ p . By Lemma 4.8 there exists a sur jectiv e contin uous h omomorphism f : G → H . Clearly , G = Q i ∈ I G i , where | I | = κ and eac h G i is either b Q or Z p for a su itable p ∈ P . By Lemmas 4.7 and 4.6, for ev ery i ∈ I there exists a sequence X i con v erging to 0 whic h is qc-dense in G i . Applyin g L emmas 4.3 and 4.4, w e conclude that X = fan i ∈ I ( X i , G i ) is a s up er-sequ en ce in G conv erging to 0 such that X is qc-dense in G . Sin ce f : G → H is a s urjection, S = f ( X ) is qc-dense in H by F act 1.4(ii). Being the image of a sup er-sequence X in G con v erging to 0, S is s u p er-sequence in H con v erging to 0 [10, F act 4.3]. Finally , | S | ≤ | X | ≤ ω · | I | = κ .  5 Pro ofs of F acts 1.1, 1.3, 1.4 , 1.6 and 1.8 Pro of of F act 1.1 : Ind eed, let B and C b e bases for X and Y resp ectiv ely such that |B | ≤ w ( X ) and |C | ≤ w ( Y ). It suffices to sho w that the family E = { [ V , U ] : V ∈ B , U ∈ C } is a su b base 9 for the top ology of C ( X , Y ), since th is wo uld imply w ( C ( X, Y )) ≤ |B × C | ≤ w ( X ) + w ( Y ) + ω . Indeed, assum e that f ∈ C ( X, Y ), K is a compact sub set of X , O is an op en subset of Y and f ∈ [ K, O ]. Let x ∈ K . Sin ce f ( x ) ∈ O and C is a b ase for Y , there exists U x ∈ U suc h that f ( x ) ∈ U x ⊆ O . Using con tin uit y of f and regularit y of X one can find V x ∈ B such that f ( V x ) ⊆ U x . Since K is compact, K ⊆ S x ∈ F V x for some fi nite subs et F of K . Now f ∈ T x ∈ F [ V x , U x ] ⊆ [ K , O ].  Pro of of F act 1.3 : (i) Th ere exists n ∈ N s uc h that V n = { x ∈ T : k x ∈ T + for all k = 1 , 2 , . . . , n } ⊆ U. (11) Let χ ∈ K ⊲ n . Fix x ∈ X . L et k = 1 , 2 , . . . , n b e arbitrary . Since 0 ∈ X ∪ { 0 } , on e has k x ∈ K n , and so k χ ( x ) = χ ( k x ) ∈ T + . This yields χ ( x ) ∈ V n ⊆ U b y (11). Since x ∈ X was chose n arbitrarily , it follo ws that χ ∈ W ( X, U ). S ince W ( X , U ) = { 0 } , this give s χ = 0. Therefore, K ⊲ n = { 0 } , and so K n is qc-dense in G . (ii) Sup p ose that th e smallest subgroup N of G conta ining X is not dense in G . Th en we can c ho ose χ ∈ b G s uc h that χ ( N ) = { 0 } and χ ( y ) 6 = 0 for some y ∈ b G \ N . S o χ ∈ W ( X , U ) and y et χ 6 = 0, in con tradiction with our assumption.  Pro of of F act 1.4 : (i) Assum e that x ∈ Q G ( X ) b ut f ( x ) 6∈ Q H ( f ( X )). Th en there exists ξ ∈ b H s uc h th at ξ ( f ( X )) ⊆ T + and ξ ( f ( x )) 6∈ T + . Th en χ = ξ ◦ f ∈ b G and χ ( X ) ⊆ T + , wh ile χ ( x ) 6∈ T + . Therefore, x 6∈ Q G ( X ), a cont radiction. (ii) By our assu m ption, Q G ( X ) = G . Therefore, f ( G ) = f ( Q G ( X )) ⊆ Q H ( f ( X )) by item (i). Since Q H ( f ( X )) must b e closed in H and f ( G ) is dense in H , this yields Q H ( f ( X )) = H , that is, f ( X ) is qc-dense in H .  Pro of of F act 1.6 : Since b G is d iscr ete, (i) is equiv alen t to b D b eing discrete. Since b D carries the compact op en top ology , this is equ iv ale nt to ha ving W ( K, U ) = { 0 } for an app r opriate pair of a compact subset K of D and an op en neigh b orho o d U of 0 in T . Ha ving this in mind, we are going to p ro ve that (i) and (ii) are equ iv alent. (ii) → (i) Su pp ose that K is a compact subset of D that is qc-dense in G . T ak e an op en neigh b our ho o d of 0 in T with U ⊆ T + . Then W ( K, U ) ⊆ K ⊲ = { 0 } , wh ic h giv es W ( K , U ) = { 0 } . Th us , (i) holds . (i) → (ii) By our assum ption, there exist a compact su bset X of D and an op en neigh b orho o d U of 0 in T su c h that W ( X , U ) = { 0 } . Let K n b e as in F act 1.3 . Then K n ⊆ D is compact and qc-dense in G .  Pro of of F act 1.8 : Let D b e a dense subgroup of H . Th en f − 1 ( D ) is a d en se su bgroup of G . Since G is determined, by F act 1.6 w e can find a compact s ubset X of f − 1 ( D ) that is qc-dense in G . Then f ( X ) is a compact subset of D wh ic h is qc-dense in H by F act 1.4(ii). Applying F act 1.6 once again, we conclude that D determines H .  Ac kno wledgemen t. T he authors would lik e to thank Lydia Au ßenhofer for attracting their atten tion to Theorem 4.3 and Corollary 4.4 of her man uscrip t [1], and G´ ab or Luk´ acs for nu- merous commen ts and h elpful suggestions. T he author’s collab oration on this man uscript has started d uring the 49th W orksh op “Adv ances in Set-Theoretic T op ology: Conf er en ce in Hon- our of Tsugunori Nogura on his 60th Birthday” of the In ternational S c ho ol of Mathematics “G. S tamp acc hia” held on Jun e 9–19, 2008 at the Center for S cien tific Cu lture “Ettore Ma jo- rana” in Erice, S icily (Italy). The authors w ould like to express their warmest gratitude to the Ettore Ma jorana F oundation and C en ter for Scien tific Cu lture for pr o viding excellen t conditions whic h insp ired this researc h endeav or. 10 References [1] L. A ußenhof er , C on tributions to th e dualit y theory of ab elian topological groups and to the theory of nucle ar groups, Dissertationes Math. 384 (1999), 113 pp. [2] W. 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