Super-sequences in the arc component of a compact connected group

SUPER-SEQUENCE S IN T H E A R C COMPONE NT OF A COMP A CT CONNECTED GR OUP DIKRAN DIKRANJAN AND DMITRI SHAKHMA TOV De di c a te d to Karl H. Hofmann on the o c c asion of his 76th anniversary Abstract. Let G be an abelia n top ologica l group. The symbol b G denotes the group of all contin uous characters χ : G → T endow ed with the co mpact op en top olog y . A subset E of G is said to b e qc-dense in G provided that χ ( E ) ⊆ ϕ ([ − 1 / 4 , 1 / 4]) holds only for the trivial character χ ∈ b G , where ϕ : R → T = R / Z is the canonical homomorphism. A sup er-se quenc e is a non-e mpty compa ct Hausdorff spa c e S with at mos t one non-iso lated p oint (to which S c onver ges ). W e pro ve that an infinite compact abelian gr oup G is connected if and only if its arc co mpo nent G a contains a sup er-seq uence conv erging to 0 that is qc-dense in G . This gives as a corolla r y a recent theore m of Außenhofer: F o r a connected lo cally compact ab elian gro up G , the restr ic tio n homomorphism r : b G → b G a defined by r ( χ ) = χ ↾ G a for χ ∈ b G , is a top ologic al iso morphism. W e also show that a n infinite compac t gro up G is connected if and only if its ar c comp onent G a contains a sup er- sequence S converging to the ident it y e that generates a dense subgr oup o f G (equiv a lently , S \ { e } is a n infinite suitable set for G in the sense of Ho fmann a nd Morr is). 1. Intr oduction A l l top olo gic al gr oups ar e assume d to b e Hausdorff . Let G b e a top olog ical group. W e denote b y b G the group of all contin uous characters χ : G → T endo w ed with the compact op en to p ology . A subgroup D of G determines G if the restriction homomorphism r : b G → c D defined by r ( χ ) = χ ↾ D for χ ∈ b G , is a top ological isomorphism [5]. If G is lo cally compact and ab elian, then ev ery s ubgroup D that determines G m ust b e dense in G . (When D is dense in G , the map r : b G → c D is a con tin uous isomorphism.) The follow ing t w o theorems are the cornerstone results in t he topic of determining subgroups: Theorem 1.1. [1, 4] A m e triza b l e ab elian gr oup G is determine d by e ach dense sub gr oup o f G . Theorem 1.2. [12] Every non-metrizable c omp act gr oup G c ontains a dense sub gr oup that do es not determine G . 1 According to a w ell-known classical resu lt of Eilen b e rg and P o n try agin, in a connected lo cally compact ab elian group G t he arc comp onen t G a is dense. Since a subgroup o f a 1991 Mathematics Subje ct Classific ation. Prima ry: 22 C05; Secondary : 22B05 , 5 4 D05. Key wor ds and phr ases. compact group, ab elian group, connected space, path wise connected space, arc comp onent, sup er -sequence, suitable set, dual gro up. The first author w as pa rtially s uppo rted by MEC. MTM2006 -0203 6 and FEDER FUNDS. The sec o nd author was par tially supp or ted by the Grant-in-Aid for Scientific Research no. 1 9540 0 92 by the J apan So cie t y for the Promotion o f Science (JSPS). 1 A pr o of o f this theo rem under the assumption of CH can b e found in [5]. 1 2 D. DIKRAN JAN AND D. SH AKHMA TOV lo cally compact abelian group determining it must b e dense, the f o llo wing theorem, r ecen tly pro v ed b y Außenhofer, is a strengthening of this classical result: Theorem 1.3. [3] The ar c c omp onent G a of a c onne cte d lo c al ly c omp ac t ab elian gr oup G determines G . While this theorem is a corollary of Theorem 1.1 for a metrizable g roup G , in the non- metrizable case t he mere densit y of G a in G ensured b y the classical result of Eilen b erg and P o ntry agin men tioned abov e, need not guarantee that G a determines G (as witnessed b y Theorem 1.2). Theorem 1.3 is used in [2] to pro v e that the unc oun table pow ers of Z are no t strongly reflexiv e, thereby resolving a problem raised b y Banaszczyk on whether uncoun table p o w ers of the reals R are strongly reflexiv e. Let ϕ : R → T = R / Z b e the canonical ho momorphism and T + = ϕ ([ − 1 / 4 , 1 / 4]). W e will sa y that a subset E of a top ological group G is qc-de nse in G (a n abbreviation for quasi- c onvexly dense ) pr ovided that χ ( E ) ⊆ T + only f o r the trivial contin uous homomorphism χ : G → T . This notion was in tro duce d in [6] in the a b elian con text, and its significance for applications has b een recen tly demonstrated in [10]. In particular, qc-densit y was used in [10] to establish essen tial prop erties of determining subgroups of compact ab elian gro ups, thereb y allowing to get a short elemen tar y pro of of Theorem 1.2. The host of applications of qc-dense sets is made p ossible b y the ultimate connection b et w een the notions of determining subgroup and qc-densit y describ ed in the next fact (pro v ed in [10, F act 1.4]). It is a particular case of a mor e general fa ct stated without pro of (and in equiv alen t terms) in [5, Remark 1.2 ( a)] and [12, Corollary 2.2]. F act 1.4. A sub gr oup D of a c omp act ab elian gr oup G determines it if and onl y if ther e exists a c omp act subset o f D that is qc-de nse in G . It has b een recen tly sho wn in [10] tha t qc-dense compact subsets (and th us determining subgroups) of a compact ab elian group m ust b e rather big. Theorem 1.5. [10, Corollar y 2.2] If a cl o s e d subset X of an infinite c o m p act ab elian gr oup G is qc-dense in G , then w ( X ) = w ( G ) . (Her e w ( X ) denotes the weight of a s p ac e X .) A sup er-se quenc e is a non-empt y compact Hausdorff space X with a t most one non-isolated p oin t x ∗ [9]. W e will call x ∗ the limit of X and say that X c o n ver ges to x ∗ . Observ e tha t a coun t a bly infinite sup er-sequence is a con v ergent sequence (together with its limit). Außenhofer [1] essen t ia lly pro v ed that ev ery infinite compact metric ab elian group has a qc-dense seq uence con v erging to 0. 2 This result has b een recen tly extended to all compact groups by replacing con v ergent sequences with sup er-sequences : Theorem 1.6. [10] Every in fi nite c o m p act ab elia n gr oup c ontains a qc-dense sup er-se quenc e c onver ging to 0 . A subspace X of a top olo gical group G top olo gic al ly gener ates G if the subgroup of G generated b y X is dense in G . The pro of of the follo wing fact is stra ig h tforw a rd. 2 This is an immediate co ns equence of [1 , Theorem 4.3 or Cor ollary 4.4 ]. In fact, a more genera l statement immediately follows from these results: Ev ery dense subgr oup D of a compact metr ic ab elia n gr oup G contains a seque nce conv erging to 0 that is qc-dense in G . SUPER-SEQU ENCES IN THE A RC COMPONENT 3 F act 1.7. [10, F act 1.3(ii)] Every qc-dense subset of a c om p act ab elian gr oup top olo gic al ly gener ates it. Let G b e a to p ological group with the iden tit y e . If a discrete subs et S of G top ologically generates G and S ∪ { e } is closed in G , then S is called a suitable set f o r G [13]. Remark 1.8. Clearly , if S is a sup er-sequence in G that conv erges to e and top ologically generates G , then S \ { e } is a suitable set for G . C on v ersely , if G is compact and S is a suitable set f o r G , then S ∪ { e } m ust b e a sup er-sequence. It follow s from this remark that a subgroup D of a compact gr o up G con tains a sup er- sequence con v erging to the iden tit y that top olo gically generates G if and only if D con tains an infinite suitable set for G . Hofmann and Morris disco ve red the following fundamen ta l result: Theorem 1.9. ([1 3]; see also [14]) Every c omp act gr o up has a suitable set. 3 Remark 1.10. (i) The or em 1.6 impli e s the p articular c ase of The or em 1.9 fo r ab elian gr oups. Indeed, let G b e a compact ab elian group. If G is finite, then G is discrete, and so G itself is a suitable set for G . Ass ume now that G is infinite. By Theorem 1.6, G con tains a qc-dense sup er-sequence S conv erging to 0. By F act 1.7 , S top olo gically generates G . According to R emark 1.8, S \ { 0 } is a suitable set for G . (ii) A suitable set for a c om p act ab elian gr oup G ne e d not b e qc-dens e in G . Indeed, it is w ell-kno wn that the group T c is monot hetic, that is , top olo gically generated by a singleton S . Cle arly , S is a suitable set for T c . Sinc e w ( S ) ≤ ω < c = w ( T c ), S cannot b e qc-dense in T c b y Theorem 1 .5. (iii) It follo ws from item (ii) that the p articular c a s e of Th e or em 1.9 for ab elian gr oups do es not imply The or em 1.6 . 2. Resul ts Our first result is a particular v ersion of Theorem 1.6 t ha t characterize s connected compact ab elian g roups. Theorem 2.1. F or an infinite c omp act ab elian gr oup G the fol low ing c ond itions ar e e quiv- alent: (i) the ar c c omp onent G a of G c ontains a sup er-se quenc e c on ver ging to 0 that is qc-den s e in G ; (ii) G is c o nne cte d. In view o f F act 1.4, the implication (ii) → (i) of Theorem 2.1 yields Theorem 1.3 when G is compact. The g eneral case of Theorem 1.3 easily follow s from the compact case, see the pro of in the end of Section 4. Therefore, as a b y-pro duct, our pro of of Theorem 2.1 also pro vides an alt ernat ive short and self-contained pro of of the theorem of Außenhofer. Remark 2.2. In view of Theorems 1.6 and 2.1, as w ell a s Außenhofer’s result cited in fo otnote 2, the reader ma y w onder if ev ery dense subgroup o f a compact ab elian group G con t ains a sup er-sequence conv erging to 0 that is qc-dense in G . The answ er to this question is negativ e: Every non-metriza b l e c om p act ab elian gr oup G c ontains a dense sub gr oup H such that no sup er-se q uen c e S ⊆ H is q c-dense in G . Indeed, apply Theorem 1.2 to get a 3 A “ pur ely top olo gical” pro of of this r esult based on Michael’s sele c tion theore m can b e found in [16]. 4 D. DIKRAN JAN AND D. SH AKHMA TOV dense subgroup H of G that do es not determine G . If S ⊆ H is a super-sequence, then (b eing compact) S cannot b e qc-dense in G b y F act 1.4. Our second result is a particular v ersion of Theorem 1.9 that c har a cterizes connected compact groups: Theorem 2.3. F or an infinite c omp act gr oup G the fol lowing c ondi tion s ar e e quivalent: (i) the ar c c omp onent G a of G c ontains a suitable set for G ; (ii) G a c ontains an in fi nite suitable set for G that is qc-dense in G ; (iii) G is c o nne cte d. Remark 2.4. One c annot add the follow ing item to the list of equiv alen t conditions in Theorem 2.3: (iv) G a con t ains a sup er- sequence con v erging to the identit y that is qc-dense in G . Indeed, for ev ery finite simple non-comm utativ e group L the compact group G = T × L has a sequence S ⊆ G a = T × { e } con v erging to the identit y (0 , e ) o f G that is qc-dens e in G , see Example 5 .3. Since G is not connected, the implication (iv) → (iii) fails. Remark 2.5. Let c denote the cardinality of the con tin uum. Ther e exists a dense (c on ne cte d, lo c al ly c onn e cte d, c ountably c o m p act) sub g r oup H of ( the c o mp act, c onne cte d ab e lian gr o up) G = T 2 c such that H c ontains no suitable set for G . Indeed, one can tak e as H the dense subgroup o f G without a suitable set f or H constructed in [11, Corolla ry 2 .9]. Assume tha t S ⊆ H is a suitable set for G . Then S is discrete and S ∪ { 0 } is closed in G . Since S ∪{ 0 } ⊆ H , it follows that S is closed in H . Since S top o logically generates G , it top olog ically g enerates H as w ell. Hence, S is a suitable set for H , a con tradiction. 3. A qc-dense super-s equence in the ar c component of b Q Our main result in this section is Lemma 3.4. It follo ws from the density of b Q a in b Q and the general result of Außenhofer quoted in the fo otnot e 2. Ho wev er, Außenhofer’s pro of relies on Arzela-Ascoli theorem a nd an inductiv e construction, so the qc-dense sequenc e she constructs in her pro of is “generic”. T o kee p this man uscript self-con tained, w e provide a “constructiv e” example of a “concrete” qc-dense sequence in Q a . The pro of o f the following fact is straigh tforw ard from the definition. F act 3.1. L et G and H b e top olo g i c al gr oups an d π : H → G a c on tinuous surje ctive gr oup homomorphism . If a subset E of H is qc-dense in H , then π ( E ) is qc-dens e in G . In the sequel N denotes the set of natural n umbers. Example 3.2. Let T = n 1 2 n : n ∈ N , n ≥ 1 o ∪ { 0 } . The set ϕ ( T ) is a qc-dense se quenc e in T c onver g ing to 0. Indeed, let χ ∈ b T b e a non-zero c haracter. Then there exists m ∈ Z \ { 0 } suc h that χ ( x ) = mx fo r all x ∈ T . Let n = | m | . Then 1 2 n ∈ T a nd so x = ϕ  1 2 n  ∈ ϕ ( T ). Since χ ( x ) = mx = ϕ  m 2 n  = ϕ  1 2  6∈ T + , w e hav e χ ( ϕ ( T )) \ T + 6 = ∅ . This pro v es that ϕ ( T ) is qc-dense in T . F or g ∈ G the sym b ol h g i denotes the cyclic subgroup of G generated by g . SUPER-SEQU ENCES IN THE A RC COMPONENT 5 Lemma 3.3. L et P = { p n : n ∈ N } b e a faithful enumer ation of the set P of prime numb ers. Define H = Y n ∈ N Z p n , and let v = { 1 p n } n ∈ N ∈ H , wher e e ach 1 p n is the iden tity of Z p n . F or n ∈ N defin e k n = ( p 0 p 1 . . . p n − 1 ) n . Then the se t (1) S = { mk n v : n ∈ N , m ≤ k n +1 } ∪ { 0 } ⊆ h v i is a se quenc e c onver ging to 0 that is qc-de n se in H . Pr o of. F or n ∈ N define (2) W n = k n H = p n 0 Z p 0 × p n 1 Z p 1 × . . . × p n n − 1 Z p n − 1 × ∞ Y i = n Z p i . (Note that k 0 = 1.) Then { W n : n ∈ N } forms a base of H at 0 consisting of clop en subgroups. It is easy to see that eac h W n ma y miss only finitely man y mem b ers of S , so S is a sequence con v erging to 0 in H . Let us sho w that S is qc-dense in H . Let χ ∈ c H and χ 6 = 0. W e need to pro v e that χ ( S ) \ T + 6 = ∅ . Being a contin uous homomorphic imag e of the compact totally disconnected group H , χ ( H ) is a closed totally disconnected subgro up of T . Therefore, χ ( H ) m ust b e finite. Hence ke r χ is an o p en subgroup of H , a nd consequen tly it contains a subgroup W n for some n ∈ N . Without loss of generality w e will a ssume that (3) n = min { m ∈ N : W m ⊆ ke r χ } . Since k er χ 6 = H = W 0 b y our assumption, we ha v e n ≥ 1, a nd so n − 1 ∈ N . Claim : χ ( k n − 1 v ) 6 = 0 . Pr o of. As sume the con tr ary . Then χ ↾ h k n − 1 v i = 0. Since h v i is dense in H and W n − 1 is a n op en subset of H , it follo ws that h v i ∩ W n − 1 = h k n − 1 v i is dense in W n − 1 . No w from χ ↾ h k n − 1 v i = 0 and con tinuit y of χ w e conclude that χ ↾ W n − 1 = 0. This give s W n − 1 ⊆ ke r χ , in contradiction with (3).  Since k n v ∈ W n ⊆ k er χ b y (2) and (3), w e hav e k n χ ( v ) = χ ( k n v ) = 0 . That is, h χ ( v ) i is a cyclic g roup of order at most k n . Since χ ( k n − 1 v ) = k n − 1 χ ( v ) ∈ h χ ( v ) i , the order of the elemen t χ ( k n − 1 v ) of T is also at most k n . Since χ ( k n − 1 v ) 6 = 0 b y claim, w e can choose an integer m ≤ k n suc h that χ ( mk n − 1 v ) = mχ ( k n − 1 v ) 6∈ T + . F rom (1) we conclude that mk n − 1 v ∈ S , and so χ ( S ) \ T + 6 = ∅ .  An explicit qc-dense sequence in b Q con v erging to 0 can b e found in [1 0, Lemma 4.7 ]. Ho w ev er, that sequenc e is not contained in b Q a . In our next lemma w e produce a qc-dens e sequence con v erging to 0 inside b Q a . Lemma 3.4. b Q a c ontains a se quenc e c onver ging to 0 that is qc-dense in b Q . Pr o of. W e contin ue using notations from Lemma 3 .3. Let K = R × H and u = (1 , v ) ∈ K . Then the cyclic subgroup h u i of K is discrete a nd the quotien t g r o up C = K/ h u i is isomorphic to b Q [7, § 2.1]. Therefore, it suffices to prov e that C a con t ains a sequence con v erging to 0 that is qc-dense in C . 6 D. DIKRAN JAN AND D. SH AKHMA TOV Let π : K → C = K / h u i b e the quotien t homomorphism. Since π is a lo cal ho meomor- phism, ev ery contin uous map f : [0 , 1] → C with f (0) = 0 C can b e lift ed to a con tin uous map e f : [0 , 1] → K with e f (0) = 0 K and π ◦ e f = f . (A mor e general statemen t can b e fo und in [15, Lemma 1].) Th erefore, C a = π ( K a ). Since H is zero- dimensional and R × { 0 } is arcwise connected, one has K a = R × { 0 } , a nd so C a = π ( R × { 0 } ). Define N = π ( { 0 } × H ), a nd let f : C → C / N b e t he quotient homomorphism. By Lemma 3.3 and F act 3.1, there exists a con v erging to 0 sequence S ′ in the subgroup h π ( 0 , v ) i of N suc h that S ′ is qc-dense in N . As π (0 , v ) = π (( − 1 , 0) + ( 1 , v )) = π ( − 1 , 0) + π ( u ) = − π (1 , 0) ∈ π ( R × { 0 } ) = C a , one has S ′ ⊆ π ( h (0 , v ) i ) ⊆ C a . With T = n 1 2 n : n ∈ N , n ≥ 1 o define S ′′ = π ( T × { 0 } ) ⊆ π ( R × { 0 } ) = C a . Clearly , S ′′ is a sequenc e con v erging to 0. Since C / N ∼ = K/ ( Z × H ) ∼ = T and the comp osed isomorphism C / N → T sends f ( S ′′ ) to ϕ ( T ), from Example 3.2 w e conclude that f ( S ′′ ) is qc-dense in C / N . Since S ′ and S ′′ are sequence s conv erging to 0 in C , so is X = S ′ ∪ S ′′ . By our construction, X ⊆ C a . So it r emains only to prov e tha t X is qc-dense in C . Sup p ose that χ ∈ b C and χ ( X ) ⊆ T + . Since χ ↾ N ∈ c N , χ ↾ N ( S ′ ) = χ ( S ′ ) ⊆ χ ( X ) ⊆ T + and S ′ is qc-dense in N , we ha v e χ ↾ N = 0. Therefore, χ = ξ ◦ f for some ξ ∈ [ C / N . In particular, ξ ( f ( S ′′ )) ⊆ ξ ( f ( X )) = χ ( X ) ⊆ T + . Since f ( S ′′ ) is q c-dense in C / N , it follows that ξ = 0. This giv es χ = 0. Therefore, X is qc-dense in C .  4. Pro of of Theorems 2.1 and 1.3 The follow ing definition is an adapta tion to the ab elian case of [9, Definition 4.5]: Definition 4.1. Let { G i : i ∈ I } b e a family of ab elian top o logical g r o ups. F or eve ry i ∈ I let X i b e a subset of G i . Identifying eac h G i with a subgroup of the direct pro duct G = Q i ∈ I G i in the ob vious w ay , define X = S i ∈ I X i ∪ { 0 } , where 0 is the zero elemen t of H . W e will call X the fan of the family { X i : i ∈ I } and will denote it b y fan i ∈ I ( X i , G i ). The pro of o f the following lemma is straightforw ard. Lemma 4.2. [10, Lemmas 4.3 and 4.4] L et { G i : i ∈ I } b e a family of ab elian top olo gic al gr oups, and let G = Q i ∈ I G i . F or every i ∈ I let X i b e a subset of G i , and let X = fan i ∈ I ( X i , G i ) . Then: (i) if X i is a se quenc e c onver ging to 0 in G i , then X is a s up er-se quenc e in G c onver g i ng to 0 . (ii) if X i is a qc - dense s ubse t of G i for e ach i ∈ I , then X is qc-dense in G . Lemma 4.3. F or ev e ry c ar dinal κ ther e e x i s ts a sup er-se quenc e S ⊆ ( b Q κ ) a c onver ging to 0 that is qc-de n se in b Q κ . Pr o of. W rite b Q κ as b Q κ = Q α<κ G α , where G α is the α ’s copy of b Q . By Lemma 3 .4, for ev ery α ∈ κ there is a sequence S α in ( G α ) a con verging to 0 that is qc-dense in G α . By Lemma 4.2(i), S = fan α ∈ κ ( X α , G α ) is a sup er- sequence in b Q κ con verging t o 0. By Lemma 4.2(ii), S is qc-dense in b Q κ . Finally , note that S ⊆ L α ∈ κ ( G α ) a ⊆ ( b Q κ ) a .  SUPER-SEQU ENCES IN THE A RC COMPONENT 7 Pro of of Theorem 2.1. (i) → (ii) Let S ⊆ G a b e a sup er-sequence that is qc-dense in G . Then the subgroup H of G generated by S is dense in G by F act 1.7. Since H ⊆ G a , it follo ws that G a is dense in G as w ell. Th us G is connected. (ii) → (i) There exits a contin uous surjectiv e ho momorphism π : b Q κ → G fo r some cardinal κ (see, for example, the pro of of [10, Theorem 3.3]). Let S b e as in the conclusion of Lemma 4.3. Since S is qc-dense in b Q κ , π ( S ) is qc-dense in G by F act 3.1. Since a finite set cannot b e qc-dense in an infinite compact gro up ([1]; this also f o llo ws fro m Theorem 1.5), π ( S ) m ust b e infinite. Since S is a sup er-sequence con ve rging to 0 suc h that π ( S ) is infinite, π ( S ) m ust b e a super-sequence con ve rging to 0 b y [9, F a ct 4.3] (see [16, F act 12] fo r the pro of ). Finally note that π ( S ) ⊆ π ( b Q κ a ) ⊆ G a .  Pro of of Theorem 1.3. W e hav e G = R n × K , where K is a compact connected group [8]. Since R n is arcwise connected, one has G a = R n × K a . F rom Theorem 2.1 and F a ct 1.4 w e conclude that K a determines K . Hence G a = R n × K a determines G = R n × K .  5. Pro of of Theorem 2.3 In the sequel we denote b y H ′ the comm utator subgroup of a group H . Our next lemma sho ws that qc-densit y can b e essen tially studied in the a b elian context. Lemma 5.1. L et H b e a top olo gic al gr oup, and let G d e note the quotient H/ H ′ , wher e H ′ is the closur e o f H ′ in H . L et π : H → G denote the c anonic al map. Then a subset E of H is qc-dens e in H if and only if π ( E ) is q c-dense in G . Pr o of. T he “o nly if ” pa rt follow s from Lemma 3.1. T o pro v e the “if ” part, assume tha t π ( E ) is qc-dense in G , and let χ : H → T b e a con tin uous ho momorphism suc h that χ ( E ) ⊆ T + . Since T is ab elian and Hausdorff, H ′ ⊆ k er χ , so χ = ξ ◦ π fo r some character ξ : G → T . Since ξ ( π ( E )) = χ ( E ) ⊆ T + and π ( E ) is qc-dense in G , w e conclude that ξ is trivial, and so χ is trivial to o. This prov es tha t E is qc-dense in H .  Recall that a g r o up L is called p erfe ct if L ′ = L . Corollary 5.2. L et L b e a p erfe c t top olo gi c al gr oup, G an ab elian top o l o gic al gr oup and H = G × L . Then a s ubse t E of G i s qc-dense in G if and only if the subset E × { e L } of the gr oup H is qc-den se in H . Pr o of. Sinc e H ′ = { 0 G } × L , we hav e H ′ = H ′ and G ∼ = H /H ′ = H/ H ′ . No w the conclusion of our coro llary follow s fro m Lemma 5.1 applied to the pro jection π : H → G .  Example 5.3. Let T = n 1 2 n : n ∈ N , n ≥ 1 o ∪ { 0 } . Assume tha t L is a finite simple non- ab elian gr o up and G = T × L . Then S = ϕ ( T ) × { e L } is a qc-dense se q uen c e in G c o n ver ging to e G . Indeed, since a ll simple non-ab elian groups a re p erfect, this follows from Example 3.2 and Corollary 5.2. Pro of of Theorem 2.3. (i) → (iii) Let S ⊆ G a b e a suitable set f o r G . Then the subgroup H of G generated by S is dense in G . Since H ⊆ G a , it follows that G a is dense in G as w ell. Th us G is connected. (iii) → (ii) According to [9, Theorem 3.3], there exist a cardinal κ and a con tin uous surjec- tiv e group homomorphism π : b Q κ × L → G , where L is a direct pro duct of simple Lie gro ups. In part icular, L is p erfect. By Lemma 4.3 , there exists a sup er- sequence E in ( b Q κ ) a con verg- ing to 0 that is qc-dense in b Q κ . By Corollary 5.2, S ′ = E × { e L } is a conv erging to (0 , e L ) 8 D. DIKRAN JAN AND D. SH AKHMA TOV sup er-sequence in ( b Q κ ) a × { e L } that is qc-dense in b Q κ × L . Since S ′ is qc-dense in b Q κ × { e } , it top ologically generates b Q κ × { e } b y F act 1.7. By Theorem 1.9 and Remark 1.8, there exis ts a sup er-sequence S ′′ ⊆ { 0 } × L con v erging to (0 , e L ) tha t top ologically generates { 0 } × L . Then S = S ′ ∪ S ′′ is a sup er-sequence conv erging to (0 , e L ) that is qc-dense in b Q κ × L and top ologically generates b Q κ × L . Therefore, X = π ( S ) is a sup er-sequence (by [9, F act 4 .3]; see [16, F act 12] for the pro of ) that is qc-dense in G (b y F act 3.1) and top o logically generates G . Since L is arcwise connected and S ⊆ ( b Q κ ) a × L , we hav e X = π ( S ) ⊆ π (( b Q κ ) a × L ) ⊆ G a . Applying Remark 1.8, w e conclude that X \ { e } is a suitable set f or G . Since X is qc-dense in G , so is X \ { e } . If X is infinite, w e ar e done. Assume no w tha t X is finite. S ince S ′ is qc-dense in b Q κ × { e } , π ( S ′ ) is qc-dense in K = π ( b Q κ × { e } ) b y F a ct 3.1. Since π ( S ′ ) ⊆ π ( S ) = X , the set π ( S ′ ) is finite. F rom Theorem 1.5 w e conclude that t he (compact ab elian) gro up K must finite as w ell. Being a con t in uous imag e of the connected gr o up b Q κ × { e } , the group K is connected. Hence, K is trivial, and so G = π ( b Q κ × L ) = π ( { 0 } × L ) ⊆ G a ⊆ G b ecause L is path wise connected. This yields G = G a . Let f : [0 , 1] → G a b e a con tinuous map suc h that f (0) = e G and f (1) 6 = e G . Define c = inf { t ∈ [0 , 1] : f ( t ) 6 = e G } . Then f ( c ) = e G b y contin uity of f and t he c hoice of c . F or ev ery n ∈ N c ho ose c n ∈ [0 , 1] s uc h that c < c n < c + 1 /n and f ( c n ) 6 = e G . Since { c n : n ∈ N } conv erges to c and f is con tin uous, the sequence X 0 = { f ( c n ) : n ∈ N } con verges to f ( c ) = e G . Since f ( c n ) 6 = e G for ev ery n ∈ N , w e conclude that X 0 is infinite. No w X ′ = X ∪ X 0 ∪ { e G } is an infinite sup er-sequence con verging to e G , and so X ′ \ { e G } ⊆ G a is an infinite suitable set for G b y Remark 1.8. Since X is qc-dense in G and top ologically generates G , X ′ has the same pro p erties. The implication (ii) → (i) is trivial.  Reference s [1] L. Außenhofer, Con tributions to the duality theor y of abelian top ological groups and to the theory of nu clear gr oups . Diss. Math. CCCLXXXIV. W a rsaw, 1 999. [2] L. Außenhofer, A duality pro p erty of an unco untable pr o duct of Z , Math. Z. 257 (2007) 2 31–23 7. [3] L. Außenhofer, On the arc co mp one nt of a lo cally compact ab elian gro up , Math. Z. 25 7 (200 7 ) 239– 250. [4] M. J. Chasco, Pontry a gin dua lity for metriz a ble gr oups , Ar ch. Math. (Basel) 70 (1998 ) 22–28 . [5] W. W. Co mfor t, S. U. Raczkowski and F. J . T rigos-Ar r ieta, The dual gro up of a dense subgroup , Cze choslovak Math. J. 5 4 (12 9) (2004) 509– 5 33. [6] D. Dikranjan a nd L. De Leo , Countably infinite quasi-co nv ex sets in some lo cally compact ab elian groups , submitted. [7] D. Dikr anjan and C. Milan, Dualities of lo c a lly compact mo dules ov er the ra tionals , J. Algebr a 25 6 (2002) 43 3–46 6. [8] D. Dikranjan, Iv. Pro danov and L. Sto yano v, T op olo gic al Gr oups: Char acters, Dualities and Minimal Gr oup T op olo gies , Pur e a nd Applied Mathematics, vol. 130, Marcel Dekker Inc., New Y o rk-Ba sel (198 9). [9] D. Dikranjan and D. Shakhmatov, W eigh t o f clo sed subsets top ologica lly generating a compact gro up , Math. Nachr. 280 (2 007) 50 5–522 . [10] D . Dikranjan and D. Shakhmatov, Qua si-conv ex densit y and deter mining s ubgroups o f compact ab elia n groups , submitted (av ailable as ArXiv preprint no. ar Xiv:0807 .3 846 v 3). [11] D . Dikra nja n, M. Tk aˇ cenko, and V. Tk ach uk, So me top o logical gro ups with and some without suita ble sets , T op ol. Appl. 98 (1999) 1 31–14 8. [12] S. Hern´ andez, S. Macario and F. J. T rigos- Arrieta, Uncountable products of determined gr oups need not b e determined , J. Math. Anal. A ppl. 34 8 (2008 ) 834 –842 . [13] K.-H. Hofmann a nd S. A. Mo rris, W eigh t a nd c , J. Pu r e Appl. A lgebr a 68 (1990 ) 181–1 94. SUPER-SEQU ENCES IN THE A RC COMPONENT 9 [14] K.-H. Hofmann and S. A. Mo rris, The structure of co mpact groups. A primer for the student—a hand- bo ok for the exp ert , de Gruyter Studies in Ma thematics, 25 , W alter de Gruyter & Co., Ber lin, 199 8. [15] N. W. Rick ert, Arcs in lo ca lly compa ct gro ups , Math. Ann. 172 (1 9 67) 2 22–2 28. [16] D . Shakhma tov, Building suitable se ts for lo cally compact gro ups by means of contin uous se- lections , T op ol. Appl. (20 0 8), do i:1 0.101 6/j.top ol.20 08.12.009 (av ailable as ArXiv preprint no. arXiv:081 2.048 9 v2). (Dikran Dikranjan) Universit ` a di Udine, Dip ar timento di Ma tema tica e Informa tica, via delle Scienze, 206 - 3 3100 Udine, It a l y E-mail addr ess : di kran. dikra njan@dimi.uniud.it (Dmitri Shakhmatov) Gradua te School of Science and E n gineering, Division of Ma thema tics, Physics and Ear th Sciences, E hime University, Ma tsu y ama 790-8577, Jap an E-mail addr ess : dm itri@ dpc.e hime-u.ac.jp