Locally minimal topological groups

Lo cally minimal top ological groups ∗ Lydia Außenhofer Institut f¨ ur A lgebr aische Ge ometrie, Welfengarten 1 D-3016 7 Hannover, e-mail: aussenhofer@maphy.uni-hannov er.de M. J. Chasc o Dept. de F ´ ısic a y Mate m´ atic a Aplic ada, Universidad de Navarr a , e-mail : mjchasc o@unav.es Dikran Dikranjan Dip artimento di Mat ematic a e Informatic a, Universit` a di Udine, e- mail : dikr anja@dimi. uniud.it Xabier Dom ´ ınguez Dep artamento de M´ eto dos Matem´ atic os y de R epr esentaci´ on, Uni versidad de A Coru˜ na, e-mail: x dominguez@udc.es Octob er 15, 2009 Abstract The aim of th is paper is to go deep er into the study of lo cal minimality and its conn ection to some naturally related properties. A H ausdorﬀ top ological group ( G, τ ) is called locally minimal if th ere exists a neigh b orho o d U of 0 in τ such that U fails to b e a neighborho od of zero in any Hausdorﬀ group t op ology on G which is strictly coarser than τ . Examples of lo cally minimal groups are all subgroups of Banach-Lie groups, all lo cally comp act groups and all minimal groups. Motiv ated by the fact that locally compact N SS groups are Lie groups, we study the connection b etw een local minimalit y and th e NSS prop erty , establishing that under certain conditions, locally minimal NS S groups are metrizable. A symmetric subset of an ab elian group con taining zero i s said to b e a G TG set if it generates a group top ology in an analogous wa y as conv ex and symmetric sub sets are un it balls for p seudonorms on a vector space. W e consider topological groups which hav e a neigh b orho o d basis at zero consisting of GTG sets. Examples of these lo cally GTG groups are: locally pseud o–conve x spaces, groups uniformly free from small subgroups (UFS S groups) and lo cally compact abelian groups. The precise relation b etw een these classes of groups is obtained: a topological ab elian group is UFSS i f and only if it is lo cally minimal, lo cally GTG and NSS. W e d evelo p a universal construction of GTG sets in arbitrary non-discrete metric ab elian groups, that generates a strictly ﬁner non-discrete UFSS top ology and we characterize the metrizable ab elian groups admitt in g a strictly ﬁner non-discrete UFSS group t opology . Unlike the minimal top ologies, the locally minimal ones are alw ays av ailable on “large” groups. T o supp ort this line, we pro ve that a boun d ed abelian group G admits a non- discrete locally minimal and lo cally GTG group top ology iﬀ | G | ≥ c . Keywo rds: locally minimal group, minimal group, group without small subgroups, group uniformly free from small sub groups, pseudo–conv ex set, GTG set, locally GTG group, locally b ounded group, b ounded group. MSC 22A05, 22B05, 54H11, 52A30 ∗ The ﬁrst named author was partially supp orted by MTM 2008-04599. The third author was partially supp orted by SRA, grants P1-0292-0101 and J1-9643-0101. The other authors were partially supp orted by MTM 2006-0303 6 and FEDER funds. 1 1 In tro duction Minimal top olog ical spaces hav e bee n largely studied in the literatur e ([7]). Minimal top ologica l gro ups were intro- duced indep endently b y Cho quet, Do ¨ ıtchino v [14] and Stephenson [3 6]: a Hausdorﬀ top ologica l group ( G, τ ) is called minimal if there exists no Hausdorﬀ group top ology on G which is strictly coar ser than τ . The ma j or problem that determined the theory of minimal ab elian gr oups was es tablishing pr e c omp actness of the ab elia n minima l g roups (Pro danov-Stoy anov’s theorem [13, Theorem 2.7.7]; for rece nt adv ances in this ﬁeld s ee [9, 10, 13]). Generalizations o f minimality were recently pr op osed by v arious a uthors. Relative minimality and co-minimality were introduced by Megr elishvili in [24] (see also [11, 34]). The notion of loc al minimality (see Deﬁnition 2 .1) was int ro duced by Morris and Pestov in [26] (see also Bana kh [5 ]). A stronger version o f this notion was used in [12] to characterize the lo cally compact subgroups of inﬁnite pr o ducts of lo cally compact g roups. W e start Section 2 with some p er manence prop erties of lo ca l minima lity (with res pec t to tak ing clos e d or op en subgroups). W e pr ove in Theo rem 2.8 that nw ( G ) = w ( G ) for every lo cally minimal group (in particular , all countable lo ca lly minimal groups ar e metrizable). Subsection § 2 .2 is dedicated to the NSS groups. Let us reca ll, that a top olog ical gro up ( G, τ ) is called NSS gr oup (resp., NS n S gr oup ) if a suitable zero neig h b orho o d contains only the triv ial (resp., normal) s ubgroup. The relev ance o f the NSS prop erty comes fro m the fact that it ch ar acterizes the Lie g r oups within the cla ss of lo cally c o mpact gr oups. Since lo cal minimality generaliz es lo cal compactness, it is quite na tural to inv estigate lo cal minima lity c o mbin ed with the NSS prop erty . It turns out that lo cally minimal ab elian NSS groups are metrizable (P rop osition 2.13), which should b e compared with the classical fact that loca lly compact NSS gro ups ar e Lie g roups (hence, metrizable). W e do not k now whether “ab elian” can b e removed here (cf. Question 6.7). Section 3 is dedicated to a proper t y , in tro duced by Enﬂo [15] that sim ultaneously strengthens lo cal minimalit y a nd the NSS pr op erty . A Hausdorﬀ top ologica l gro up is UFSS (Uniformly F ree from Small Subgroups) if its topolo gy is generated by a sing le neighbor ho o d of zero in a natural analogo us wa y as the unit ball o f a no rmed spac e determines its to po logy (a precise deﬁnition is given in 3.1 b elow). In P rop osition 3.8 we show that lo cally minimal NSnS precompact gr oups are UFSS (hence minima l NSS ab elian g roups are UFSS). Lo ca l minimalit y presents a common generaliza tion of lo cal compac tnes s, minimality and UFSS. Since the latter prop er t y is not suﬃciently studied, in contrast with the former tw o, we dedicate § 3.2 to a detailed study of the per manence prop erties of this remar k able class. W e s how in Pr op osition 3.12 that UFSS is stable under taking subgr o ups, extensio ns (in pa rticular, ﬁnite pro ducts), completions and lo c a l is omorphisms. In § 4 we introduce the concept of a GTG set that, roug hly sp eaking, is a sy mmetric subset U of a g roup G containing 0, with a n appropriate co nv exit y-like pr op erty (i. e., these sets a re ge neralizations o f the symmetric conv ex sets in re a l vector spaces, see Deﬁnition 4.2). A top olo gical group is called lo cally GTG, if it ha s a bas e o f neighborho o ds of 0 that are GTG s ets. Since lo cally precompact ab elian groups, as well a s UFSS gr oups, are lo cally GTG, this explains the imp or tance of this new class. On the other ha nd, minimal ab elia n g roups a re pr ecompact, so minimal ab elian gr oups a re b oth lo cally minimal a nd lo ca lly GTG. W e prov e in Theore m 5.10 that a Hausdor ﬀ ab elian top olo gical gro up is UFSS iﬀ it is lo cally minimal, NSS a nd lo c a lly GTG. According to a theor em o f Hewitt [20], the usual top ologie s on the group T and the g r oup R hav e the prop erty that the only strictly ﬁner lo cally compact gr oup top ologies a re the dis crete top olo gies. Since lo cally minimal lo cally GTG top olog ies generaliz e the lo cally compact gro up top olog ies, it would b e na tural to ask whether the groups T and R admit stronger non-discrete lo cally minimal lo ca lly GTG top o lo gies. In Co rollary 4.24 we give a strongly p ositive answer to this question for the large clas s of all non-to tally disconnec ted lo ca lly compac t metriza ble ab elian groups and for the strong e r pro per ty of UFSS top o lo gies. T o this end we develop, in Theorem 4.21, a universal constructio n o f GTG sets in arbitra r y 2 non-discrete metric ab elian groups, that g enerates a strictly ﬁner no n- discrete UFSS to po logy . The description o f the algebr aic s t ructur e of lo cally minimal abelia n groups seems to be an impo rtant problem. Its solution for the class of compact g roups b y the end o f the ﬁfties of the last century brought a signiﬁcant dev elopment of the theory o f inﬁnite ab elian gr oups. This line was follow ed la ter also in the theo ry o f minimal gr oups, but here the pr oblem is still op en even if solutions in the case of many smaller class es of ab elian groups a re av aila ble ([9, §§ 4.3, 7.5], [13, chapter 5]). Unlike the minimal topo lo gies, the lo cally minimal ones are alwa ys av aila ble on “larg e” groups. T o suppor t this line, we prove in Theorem 5.18 that a bounded ab elian gr oup G admits a non-disc r ete loca lly minimal and lo c a lly GTG gro up top olo gy iﬀ | G | ≥ c (and this o ccur s pr ecisely when G a dmits a no n-discrete lo cally compact group to po logy). Analogously , in a nother small group (namely , Z ), the non-discr ete lo ca lly minimal and lo cally GTG g roup top olog ies are no t muc h more tha n the minimal ones (i. e., they ar e either UFSS or hav e an op en minimal s ubgroup, se e Example 5.16). This line will b e pur sued further and in more detail in the forthcoming pap er [4] where we study also the lo c a lly minimal groups that can b e obtained a s extensions of a minimal gr oup via a UFSS quotient g roup. In the next diagram w e collect all implica tio ns be t ween all pro per ties introduce d s o fa r: discrete   compact   u u k k k k k k k k k k k k k k k k k ) ) S S S S S S S S S S S S S S S S S S normed spaces { { v v v v v v v v v v v v v v v v v v v v v v v v v lo cally co mpact   / / lo c.min. & lo c.GTG u u k k k k k k k k k k k k k k k k k k minimal o o lo c. min. countable      lo c.GTG & loc .min. & NSS O O o o   / / UFSS i i S S S S S S S S S S S S S S S S S S o o u u k k k k k k k k k k k k k k k k k k minimal & NSS o o i i S S S S S S S S S S S S S S S S S S metrizable lo c.min. & NSS i i S S S S S S S S S S S S S S S S S S o o All the implica tions denoted by a solid a rrow a re true for arbitrar y ab elian g roups, those that require some additional condition on the group are g iven by dotted arr ows a ccompanied by the additional condition in que s tion. W e give sepa rately in the next diag ram only those arr ows that are v alid for all, not necessar ily ab elian, to po logical groups. discrete   compact   u u k k k k k k k k k k k k k k k k k k k k k k k k k ) ) S S S S S S S S S S S S S S S S S S S S S S S S S S lo cally compact / / lo c.min. countable u u k k k k k k k k k k k k k k minimal o o metr. lo c.min. & NSS O O ??? o o UFSS i i S S S S S S S S S S S S S S S S S S S S S S S S S S o o lo c.min. & NSnS & pr ecompact 5 5 k k k k k k k k k k k k k k k k k k k k k k k k k k O O i i S S S S S S S S S S S S S S S S S S S S S S S S S S 3 Notation and termi nology The subgr oup generated by a subset X of a gro up G is deno ted by h X i , and h x i is the cyclic subgroup of G genera ted by an element x ∈ G . The abbreviation K ≤ G is used to denote a subgroup K of G . W e use additive notatio n for a not necessar ily ab elian group, a nd denote b y 0 its neutral element. W e denote by N , N 0 and P the sets of p ositive natural num ber s, non-negative int eger s and primes , resp ectively; by Z the integers, by Q the rationals , b y R the reals, and b y T the unit circle g roup which is ident iﬁed with R / Z . The cyclic group of order n > 1 is denoted by Z ( n ). F or a prime p the s ymbol Z ( p ∞ ) sta nds for the quasicyclic p - group and Z p stands for the p -adic in tegers . The torsion p art t ( G ) of an ab elian group G is the set { x ∈ G : nx = 0 for some n ∈ N } . Clearly , t ( G ) is a subgroup of G . F or any p ∈ P , the p -primary c omp onent G p of G is the subgro up of G that consists of all x ∈ G satisfying p n x = 0 for some p ositive integer n . F or every n ∈ N , we put G [ n ] = { x ∈ G : nx = 0 } . W e say that G is b oun de d if G [ n ] = { 0 } for some n ∈ N . If p ∈ P , the p - r ank o f G , r p ( G ), is deﬁned a s the c a rdinality o f a ma ximal independent subset of G [ p ] (se e [32, Section 4.2]). The group G is divisible if n G = G for every n ∈ N , and r e duc e d , if it has no divisible subg roups beyond { 0 } . The fr e e r ank r ( G ) of the g roup G is the cardinality of a maximal independent subs e t o f G . The so cle of G, S oc ( G ) , is the subgro up of G genera ted b y all elements of prime order , i. e. S o c ( G ) = L p ∈ P G [ p ] . W e denote b y V τ (0) (or simply b y V (0)) the ﬁlter o f neighborho o ds of the neutral elemen t 0 in a topo logical group ( G, τ ). Neighbor ho o ds ar e not necessa rily op en. F or a top olo gical group G we denote by e G the Ra ˘ ıko v completion of G . W e recall her e that a group G is pr e c omp act if e G is compact (some a uthors pr e fer the ter m “totally bo unded”). W e say a top ologica l g roup G is line ar or is line arly top olo gize d if it has a neighbo rho o d basis at 0 formed by op en subgroups. The cardinality o f the co n tinuum 2 ω will be also denoted b y c . T he weight of a top olo g ical space X is the minimal cardinality of a basis for its top ology; it will b e denoted by w ( X ) . The net weight of X is the minimal cardinality of a netw ork in X (that is, a family N o f subs e ts of X such that for any x ∈ X a nd any op en set U containing x there exists N ∈ N with x ∈ N ⊆ U ). The netw e ight o f a s pace X will b e denoted by nw ( X ). The pseudo char acter ψ ( X , x ) of a space X at a point x is the minimal cardinality of a family o f open neighborho o ds of x whose intersection is { x } ; if X is a homogeneous space, its pseudo character is the sa me a t every po in t a nd we denote it b y ψ ( X ) . The Lindel¨ of nu m b er l ( X ) of a s pace X is the minima l cardinal κ such that any op en cover of X admits a s ub cover of cardinality not greater than κ. By a char acter on an ab elian top olo gical g roup G it is commonly unders to o d a co nt inuous homo mo rphism from G in to the unit circle group T . Let U b e a symmetric subs e t of a group ( G, +) suc h that 0 ∈ U, a nd n ∈ N . W e deﬁne (1 /n ) U := { x ∈ G : k x ∈ U ∀ k ∈ { 1 , 2 , · · · , n }} and U ∞ := { x ∈ G : nx ∈ U ∀ n ∈ N } . Recall that a nonempt y s ubs et U o f a real vector s pa ce is starlike w he never [0 , 1] U ⊆ U. No te that if U is star like and s ymmetric then (1 /n ) U = 1 n U ; in general, for s ymmetric U : (1 / n ) U = n \ k =1 1 k U. All unexplained top ologica l terms can be found in [16]. F or background on ab elian groups, s ee [17] and [3 2]. 4 2 Lo cal minimalit y 2.1 The notion of a lo cally minimal top ological group In this section we r ecall the deﬁnition and basic examples of lo cally minimal g roups, a nd pr ov e that for lo ca lly minimal groups the w eight and the net weigh t coincide. Deﬁnition 2.1 A Hausdor ﬀ top ologica l group ( G, τ ) is lo c al ly minimal if there exists a neighborho o d V of 0 such that whene ver σ ≤ τ is a Ha us dorﬀ gr oup top olo gy on G such that V is a σ - neighborho o d of 0, then σ = τ . If we wan t to po in t out that the neig h b orho o d V witnesses lo cal minimality for ( G, τ ) in this sense, we say that ( G, τ ) is V –lo c al ly minimal. Remark 2 .2 As mentioned in [12], one obta ins an equiv a lent deﬁnition replacing “ V is a σ -neig hbo rho o d of 0 ” with “ V has a non-empty σ -interior” ab ove. It is easy to see that if lo ca l minimality of a gr oup G is witnessed b y s o me V ∈ V τ (0), then every smaller U ∈ V τ (0) witnesses lo cal minimalit y of G as w ell. Example 2. 3 Exa mples for lo cally minimal gro ups: (a) If G is a minimal top ological group, G is lo c ally minimal [ G witnes ses loc a l minimality of G ]. (b) If G is a lo cally co mpact gr oup, G is lo ca lly minimal [every compact neighborho o d of zero witnesses lo cal minimalit y of G , [12]]. (c) It is easy to chec k that a normed space ( E , τ ) with unit ball B is B -lo cally minimal. W e sta rt with so me p erma nence pr op erties of lo c a lly minimal groups. Prop ositio n 2 .4 A gr oup having an op en lo c al ly m inimal sub gr oup is lo c al ly minimal. Pr o of . Let H b e a lo cally minimal g r oup witnessed by U ∈ V H (0) a nd s uppo s e that H is an op en subgro up of the Hausdorﬀ gro up ( G, τ ). Then U is a neighborho o d of 0 in G . Assume that σ is a Hausdor ﬀ g r oup top ology o n G coarser than τ such that U is a neighborho o d o f 0 in ( G, σ ). Then τ | H ≥ σ | H and s ince U is a neighborho o d o f 0 in ( H, σ | H ), we o bta in τ | H = σ | H . Since U is a neighbor ho o d of 0 in ( G, σ ), the subg roup H is op en in σ and hence σ = τ . QED In the other direction we ca n weak en the h yp othesis “ op en s ubgroup” to the muc h weak er “ closed subgroup” , but w e need to further impos e the restraint on H to b e central. Prop ositio n 2 .5 L et G b e a lo c al ly minimal gr oup and let H b e a close d c ent r al sub gr oup of G . Then H is lo c al ly minimal. Pr o of . Let τ denote the top olog y of G and le t V 0 ∈ V ( G,τ ) (0) witness lo cal minimality of ( G, τ ). Cho ose V 1 ∈ V ( G,τ ) (0) such that V 1 + V 1 ⊆ V 0 . W e show that V 1 ∩ H witnesses lo ca l minimalit y o f H . Suppose σ is a Hausdo rﬀ g roup top ology on H co arser than τ | H such that V 1 ∩ H is σ - neighborho o d of 0. It is ea s y to verify tha t the family of sets ( U + V ) where U is a σ -neighbo r ho o d of 0 in H and V is a τ -neighborho o d of 0, form a neighbo rho o d bas is of a group topo lo gy τ ′ on G which is coar ser than τ . Let us prov e that τ ′ is Hausdorﬀ: Ther efore, obse r ve that for a subset A ⊆ H we hav e A τ ⊆ A σ , sinc e H is c losed in τ . Hence w e o btain { 0 } τ ′ = T { U + V : U ∈ V ( H,σ ) (0) , V ∈ V ( G,τ ) } = T U ∈V ( H,σ ) (0) T V ∈V ( G,τ ) (0) U + V = T U ∈V ( H,σ ) (0) U τ ⊆ T U ∈V ( H,σ ) (0) U σ = { 0 } since σ was assumed to b e Hausdorﬀ. 5 Moreov er, if W ∈ V σ (0) such that W ⊆ V 1 ∩ H , then W + V 1 ⊆ V 0 implies tha t V 0 ∈ V ( G,τ ′ ) (0). B y the choice of V 0 this yields τ ′ = τ . Hence σ = τ . QED Corollary 2.6 An op en c entra l sub gr oup U of a top olo gic al gr oup G is lo c al ly m inimal iﬀ G itself is lo c al ly minimal. These results leave op en the question on whether “central” can be o mitted in the ab ov e corollary and Pr op osition 2.5 (see Q uestion 6.8). The questio n whether the pro duct of tw o minimal (ab elian) g roups is again minimal was answered negatively by Do ¨ ıtchino v in [14] where he prov ed that ( Z , τ 2 ) × ( Z , τ 2 ) is no t minimal although the 2– adic top olo gy τ 2 on the int eger s is minimal. W e will show in P rop. 5.17 that ( Z , τ 2 ) × ( Z , τ 2 ) is not even lo cally minimal. Next we a re going to see some cas e s wher e metrizability ca n be deduced from lo ca l minimality . W e start with a generaliza tion to lo ca lly minimal g roups of the following theo rem of Arhangel ′ skij: w ( G ) = nw ( G ) for every minimal group; in par ticular, every minimal group with count able net weigh t is metrizable. F or that we need the following result from [1]: Lemma 2.7 L et κ b e an inﬁn it e c ar dinal and let G b e a top olo gic al gr oup with (a) ψ ( G ) ≤ κ ; (b) G has a s ubset X with h X i = G and l ( X ) ≤ κ . Then for every family of neighb orho o ds B of the neutr al element 0 of G with |B | ≤ κ t her e exists a c o arser gr oup top olo gy τ ′ on G such that w ( G, τ ′ ) ≤ κ and every U ∈ B is a τ ′ -neighb orho o d of 0 . Theorem 2. 8 F or a lo c al ly minimal gr oup ( G, τ ) one has w ( G ) = nw ( G ) . In p articular, every c ountable lo c al ly minimal gr oup is metrizable. Pr o of . Let κ = nw ( G ) and let N be a netw ork of G of size κ . Then also ψ ( G ) ≤ κ as \ { G \ B : 0 6∈ B , B ∈ N } = { 0 } . Moreov er, the Lindel¨ of num b er l ( G ) of G is ≤ κ . Indeed, if G = S i ∈ I U i and e ach U i is a non-empty op en set, then by the deﬁnition of a netw ork for every x ∈ G there e xists i x ∈ I and B x ∈ N such that x ∈ B x ⊆ U i x . [F or z ∈ G we choose y ∈ Y suc h that B z = B y and obtain z ∈ B z = B y ⊆ U i y .] Let N 1 = { B x : x ∈ G } and Y ⊆ G such that the assignment Y → N 1 , deﬁned by Y ∋ x 7→ B x , is bijective. Then | Y | ≤ κ and G = S y ∈ Y U i y . This proves l ( G ) ≤ κ . T o end the pr o of of the theorem apply Lemma 2.7 taking X = G a nd any family B of size κ of τ -neighborho o ds of 0 containing U as a member and witnes s ing ψ ( G ) ≤ κ (i.e., T B = { 0 } ). This gives a Hausdor ﬀ topo logy τ ′ ≤ τ on G sa tisfying the co nclusion of the lemma. By the lo cal minimality of ( G, τ ) we conclude τ ′ = τ . In par ticular, w ( G, τ ) ≤ κ . Since alwa ys nw ( G ) ≤ w ( G ), this proves the r e quired equality w ( G ) = nw ( G ). Now supp o se that G is countable. Then nw ( G ) = ω , so the equality w ( G ) = nw ( G ) implies that G is seco nd countable, in particular metrizable. QED Remark 2 .9 (a) The fact that every co unt able loc ally minimal group ( G, τ ) is metriz able admits also a str aight- forward pro of. Indeed, let { x n : n ∈ N } = G \ { 0 } and let U 0 be a neighbor ho o d of 0 such that G is U 0 -lo cally minimal. Then ther e ex ists a sequence of symmetric neighbo rho o ds o f ze r o ( U n ) satisfying U n + U n ⊆ U n − 1 , x n 6∈ U n , U n ⊆ T n − 1 k =1 ( x k + U n − 1 − x k ) for all n ∈ N . Since T ∞ n =1 U n = { 0 } , the family ( U n ) forms a base of neighborho o ds of 0 of a metrizable group top olog y σ ≤ τ on G with U 0 ∈ σ . Hence τ = σ is metrizable. 6 (b) A s imilar direct pr o of shows that every lo cally minimal abe lia n g roup ( G, τ ) of countable pseudo character is metrizable. Here “a belia n” cannot b e r e moved, since examples of minimal (necessa rily non-ab elian) gro ups of countable pseudo character and ar bitrarily high character (in particular, non-metrizable) w ere built b y Shakhma- tov ([33]). 2.2 Groups with no small (normal) subgroups In this subse ction we show that g roups with no small (normal) subgro ups are clos e ly rela ted to lo cally minimal groups and study so me o f their prop erties. Deﬁnition 2.10 A top olog ical g roup ( G, τ ) is called NSS gr oup (No Sma ll Subgroups) if a suitable neighborho o d V ∈ V (0) contains only the triv ia l s ubg roup. A top olo g ical gr oup ( G, τ ) is called NSnS gr oup (No Small normal Subgroups) if a s uitable neig h b orho o d V ∈ V (0) contains only the trivial normal subgroup. The distinction b etw een NSS and NSnS will b e necessar y only when we co ns ider non-ab elian gro ups (or non- compact groups, see Remark 3 .6 b elow). Example 2. 11 Examples for NSS and non-NSS groups. (a) The unit circle T is a NSS g roup. (b) Mon tgomer y a nd Z ippin’s s o lution to Hilbe r t’s ﬁfth problem a sserts that every lo cally compact NSS group is a Lie gr oup. (c) Any free ab elian top olo g ical gr oup on a metric space is a NSS group ([27 ]). (d) A dichotomy of Haus do rﬀ group top olog ies on the integers: An y Haus dorﬀ group top olo gy τ on the integers is NSS if and only if it is not linear . Indeed, s uppo s e that τ is no t NSS; let U b e a clo sed neighbo rho o d of 0. By assumption, U cont ains a nontrivial closed subgr oup H which is of the form n Z ( n ≥ 1). Since Z /n Z is a ﬁnite Hausdorﬀ group, it is discr ete a nd hence n Z is op en in Z . This shows that τ is line a r. (e) A gr oup G is t op olo gic al ly simple if G has no prop er closed normal subgro ups. E very Hausdorﬀ top olo g ically simple gro up is NSnS. [Suppose that G is top olo gically simple and Hausdo rﬀ and le t U 6 = G b e a closed neighborho o d of 0. Let N be a normal subgroup of G contained in U . Then N is also a closed subgroup of G contained in U a nd hence { 0 } = N = N . So G is an NSnS gr oup. Actually a s tronger pro pe rty is true: if G is Hausdorﬀ and every closed nor mal subgroup of G is ﬁnite, then G is NSnS (this provides a pro o f o f item (a)).] The inﬁnite p ermutation group G = S ( N ) is an e xample of a top olo gically simple group ([13, 7.1 .2]). W e o mit the eas y pro of of the next lemma: Lemma 2.12 (a) The classes of NSnS gr oups and NS S gr oups ar e stable u nder taking ﬁnite dir e ct pr o ducts and ﬁner gr oup top olo gies. (b) The class of NSS gr oups is stable u n der taking sub gr oups. (c) The class of NSnS gr oups is stable under taking dense sub gr oups. (d) N o inﬁnite pr o duct of non-trivial gr oups is NSnS . 7 Recall that a SIN group (SIN sta nds for Small Inv ariant Neighborho o ds) is a top ologica l group G such that for every U ∈ V (0) there exis ts V ∈ V (0) with − x + V + x ⊆ U for all x ∈ G. Prop ositio n 2 .13 Every lo c al ly minimal SIN gr oup G is metrizable pr ovide d it is NSn S. Pr o of . Le t us assume that ( G, τ ) is V –lo cally minimal and NSnS, where V is a neighbo rho o d of 0 in ( G, τ ) containing no non–tr ivial normal subgr oups. Since τ is a group top ology , it is p ossible to construct inductively a sequence ( V n ) of symmetric neig h b orho o ds of 0 in τ whic h satisfy V n + V n ⊆ V n − 1 (where V 0 := V ) and − x + V n + x ⊆ V n − 1 for all x ∈ G . Let σ b e the gr oup top olo gy genera ted by the neighbo rho o d bas is ( V n ) n ∈ N . O bviously , σ is coar ser than τ and V ∈ V σ (0). In order to co nc lude that σ = τ , it only remains to s how that σ is a Hausdorﬀ top ology , which is equiv ale n t to \ n ∈ N V n = { 0 } . This is trivia l, since the intersection is a nor mal s ubgroup co nt ained in V . QED Example 2. 14 One ca nnot rela x the “SIN” condition even w he n G is minimal. Indeed, for e very inﬁnite set X the symmetric group G = S ( X ) is minimal and NSnS. O n the other hand, S ( X ) is metrizable only when X is co unt able ([13, § 7 .1 ]). Note that this gro up strongly fails to b e NSS, as V (0) has a base co nsisting o f op en subgroups (namely , the po in twise stabilizer s of ﬁnite subsets of X ). Remark 2 .15 (a) The co mpletion of a NSS group is not NSS in ge neral: F or example the group T N is monothetic, i.e. it has a dense subgro up H algebr aically isomor phic to Z . Since the completio n o f a linear g roup top ology is again linear, and the pro duct top ology on T N is not linear, H is not linear either. So Example 2.11(d) implies that H is NSS. But H is dense in T N which is not NSS. (b) It was a problem o f I. Kaplansk y whether the NSS prop erty is preser ved under tak ing ar bitrary quotients. A counter-example was g iven by S. Morris ([2 5]) and P r otasov ([30]); the latter proved that NSS is pr eserved under taking q uotients with resp ect to discrete normal subgroups. (c) In co n tras t with the NSS pro per ty , a subgro up of an NSnS gr oup nee d no t b e NSnS. Indeed, take the pe rmu- tation group G = S ( N ). Let N = S n F n be a partition of the naturals into ﬁnite sets F n such that ea ch F n has size 2 n . Let σ n be a c y clic per mut ation of length 2 n of the ﬁnite set F n and let σ b e the p er mutation of N that acts on e a ch F n as σ n . O bviously , σ is a non-to r sion element of G , so it g e ne r ates an inﬁnite cyclic subgroup C ∼ = Z . F or conv enience iden tify C with Z . Then, while G is NSnS by E xample 2.11(e), the induced top ology of C coincide s with the 2-adic to po logy of C = Z , so it is linear and certainly non-NSnS. Indeed, a prebasic neighborho o d of the identit y element id N in C has the form U x = C ∩ Stab x , where Stab x is the stabilizer of the po int x ∈ N . If x ∈ F n , then obviously a ll p ow ers of σ 2 n stabilize x , so U x contains the subgroup V n = h σ 2 n i . This proves that the induced top olo gy of C ∼ = Z is co arser than the 2 - adic top ology . Since the la tter is minimal ([13, 2.5.6]), we conclude that C has the 2-adic top olo gy . 3 Groups un iformly free from small subgroups 3.1 Lo cal minimalit y and t he UFSS prop ert y W e have seen (Example 2.3(c)) that all no rmed spaces a r e lo ca lly minimal when reg arded as top o lo gical ab elia n groups. The following group analog o f a normed space was int ro duce d by Enﬂo ([15 ]); we will s how in F acts 3.3(a) that every such group is loca lly minimal: 8 Deﬁnition 3.1 A Hausdorﬀ top ological group ( G, τ ) is uniformly fr e e fr om smal l sub gr oups (UFSS for short) if for some neighborho o d U of 0, the sets (1 /n ) U form a neighborho o d basis at 0 for τ . Neighborho o ds U satisfying the co ndition desc rib ed in Def. 3 .1 will be said to b e distinguishe d. It is eas y to s ee that any neighborho o d of zero co ntained in a distinguished one is distinguishe d, a s well. Obviously , discr ete g roups are UFSS. Now we give some non-trivia l examples. Example 3. 2 (a) R is a UFSS gro up with res p ect to [ − 1 , 1] . (b) T = R / Z is a UFSS group with resp ect to T + , the image of [ − 1 / 2 , 1 / 2] under the quotient map R → T . (c) A top olog ical vector space is UFSS a s a top olo gical ab elian gr oup if a nd o nly if it is loca lly b ounded. In particular every no rmed space is a UFSS gr o up. Recall that a subset B of a (r e a l or co mplex) top ologica l vector space E is usually referred to a s b ounde d if for every neighborho o d of zer o U in E there exists α > 0 with B ⊆ λU fo r every λ with | λ | > α, and b alanc e d whenever λB ⊆ B for every λ with | λ | ≤ 1 . The space E is lo c al ly b ounde d if it ha s a b ounded neighbor ho o d of zero. It is straightf or ward that any loca lly b ounded spac e is UFSS when rega r ded as a top ological ab elia n group, and any of its b ounded neighborho o ds of zer o is a distinguished neig hborho o d. Conv ers e ly , if a top olo gical vector space is UFSS as a top olo gical ab elian group, then any distinguis hed balanced neig hborho o d of zero is bo unded in this sense . This, of co urse, includes unit balls of normed spaces, but there are some impor ta nt non-lo cally-convex examples as well (see E xample 3.5(b)). (d) Every Banach-Lie gr oup is UFSS, [2 6, Theorem 2.7 ]. F acts 3. 3 (a) Every UFSS gr oup with distinguishe d neighb orho o d U is U -lo c al ly minimal ([26, Pr op osition 2.5]) . Inde e d, one c an se e that a UFSS gr oup ( G, τ ) with distinguishe d neighb orho o d U has the fol lowing pr op erty, which trivial ly implies that ( G, τ ) is U -lo c al ly minimal: if T is a gr oup top olo gy on G such that U is a T - neighb orho o d of 0 , then τ ≤ T . (b) Al l UFSS gr oups ar e NSS gr oups. Next w e give some examples of NSS g roups that are no t UFSS. Example 3. 4 (a) Co nsider the group R ( N ) = { ( x n ) ∈ R N : x n = 0 for almo st all n ∈ N } , endowed with the rectangular topolo gy , which admits a s a basis of neighborho o ds of zero the following family of sets: U ( ε n ) := { ( x n ) ∈ R ( N ) : | x n | < ε n ∀ n ∈ N } , ( ε n ) n ∈ N ∈ (0 , ∞ ) N . This gro up is not metrizable, hence it canno t b e a UFSS group. On the other hand, any of the neighborho o ds U ( ε n ) contains only the trivial subgroup, so it is a NSS gro up. (b) All fre e ab elian top ologic a l gro ups on a metric space are NSS groups (se e Example 2.11(c)). T ak e a non-lo cally compac t metric space X , then A ( X ) is NSS, but no t UFSS (indeed, if A ( X ) is a k -spa c e for some metrizable X , then X is lo cally compact b y [2, Prop o s ition 2.8]). Example 2.3, F acts 3.3 and Example 3.2 give us a stro ng motiv ation to study lo cally minimal groups, whic h put under the s ame umbrella three extremely relev ant pr op erties a s minimality , UFSS and lo ca l compactnes s . 9 Example 3.2 shows that a lo cally minimal ab elian g roup need not b e precompact, in co nt ra s t with Pro danov- Stoy anov’s theorem. W e see in the follo wing example tha t a ctually ther e ex is t abelian loc ally minimal gr oups witho ut nontrivial contin uous character s. Example 3. 5 (a) Acco rding to a re s ult of W. Banasz c zyk ([6]), every inﬁnite dimensional B a nach spa ce E has a discr ete and free subgroup H such that the quotient gr oup E /H a dmits o nly the triv ia l character. E /H is lo cally iso morphic with E , hence it is a Banach-Lie gr oup and then UFSS. (b) Fix any s ∈ (0 , 1 ) and consider the top ologica l vector space L s of all classes of Le bes gue measurable functions f on [0 , 1] (mo dulo almost everywhere equality) such that R 1 0 | f | s dλ is ﬁnite, with the top o logy given by the following ba sis of neighborho o ds of zer o: U r =  f : Z 1 0 | f | s dλ ≤ r  , r > 0 . (F ollo wing a customa ry abuse of no tation, we use here (and in Example 5.4 and Remark 6.6) the same sy mbo l to deno te b oth a function and its class under the e q uiv alence rela tio n of almost everywhere equality .) In [8] it was proved that L s has no nontrivial contin uous linea r functionals. It is known that every character deﬁned on the top olog ical ab elian g roup under lying a top olog ical vector s pace ca n b e lifted to a contin uous linear functional on the space ([35]). Thus as a top olog ical gr oup, L s has trivial dual. On the other hand L s is a lo cally b ounded space (note that for every r > 0 o ne has U 1 ⊆ r − 1 /s U r ), hence it is a UFSS gro up (Example 3.2(c)). Remark 3 .6 It is a well known fa c t (see for instance [3 7, 32 .1]) that for every compact g roup K and U ∈ V (0) there exists a closed no rmal subgro up N of K contained in U such that K / N is a Lie gro up, hence UFSS. This implies that the following a ssertions are equiv alent: (a) K is UFSS, (b) K is NSS, (c) K is NSnS, (d) K is a Lie gr oup. In case K is ab e lia n, they a re equiv alent to: K is a closed s ubgroup o f a ﬁnite-dimens io nal torus. (The same equiv alences ar e known to b e true for lo cally compact groups which are either connected o r abelia n.) In order to extend the ab ove equiv ale nc e s to lo cally minimal preco mpact gro ups, we need the following Lemma: Lemma 3.7 L et ( G, τ ) b e a pr e c omp act gr oup. Then the fol lowing ar e e quivalent: (a) ( G, τ ) is NSn S; (b) F or every U ∈ V (0) ther e exists a c ontinuous inje ctive homomorphism f : G → L such t hat L is a c omp act Lie gr oup and f ( U ) is a neighb orho o d of 0 in f ( G ) . (c) Ther e exist a c omp act Lie gr oup L and a c ontinuous inje ctive homomorphi sm f : G → L. (d) G admits a c o arser UFSS gr ou p top olo gy. (e) ( G, τ ) is NS S . 10 In c ase G is ab elian t hese c onditions ar e e quivalent to t he existenc e of a c ontinuous inje ctive homomorp hism G → T k for some k ∈ N . Pr o of . T o prov e that (a) implies (b) ass ume that ( G, τ ) is NSnS and ﬁx a U ∈ V (0). Let W b e a neighborho o d of 0 in the completio n K of G such that W ∩ G contains no non-tr iv ial nor mal subgroups a nd ( W + W ) ∩ G ⊆ U . As in Remar k 3.6 ther e ex ists a closed nor mal subgroup N of K cont ained in W such that L = K / N is a Lie gro up. As N ∩ G = { 0 } by our choice of W , the canonical homomorphism q : K → L restricted to G gives a co nt inuous injectiv e homo mo rphism f = q ↾ G : G → L . O bserve that f ( U ) ⊇ q (( W + W ) ∩ G ) ⊇ q ( N + W ) ∩ q ( G ) as N ⊆ W . Finally , the la tter s et is a neighbo rho o d of 0 in f ( G ) as N + W ∈ V (0 K ). (b) ⇒ (c) is trivia l. (c) ⇒ (d) is a cons equence of the fact that every Lie group is UFSS. (d) ⇒ (e) and (e) ⇒ (a) a re trivial. QED Prop ositio n 3 .8 F or a lo c al ly minimal pr e c omp act gr oup G the fol lowing ar e e quivalent: (a) G is NSnS ; (b) G is NSS; (c) G is UFSS; (d) G is isomorphic to a dense sub gr oup of a c omp act Lie gr oup. Pr o of . The implicatio n (a ) ⇒ (d) follows from (a) ⇒ (b) in Lemma 3.7, since the lo c a l minimalit y of G a nd (b) from 3.7 imply that G → L is an embedding . Note that a compact subgr oup of a compact Lie gro up is clo sed, so a Lie group its e lf. (d) ⇒ (c) ⇒ (b) ⇒ (a) ar e tr ivial. QED Remark 3 .9 F or lo cally minimal pr ecompact a be lia n g roups, condition (d) of Prop. 3.8 can b e r eplaced by: G is isomorphic to a subgroup of a tor us T n , n ∈ N . Note that the class of lo ca lly minimal preco mpact a b e lian groups contains all minimal ab elia n g roups, due to the dee p theorem of P r o danov and Stoyano v which states that such groups are precompact. Remark 3 .10 Pro po sition 3.8 shows very nea tly the diﬀerences b et ween minimality and UFSS. While all (dense) subgroups o f a torus T n are UFSS, the minimal among the dense subg roups of T n are those that contain the so cle S oc ( T n ). Indeed, S oc ( T n ) is dens e a nd every clo sed non-triv ial subgr oup N of T n is s till a Lie group, so has non-trivial torsion elemen ts (i.e., meets S oc ( T n )). Ther e fore, b y ([13, Theorem 2.5.1]) a dense subg roup H of T n is minimal iﬀ H contains S oc ( T n ). In par ticular, ther e is a smalle s t dense minimal s ubg roup o f T n , namely S oc ( T n ). Example 3. 11 Let τ b e a UFSS precompa ct top ology on Z . Then ( Z , τ ) is a dense s ubg roup o f a g roup o f the form T k × Z ( m ), where k , m ∈ N , k > 0 . Indeed, b y Prop osition 3.8 and Remark 3 .9 ( Z , τ ) is is omorphic to a subgr oup of some ﬁnite-dimensional torus T n . Then the closure C of Z in T n will b e a monothetic compact a b e lian Lie group. So the connected comp onent c ( C ) ∼ = T k for some k ∈ N , k > 0 and C / c ( C ) is a discr ete monothetic compact group, so C /c ( C ) ∼ = Z ( m ) for so me m ∈ N , so C ∼ = T k × Z ( m ) since c ( C ) splits as a div isible subgroup of C . 11 3.2 P ermanence prop ert ies of UFSS groups In the next prop osition we co llect all per ma nence pr op erties of UFSS groups w e can verify . Prop ositio n 3 .12 The class of U FSS gr oups has the fol lowing p ermanenc e pr op erties: (a) If G is a dense sub gr oup of e G and G is UFSS, then e G is UFSS. (b) Every sub gr oup of a UFSS gr oup is UFSS. (c) Every ﬁnite pr o duct of UFSS gr oups is UFSS. (d) Every gr oup lo c al ly isomorphic to a UFSS gr oup is UFSS. (e) If an ab elian t op olo gic al gr oup G has a close d su b gr oup H such that b oth H and G/H ar e UFSS , t hen G is UFSS as wel l. Pr o of . (a ) Let G be a UFSS group with distinguished neighbo rho o d U . Note that closures in e G o f the neighborho o ds of 0 in G form a basis of the neighborho o ds o f 0 in e G . Let W b e a symmetric neighbor ho o d of 0 in G whic h sa tisﬁes G ∩ ( W + W ) ⊆ U . Let us pr ov e that (1 /n ) W ⊆ (1 /n ) U ∀ n ∈ N . T o this end ﬁx x ∈ (1 / n ) W . This mea ns x, 2 x, . . . , nx ∈ W . Hence there ex ists a seq uence ( x k ) in W which tends to x and the sequences ( j x k ) conv erge to j x ∈ W for j ∈ { 1 , . . . , n } . W e ma y assume that j x k − j x ∈ W for all j ∈ { 1 , . . . , n } and all k ∈ N , which implies j x k ∈ G ∩ ( W + W ) ⊆ U for all k ∈ N and j ∈ { 1 , . . . , n } . This implies x k ∈ (1 /n ) U for all k ∈ N and hence x ∈ (1 /n ) U . The inclusion (1 /n ) W ⊆ (1 / n ) U ass ures that the sets (1 / n ) W form a neighborho o d basis of 0 in e G ; i.e. W is a distinguished neighborho o d for e G . (b) to (d) a re easy to se e . (e) By assumption, there ex ists a neighborho o d W of 0 in G such that π ( W + W ) , and ( W + W ) ∩ H , are distinguished neig hbo rho o ds of zero in G/H a nd H , resp ectively , wher e π : G → G/H deno tes the canonic a l pro jection. According to a result o f Graev ([18] or (5.38 )(e) in [2 1]), G is ﬁrst coun table, since H and G/H hav e this pr op erty . Let us s how that ∀ ( x n ) with x n ∈ (1 /n ) W ⇒ x n τ → 0 . (3) where τ is the or iginal topo logy on G . Since π ((1 /n ) W ) ⊆ (1 /n ) π ( W ), π ( W ) is a distinguished neighborho o d of zer o in G/H and G is ﬁrst countable, there exists a sequence ( h n ) in H such that x n − h n → 0. F or n 0 ∈ N , there exists n 1 ≥ n 0 such that for all n ≥ n 1 we hav e h n = x n + ( h n − x n ) ∈ ((1 /n ) W + (1 /n 0 ) W ) ∩ H ⊆ ((1 /n 0 ) W + (1 /n 0 ) W ) ∩ H ⊆ (1 /n 0 )(( W + W ) ∩ H ) . Since the sets (1 /n )((( W + W ) ∩ H ) form a basis of zero neighbo rho o ds in H , the sequence ( h n ) tends to 0 and hence ( x n ) tends to 0 as well. Condition (3) implies that the family ((1 /n ) W ) is a basis of zer o neighbo rho o ds for G . Indee d, ﬁx U ∈ V τ (0) a nd suppo se (1 /n ) W 6⊆ U for every n ∈ N . Select x n ∈ (1 /n ) W, x n 6∈ U. According to (3) the s equence ( x n ) conv erges to zero, which contradicts x n 6∈ U ∀ n ∈ N . QED 12 Remark 3 .13 (a) Items (b) and (c) imply that ﬁnite suprema of UFSS gro up top ologie s a re still UFSS. In the next section we will intro duce the lo cally GTG to po logies which, at leas t in the NSS cas e, can b e characteriz ed as ar bitrary suprema of UFSS g roup top olo gies (see Deﬁnition 5.1 and Theo rem 5.7). (b) Item (c) follows a lso from (e). Let us note, that it cannot be strengthened to countably inﬁnite pro ducts. Indeed, a n y inﬁnite pro duct of non-indiscre te groups (e. g., co pies of T ) fails to b e NSS, so c a nnot b e UFSS either. The rest of the subsectio n is dedicated to a very natural pr op erty that was miss ing in Pro po sition 3.12, namely stability under taking quotients and contin uous homomo rphic images. It follows fro m item (d) of this Prop ositio n that a quotient of a UFSS gr oup with resp ect to a discr ete subg roup is UFSS. Actually it has b een shown in [28] (Prop osition 4.5) that e very Hausdorﬀ ab elian UFSS g r oup is a quotient gro up of a subgroup of a Ba nach space. How ever, a s we see in the next example, a Hausdorﬀ quotien t of a UFSS group nee d no t b e UFSS. Example 3. 14 Let { e n : n ∈ N } deno te the canonica l ba s is of the Hilb ert spa ce ℓ 2 . Co nsider the closed subgr oup H := h{ 1 n e n : n ∈ N }i of ℓ 2 . Let us deno te b y B the unit ball in ℓ 2 and by π : ℓ 2 → ℓ 2 /H the canonical pro jectio n. F or an arbitr ary ε > 0, we will show that π ( εB ) contains a nontrivial subgr oup. This will imply that the quotient ℓ 2 /H is not NSS and, in particular, is not UFSS. Let k 0 ∈ N such that X k>k 0 1 k 2 < 4 ε 2 . Le t S b e the linear hull of the set { e k : k > k 0 } . W e will obtain π ( S ) ⊆ π ( εB ) . Indeed, ﬁx x = ( x n ) ∈ S . F or n > k 0 , there exists k n ∈ Z such that   x n − k n n   ≤ 1 2 n . Since h := P n>k 0 k n n e n ∈ H and k x − h k ≤ q P n>k 0 ( 1 2 n ) 2 < ε , w e obtain: π ( x ) = π ( h + ( x − h )) = π ( x − h ) ∈ π ( ε B ) and hence π ( S ) ⊆ π ( εB ). The next co rollar y shows that the class of pr e c omp act UFSS gro ups is clos ed under taking arbitr a ry quo tien ts. Corollary 3.15 If G is a pr e c omp act UFSS gr oup, then every c ontinuous homomorphic image of G is UFSS. Pr o of . Let f : G → G 1 be a cont inuous sur jective homomo r phism. It can b e extended to the resp ective compact completions f ′ : e G → f G 1 of G and G 1 resp ectively . Since f is surjective and each gr oup is dense in its completio n, the compactness of e G yields that f ′ is surjective. Mo reov er, f ′ is op en by the op en mapping theo r em. Hence f G 1 is isomorphic to a quotient o f e G . By P rop osition 3.1 2(a) e G is UFSS, hence (Remark 3.6) e G is a Lie gro up. Then f G 1 is a Lie group a s well, so UFSS. This prov es that G 1 is UFSS. QED 4 GTG sets and UFSS top ologies 4.1 General prop erties of GTG subsets Vilenkin [40] introduced lo ca lly quasi-convex gr oups while g eneralizing the notio n o f a lo cally conv ex space. His deﬁnition is inspired on the descr iptio n of clo sed symmetric s ubs ets o f vector s paces g iven by the Hahn-Banach theorem. Next we present a new g eneralization of lo cally conv ex spaces in the setting of to p o lo gical gr oups whic h we will call lo c al ly GTG gr oups where GTG abbr eviates g roup t op ology g enera ting (set). Simila r ly to the notion of a convex set (that dep ends only on the linear s tr ucture of the top ologic a l vector space structure , but no t o n its topo logy), the notion of a GTG set dep ends only on the algebr aic str uctur e o f the gro up. In par ticular, it does not use any dual 13 ob ject at all, whereas the notion of quasi- c onv ex set of a top ologic a l g roup G dep e nds on the top ology of G via the contin uit y of the ch ar acters to b e used for the deﬁnition of the p olar . The class of lo cally GTG gro ups will be shown to contain all loca lly quasi-conv ex groups, all lo cally pseudo convex spaces a nd a ll UFSS groups. As we will see, it ﬁts v ery well in the setting of loc a lly minimal gr oups as it g ives a connection b etw een lo ca lly minimal g roups and minimal gr oups (5.12). Moreov er, we are not aw are of a n y loc ally minimal group not ha ving this prop erty (see Question 6.2). Recall that a subset A of a vector space E is ca lled pseudo c onvex if [0 , 1] A ⊆ A a nd A + A ⊆ cA for suitable c > 0. One may ass ume that c ∈ N . (Indeed, cho o se N ∋ n > c , then cA ⊆ nA a s ca = ( c/n ) na ∈ [0 , 1] nA ⊆ n A for a ll a ∈ A .) Hence the set A is pseudoc o nv ex iﬀ [0 , 1] A ⊆ A and for s ome n ∈ N , 1 n A + 1 n A ⊆ A . If A is symmetric, this alrea dy implies that ( 1 n A ) for ms a neighborho o d ba sis of a no t necessar ily Hausdorﬀ gr oup topolo gy: 1 nm A + 1 nm A = 1 m ( 1 n A + 1 n A ) ⊆ 1 m A . A standard argument shows that scalar m ultiplication is also contin uous. It is well k nown that the unit balls of the vector spa ces ℓ s where 0 < s < 1 ar e pseudo convex but no t conv ex. The s a me c a n b e sa id of their natural ﬁnite-dimensional counterparts ℓ s n , with n ≥ 2 . Nevertheless, by far not all symmetric subsets of a vector space are pseudo c o nv ex, as we see in the next example. Example 4. 1 The subsets of R 2 : U = ([ − 1 , 1] × { 0 } ) ∪ ( { 0 } × [ − 1 , 1]) and V = ( R × { 0 } ) ∪ ( { 0 } × R ) are symmetric and not pseudo conv ex. O bserve that [0 , 1] U ⊆ U ; 1 n U = ([ − 1 /n, 1 /n ] × { 0 } ) ∪ ( { 0 } × [ − 1 /n, 1 / n ]); [0 , 1] V ⊆ V and 1 n V = V . Deﬁnition 4.2 Let G b e an ab elian g roup a nd let U b e a s ymmetric subset of G such that 0 ∈ U. W e say that U is a gr oup top olo gy gener ating subset o f G (“GTG subset of G ” for shor t) if the sequence of subsets { (1 /n ) U : n ∈ N } is a basis of ne ig hborho o ds of zero for a (not necessar ily Hausdor ﬀ ) gro up top olo gy T U on G . In case U is a GTG set in G , T U is the coarses t group topo logy on G suc h that U is a neighbor ho o d. W e do not know whether the following natural conv erse is true: Let G b e an ab elian gro up and U a symmetric subset of G which co n tains zer o and suc h that there exists the coa rsest group top olog y on G for which U is a neighborho o d of zero. Then U is a GTG set. Example 4. 3 (a) E very s ymmetric distinguished neighborho o d of zero in a UFSS gr oup is a GTG set. (b) Every subgr o up o f a g roup G is a GTG subset of G . Prop ositio n 4 .4 A symmetric subset U ⊆ G of an ab elian gr oup G is a GTG subset if and only if ∃ m ∈ N with (1 /m ) U + (1 /m ) U ⊆ U . ( ∗ ) Mor e over, if U is a GTG set, U ∞ = T ∞ n =1 (1 /n ) U is t he T U -closur e of { 0 } and in p articular, it is a close d s u b gr oup and a G δ subset of ( G, T U ) . Pr o of . The given condition is o bviously necessary . Conv ersely , to prov e that a ddition is contin uous, we a re going to see that (1 /mn ) U + (1 /mn ) U ⊆ (1 /n ) U ∀ n ∈ N . Fix x, y ∈ (1 /mn ) U a nd observe that j x, j y ∈ (1 /m ) U for all 1 ≤ j ≤ n . This implies j ( x + y ) = j x + j y ∈ (1 /m ) U + (1 /m ) U ⊆ U for all 1 ≤ j ≤ n and hence x + y ∈ (1 /n ) U . If U is a GTG set, then T U is a group top olo gy of G , hence U ∞ = { 0 } T U is a subgr oup of G . QED Prop ositio n 4.4 g ives the p ossibility to deﬁne a GTG set in a mor e precise wa y . Namely , one ca n introduce the following inv ariant for a sy mmetric subset U ⊆ G o f an ab elian gr oup G with 0 ∈ U γ ( U ) := min { m ∈ N : (1 /m ) U + (1 /m ) U ⊆ U } 14 with the usua l co nv en tion γ ( U ) = ∞ when no such m exists. According to Pro po sition 4 .4, U is a GTG se t iﬀ γ ( U ) < ∞ . Let us call γ ( U ) the GTG-de gr e e o f U , it obviously measur es the GTG-nes s of the symmetric set U containing 0 . Clearly , U has GTG-degree 1 pr ecisely when U is a subgroup. (Compare this with the mo dulus of c onc avity deﬁned in [31, 3.1].) Prop ositio n 4 .5 A symmet ric subset A of a ve ctor sp ac e E which satisﬁes [0 , 1] A ⊆ A is GTG iﬀ it is pseudo c onvex. Pr o of . By assumption, [0 , 1] A ⊆ A . This implies that (1 /n ) A = 1 n A . In the in tro duction to this section we hav e already shown that a symmetric set A is pseudo convex if and only if it satisﬁe s 1 n A + 1 n A ⊆ A for some n ∈ N . Since (1 /n ) A = 1 n A, it is a co nsequence of 4 .4 that A is pseudo convex iﬀ it is GTG. QED Example 4. 6 The subsets of U = ([ − 1 , 1] × { 0 } ) ∪ ( { 0 } × [ − 1 , 1]) and V = ( R × { 0 } ) ∪ ( { 0 } × R ) o f Example 4.1 are not GTG s e ts . Remark 4 .7 Let U be a symmetr ic subs et of an ab elian gr oup G with 0 ∈ U . W e ana lyze the b ehaviour o f the sequence (1 /n ) U in the following cas es of interest: (a) If U ∞ = { 0 } , then U is a GTG set iﬀ ( G, T U ) is UFSS. (b) No w assume t hat (1 /m ) U = U ∞ for some m . Then U is GTG iﬀ U ∞ is a subgroup. It is clear that (1 /m ) U = U ∞ is a union o f cyclic subgr oups. W e know (Prop ositio n 4.4) that if U is a GTG set, then U ∞ m ust b e a subgr oup. B ut in this circumstance, we can inv ert the implication. Indeed, if U ∞ = (1 /m ) U is a subgroup, then obviously (1 /m ) U + (1 /m ) U ⊆ (1 /m ) U ⊆ U holds true, so tha t U is a GTG set. This fact explains o nce more why the subset V = V ∞ from Example 4 .1 is not a GTG set (simply it is not a subgroup). (Note that w e are not considering he r e the third p os sibility: U ∞ 6 = { 0 } yet the chain (1 / m ) U do es not stabilize.) Remark 4 .8 Let U b e a symmetric subset of a gr o up G . Then the following holds true: (a) (1 /n )((1 /m ) U ) = (1 /m )((1 /n ) U ) fo r all n , m ∈ N . (b) F or symmetric subsets A and B of G and k ∈ N we hav e: (1 /k ) A + (1 /k ) B ⊆ (1 /k )( A + B ). (c) The following as sertions are equiv alent: (i) U is a GTG set in G . (ii) F or every k ∈ N the set (1 /k ) U is a GTG s e t in G . (iii) Ther e exis ts k ∈ N s uch that (1 /k ) U is a GTG s et in G . In this ca se T U = T (1 /k ) U for every k ∈ N . Pr o of . (a) and (b) are straightforw ard. (c) (i) ⇒ (ii): Supp ose that (1 / m ) U + (1 /m ) U ⊆ U . This yields (1 /m )(1 /k ) U + (1 / m )(1 /k ) U ( a ) = (1 / k )(1 /m ) U + (1 /k )(1 /m ) U ( b ) ⊆ (1 /k )[(1 /m ) U + (1 /m ) U ] ⊆ (1 /k ) U and hence the assertio n follows from Prop ositio n 4.4. (ii) ⇒ (iii) is tr ivial. (iii) ⇒ (i) Le t m b e such that (1 /m )((1 /k ) U ) + (1 /m )((1 /k ) U ) ⊆ (1 /k ) U . Since (1 /mk ) U ⊆ (1 / m )((1 /k ) U ) w e deduce (1 /mk ) U + (1 /mk ) U ⊆ (1 /k ) U ⊆ U 15 and the a ssertion is a co nsequence of P rop osition 4.4. Finally , ass ume that U is a GTG set. F rom (1 / mk ) U ⊆ (1 /m )((1 / k ) U ) ⊆ (1 /m ) U , we obtain the equality of the top ologies T U = T (1 /k ) U . QED Next w e give inv estigate under which co nditions in tersections and pro ducts of GTG sets are GTG. Lemma 4.9 (a) Inverse images of GTG sets by gr oup homomorphisms ar e GTG. Mor e pr e cisely, if φ : G → H is a homomorphism and A ∋ 0 is a symmetric su bset of H , then γ ( φ − 1 ( A )) ≤ γ ( A ) . If A ⊆ φ ( G ) then γ ( φ − 1 ( A )) = γ ( A ) . (b) If { A i : i ∈ I } is a family of GTG set s of a gr oup G and the subset { γ ( A i ) : i ∈ I } of N is b ounde d, then also T i ∈ I A i is a GTG subset of G . In p articular, the interse ction of any ﬁnite family of GTG sets of G is a GTG set of G . (c) Le t ( G i ) i ∈ I b e a family of gr oups and let A i b e a subset of G i for every i ∈ I . The set A := Q i ∈ I A i ⊆ Q i ∈ I G i is a GTG set of G := Q i ∈ I G i iﬀ al l A i ar e GTG sets and the s u bset { γ ( A i ) : i ∈ I } of N is b ounde d. In p articular, (c 1 ) if I is ﬁ nite then Q i ∈ I A i is GTG iﬀ al l t he sets A i ar e GTG. (c 2 ) for an arbitr ary index set I , U is a GTG set of a gr oup G iﬀ U I is a GTG set of G I . Pr o of . (a) is a consequence of the iden tity (1 /m ) φ − 1 ( A ) = φ − 1 ((1 /m ) A ) . (b) It is straightforw ard to prove that (1 /m ) T i ∈ I A i = T i ∈ I (1 /m ) A i . By our h yp othesis we may choose m so large that (1 /m ) A i + (1 /m ) A i ⊆ A i for a ll i ∈ I a nd obtain (1 /m ) T i ∈ I A i + (1 /m ) T i ∈ I A i ⊆ T i ∈ I A i . The a ssertion follows from Prop os itio n 4.4. (c) follows easily from (a), Prop ositio n 4.4 and the equa lit y (1 /n ) Q i ∈ I A i = Q i ∈ I (1 /n ) A i . QED Example 4. 10 (a) L e t P b e the set of all po sitive primes. F or each p ∈ P we deﬁne the symmetric subset of Z U p = { 0 } ∪ { ± 2 n 2 3 n 3 · · · p n p : n 2 , n 3 , · · · , n p ∈ N ∪ { 0 }} . Note that for p, q ∈ P , w e have (1 /q ) U p = U p for q ≤ p and (1 / q ) U p = { 0 } otherwise. This implies that for every p ∈ P , U p is a GTG set, ( U p ) ∞ = { 0 } and U p + U p 6⊆ U p . Hence p < γ ( U p ). The subset U = Q p ∈ P U p ⊆ Z P is s ymmetric and satisﬁes U ∞ = Q p ∈ P ( U p ) ∞ = { 0 } , but it is no t a GTG s et by Lemma 4 .9(c). Deﬁne V p := U p × Q q ∈ P , q 6 = p Z . Then for every p ∈ P the sets V p are GTG, howev er, their int erse ction T p ∈ P V p = U is no t GTG as shown ab ove. (b) A simpler example of a non- GTG intersection of GTG sets ca n b e obtained from the s et U of Example 4.1: it is the intersection of a ll || · || 1 /n -unit balls U n in R 2 , for n ∈ N . (c) If U n is the subset of G n = R 2 , as in (b), then γ ( U n ) → + ∞ . Ther efore, U = Q n ∈ N U n is not a GTG set in G = ( R 2 ) N , according to item (c) of L e mma 4.9. The next pr op osition give an in tuitive idea ab out GTG sets: Prop ositio n 4 .11 If G is a c omp act c onn e cte d ab elian gr oup and U is a GTG set of G with H aar m e asur e 1, then U = G. 16 Pr o of . F or every p ositive n the map f n : G → G deﬁned by f n ( x ) = nx is a surjective contin uous endomo r phism (such a g roup G is alwa ys divisible, see e. g. [21, 2 4.25]). Since every surjectiv e contin uous endomorphism is measure preserving ([19]), one has µ ( f − 1 n ( U )) = µ ( U ) = 1. There fore, also U ∞ = \ n f − 1 n ( U ) has measure 1. Since U ∞ is a subgr oup, this is p oss ible only when U ∞ = G . T his yields U = G. QED 4.2 Construction of GTG sets and UFSS top ologies Now we shall pro p o se a genera l construction for building inﬁnite GTG sets in ab elian gro ups. In case the group is complete metric, the GTG set can b e chosen c o mpact and totally disconnec ted. Remark 4 .12 In the construction we shall need the following sets of sequences of in teger s: Z = Z N 0 , K m =    ( k j ) ∈ Z : ∞ X j =0 | k j | 2 j ≤ 1 2 m    for m ∈ Z , a nd P = ∞ Y j =0 { 0 , ± 1 , ± 2 , ± 3 , . . . , ± 2 j +2 } (a) O bviously , K m ⊆ P when m ≥ − 2, and K m + K m ⊆ K m − 1 , for m ∈ Z . (b) W e use a ls o the direct sum Z 0 = L N 0 Z . F or ( a n ) ∈ Z 0 and any sequence ( x n ) o f elements of G the sum P ∞ j =0 a j x j makes s ense and will b e used in the sequel. In this wa y , every element x = ( x n ) ∈ G N 0 gives rise to a group homo morphism ϕ x : Z 0 − → G deﬁned b y ϕ x (( a n )) := P ∞ j =0 a j x j for ( a n ) ∈ Z 0 . (c) Z will be equipp ed with the pro duct top ology , where Z has the discrete to po logy with bas ic op en neigh b orho o ds of 0 the subgr oups W n = { ( k j ) ∈ Z : k 0 = k 1 = . . . = k n = 0 } , n ∈ N 0 . Thus, P is a compa ct zero- dimensional subspace of Z . Let us see tha t K m is closed in P for m ∈ Z , hence a compa c t zer o-dimensional space on its own acco un t. Indeed, pick ξ = ( k j ) j ≥ 0 ∈ P \ K m . Then P j ≥ 0 | k j | 2 j > 1 2 m , s o P n j =0 | k j | 2 j > 1 2 m for so me index n . Hence the neig hbo rho o d ( ξ + W n ) ∩ P o f ξ misses the set K m . A sequence ( x n ) n ≥ 0 in G will be ca lle d ne arly indep endent , if it s atisﬁes ∞ X j =0 a j x j = 0 = ⇒ ( a n ) = 0 for all ( a n ) ∈ P ∩ Z 0 . (4) This term is motiv ated by the fact, that usually a sequence ( x n ) n ≥ 0 in G is ca lle d indep endent, if k er ϕ x = 0. Claim 4 .13 If G is an ab elian gr oup and x = ( x n ) is a n e arly indep endent se quenc e of G , t hen ϕ x ↾ K − 1 ∩ Z 0 : K − 1 ∩ Z 0 → G is inje ctive. Pr o of . Ass ume that ϕ x (( k j )) = ϕ x (( l j )) for ( k j ) ∈ K − 1 and ( l j ) ∈ K − 1 . Then 0 = P n j =0 ( k j − l j ) x j and | k j − l j | ≤ 2 j +2 , co m bined with near independence, imply k j = l j for all j . QED The following lemma reveals a s uﬃcien t co ndition under which an ab elian group G admits a non-discr ete UFSS group top ology , namely the existence of a nea rly indep endent s equence. The necessity of this co ndition will b e established at a later stage (see Cor ollary 4 .25). 17 Lemma 4.14 L et G b e an ab elian gr ou p and let x = ( x n ) b e a ne arly indep endent se qu enc e of G . Then t he set X := ϕ x ( K 0 ∩ Z 0 ) is a GTG su bset of G with γ ( X ) = 2 . Mor e pr e cisely, (1 / 2 m ) X = ϕ x ( K m ∩ Z 0 ) =    n X j =0 k j x j : n ∈ N , k j ∈ Z , n X j =0 | k j | 2 j ≤ 1 2 m    , (5) X ∞ = 0 and ( x n ) t en ds to 0 in T X , so T X is a non-discr ete UFSS t op olo gy. Pr o of . The inclusion ⊇ in (5) is obvious. W e prov e the following stronger version of the reverse inclusion b y induction: if x = n X j =0 k j x j ∈ X with ( k j ) ∈ K 0 , then x ∈ (1 / 2 m ) X = ⇒ ( k j ) ∈ K m (6) F or m = 0 the assertio n is trivia l. So supp o se (6) holds true for m and let x = P n j =0 k j x j ∈ (1 / 2 m +1 ) X , w ith ( k j ) ∈ K 0 . Since x, 2 x ∈ (1 / 2 m ) X , our the induction hypo thesis gives ( k j ) ∈ K m . Mor eov er, there exists a representation 2 x = P n j =0 l j x j with ( l j ) ∈ K m , i.e., P n j =0 | l j | 2 j ≤ 1 2 m . (Observe tha t without los s of generality w e may assume that the upper index for the summatio n ma y b e as sumed to b e eq ua l for x a nd 2 x .) Then ϕ x ((2 k j )) = ϕ x (( l j )) with (2 k j ) , ( l j ) ∈ K − 1 , so Cla im 4.13 applies 2 k j = l j for all j a nd hence P n j =0 | k j | 2 j = P n j =0 | l j | 2 j +1 ≤ 1 2 m +1 . This proves (6), and c onsequently also (5). O bviously , (6 ) yields a lso X ∞ = { 0 } . F or m = 1 the equation (5) and K 1 + K 1 ⊆ K 0 give (1 / 2) X + (1 / 2) X ⊆ X . Hence γ ( X ) ≤ 2, and co nsequently X is a GTG set and T X is a UFSS top o lo gy . F or a ﬁxed N ∈ N the deﬁnition of X a nd (5) give x n ∈ (1 / 2 N ) X fo r all n ≥ N , so { x n : n ≥ N } ⊆ (1 / 2 N ) X . This shows that x n → 0 in T X and s o T X is not discrete. Finally , to prove that γ ( X ) ≥ 2 it suﬃces to observe tha t γ ( X ) = 1 would imply that X is a subg roup, so X = X ∞ . Now X ∞ = { 0 } contradicts the non-discr eteness of T X . QED Let ( G, d ) be a metric a be lian gro up, let v b e the group s eminorm as s o ciated to the metric d (i.e., v ( x ) = d ( x, 0) for x ∈ G ) and let B ε = { x ∈ G : v ( x ) ≤ ε } be the closed disk with radius ε aro und 0. F or a near ly indep endent sequence ( x n ) of G and a non-neg ative n ∈ Z let ε n := min    v   n X j =0 a j x j   : | a j | ≤ 2 j +2 , ( a j ) 6 = (0)    > 0 . (7) W e ca ll the seque nc e ( x n ) almost indep endent , if the inequality 2 n +3 v ( x n +1 ) < ε n ≤ v ( x n ) (8) holds. Note that ε n ≤ v ( x n ) obviously follows fr om the deﬁnition of ε n . More ov er, every almost indep endent sequence (rapidly) conv erges to 0 in ( G, d ). It is str a ight forward to pr ov e that a s ubsequence of a strictly , resp ectively almo s t indep endent s e quence is aga in strictly , r esp ectively almost indep endent. The motiv atio n to intro duce the sharp er no tion of almost indepe ndent sequence is given in the le mma b elow. First we need to iso late a prop erty that will b e frequently used in the sequel: Claim 4 .15 If ( G, d ) is a metric gr oup and ( x n ) is an almost indep endent se qu enc e of G , then ϕ x ( K m ∩ W n ∩ Z 0 ) ⊆ B v ( x n ) 2 m +2 for any m ∈ Z and n ≥ 0 . Pr o of . W e hav e to prove that v  P k j = n +1 k j x j  < 1 2 m +2 v ( x n ) , whenever ( k j ) ∈ K m . This follows applying (8) to the term 2 j v ( x j ) in v   k X j = n +1 k j x j   ≤ k X j = n +1 | k j | v ( x j ) = k X j = n +1 | k j | 2 j 2 j v ( x j ) < k X j = n +1 | k j | 2 j 1 4 v ( x j − 1 ) ≤ 1 4 v ( x n ) k X j = n +1 | k j | 2 j ≤ 1 2 m +2 v ( x n ) . 18 QED Lemma 4.16 L et ( G, d ) b e a metric gr oup and let ( x n ) b e an almost indep endent se quenc e of G . Then (a) t he non-discr ete UFSS top olo gy T X gener ate d by the GTG set X of G c orr esp onding to ( x n ) as in L emma 4.14, is ﬁn er than the original top olo gy of G ; (b) t he subse quenc e ( x 2 n ) is stil l almost indep endent, and for the GTG set Y of G c orr esp onding t o ( x 2 n ) as in L emm a 4.14, T X < T Y . Pr o of . (a) W e have to prove that for a given ε > 0, there exists m ∈ N such tha t (1 / 2 m ) X ⊆ B ε . Since x n → 0 in the metric top olo gy , ther e exists m ∈ N such that v ( x m − 1 ) 2 m +2 < ε . As K m ⊆ W m − 1 and (1 / 2 m ) X = ϕ x ( K m ∩ Z 0 ), from Claim 4.15 w e obtain (1 / 2 m ) X = ϕ x ( K m ∩ Z 0 ) = ϕ x ( K m ∩ W m − 1 ∩ Z 0 ) ⊆ B ε . (b) By Lemma 4.14, T Y is a UFSS to po logy on G . Since Y ⊆ X , we trivially hav e T Y ⊇ T X . It remains to b e shown that T Y is strictly ﬁner than T X . Let us prove that the T X nu ll–se q uence ( x 2 n +1 ) do es not conv erge to 0 in T Y . It is enough to show tha t { x 2 n +1 : n ∈ N } ∩ Y = ∅ . So assume x 2 m +1 = P n j =0 k j x 2 j for some m ∈ N and ( k j ) ∈ K 0 . As x 2 m +1 ∈ ϕ x ( K 0 ) as w ell, this contradicts Claim 4.13. QED In the next theorem we show that the set X from the prev ious lemmas, corresp onding to an almost indep endent sequence of G , has a co mpa ct tota lly disconnec ted closure when G is complete. Theorem 4. 1 7 L et ( G, d ) b e a c omplete metric gr oup and let ( x n ) b e an almost indep endent se quenc e of G . Then the closur e e X of the GTG set X c orr esp onding to ( x n ) as in L emma 4.14, is c omp act and total ly disc onne cte d. Mor e over, e X is a GTG set with γ ( e X ) = 2 , so T e X is a non-discr et e UFSS top olo gy ﬁ ner than the original top olo gy of G . Pr o of . W e intend to extend the map ϕ x deﬁned in item (b) o f Remark 4.1 2 to a map ϕ : S m ∈ Z K m − → G by setting ϕ (( k j )) = P j ≥ 0 k j x j (the correctness o f this deﬁnition is check ed b elow). F urthermor e, we show that ϕ ↾ K m is contin uous for each m , while ϕ ↾ K 0 is injective. Since ea ch K m is a compac t zero-dimens io nal spa ce (Remark 4.12 (c)), this will prov e that e X = ϕ ( K 0 ) itself is a co mpact z ero-dimensiona l space, while the s ubspaces ϕ ( K m ) with m < 0 are just compact. F or a ﬁxed ( k j ) j ≥ 0 ∈ K m let y n = P n j =0 k j x j ∈ G . T o see tha t ( y n ) is a Cauch y sequence in G apply Claim 4.15 to g et v ( y k − y n ) ≤ 1 2 m +2 v ( x n ) for ev ery pair n ≤ k . Since x n → 0, this pr ov es that ( y n ) is a Cauch y s equence in G . Since G is co mplete, the litmit lim y n exists and ϕ (( k j )) = P j ≥ 0 k j x j and e X := ϕ ( K 0 ) make sense . Since the norm function v : G → R is cont inuous, we obta in from Claim 4.15, after passing to the limit: ϕ ( K m ∩ W n ) ⊆ B v ( x n ) 2 m +2 . (9) Even if ϕ is no t a ho momorphism, one has ϕ ( ξ + η ) = ϕ ( ξ ) + ϕ ( η ) , whenev er ξ = ( k j ) , η = ( l j ) ∈ K m , (10) where ξ + η = ( k j + l j ) ∈ K m − 1 . Fix m ∈ Z . In order to s how that ϕ ↾ K m is continous, ﬁx ( k j ) ∈ K m and ε > 0. Ther e exists n ∈ N such that 1 2 m +1 v ( x n ) < ε . F or ( l j ) ∈ ( k j ) + W n we hav e ϕ (( l j )) ∈ ϕ (( k j )) + B ε (0) by (9 ), whic h shows that ϕ is contin uo us. In order to show that ϕ is injective, we show the following s tr onger statement , that will be nec e ssary bellow: Claim 4 .18 If ( k j ) ∈ K − 1 and ( l j ) ∈ K 0 with ( k j ) 6 = ( l j ) , then ϕ (( k j )) 6 = ϕ (( l j )) . 19 Assume for a c o nt ra diction that ϕ (( k j )) = ϕ (( l j )) with ( k j ) 6 = ( l j ). Fix m minimal with k m 6 = l m . Then ( l m − k m ) x m = X j >m k j x j − X j >m l j x j with | k j − l j | ≤ 3 · 2 j < 2 j +2 for all j ≥ m (a s k j ≤ 2 j +1 and l j ≤ 2 j ). Hence the deﬁnition of ε m gives ε m ≤ v (( l m − k m ) x m ) ≤ | k m +1 − l m +1 | v ( x m +1 ) + v   X j >m +1 k j x j   + v   X j >m +1 l j x j   . (11) T o the seco nd and the third term in the right hand side of (11 ) w e may apply (9) with m = 0, resp ectively m = − 1 and n = m + 1 and obtain v   X j >m +1 k j x j   + v   X j >m +1 l j x j   ≤ 1 2 v ( x m +1 ) + 1 4 v ( x m +1 ) = 3 4 v ( x m +1 ) . (12) Since | k m +1 − l m +1 | ≤ 3 · 2 m +1 yields | k m +1 − l m +1 | v ( x m +1 ) ≤ 3 · 2 m +1 v ( x m +1 ) , by (11 ) and (12 ), we get ε m ≤ 3 · 2 m +1 v ( x m +1 ) + 3 4 v ( x m +1 ). Along with (8), applied with n = m , we get ε m ≤ 3 · 2 m +1 v ( x m +1 ) + 3 4 v ( x m +1 ) < 2 m +3 v ( x m +1 ) < ε m , a con tra dictio n. This proves Claim 4.18. F rom Claim 4 .18 we conclude that ϕ ↾ K 0 is a contin uous bijective mapping. Since K 0 is compact, ϕ ↾ K 0 : K 0 → e X = ϕ ( K 0 ) is a ho meomorphism which implies in particular that e X is compact a nd tota lly disconnected. Since ϕ is an extens io n of ϕ x and since Z 0 ∩ K 0 is dense in K 0 , we deduce that X = ϕ x ( Z 0 ∩ K 0 ) = ϕ ( Z 0 ∩ K 0 ) is dense in e X = ϕ ( K 0 ). Since the latter set is compact, it m ust b e closed in G . Therefore, e X coincide s with the closur e of X . Next w e claim that (1 / 2 m ) e X = ϕ ( K m ) =    ∞ X j =0 k j x j : k j ∈ Z , ∞ X j =0 | k j | 2 j ≤ 1 2 m    , (13) as in the case of the set X in Le mma 4.14. The inclusion ⊇ in (13) is obvious. W e prov e the following stronge r version of the reverse inclusion b y inductio n: if x = ∞ X j =0 k j x j , with ( k j ) ∈ K 0 , then x ∈ (1 / 2 m ) e X = ⇒ ( k j ) ∈ K m . (14) F or m = 0 the ass ertion is trivial. So supp ose (14) holds true for m a nd let x = ϕ (( k j )), with ( k j ) ∈ K 0 belo ng to (1 / 2 m +1 ) e X . Since x, 2 x ∈ (1 / 2 m ) e X , by the induction hypothesis, ( k j ) ∈ K m . Moreov er, there exists a r epresentation 2 x = ϕ (( l j )) with ( l j ) ∈ K m . The n ϕ ((2 k j )) = ϕ (( l j )). Since, (2 k j ) ∈ K − 1 and ( l j ) ∈ K 0 , from Claim 4 .18 we conclude that 2 k j = l j for all j a nd hence P ∞ j =0 | k j | 2 j ≤ 1 2 m +1 . This pr ov es (14), and conseq uent ly a lso (13). In par ticular, from (13) we get (1 / 2) e X = ϕ ( K 1 ). Since K 1 + K 1 ⊆ K 0 from Remar k 4.12(b), this combined with with (13), g ives (1 / 2) e X + (1 / 2) e X ⊆ e X . Hence γ ( e X ) ≤ 2. It re mains to note that (1 4) implies also e X ∞ = { 0 } . Hence T e X is a UFSS top olo gy coar ser than T X (as X ⊆ e X ), so it is non-disc r ete. In particular , e X 6 = { 0 } = e X ∞ , so γ ( e X ) = 2. F rom (9) a nd (1 3) we conclude that T e X is ﬁner than the original topo logy o f G . QED In order to characterize thos e ab elian metrizable gr oups which admit a (strictly) ﬁner UFSS g roup top ology , w e need the following deﬁnition which will characterize these groups. Deﬁnition 4.19 An ab elian top olo gical gr oup G is called lo c al ly b ounde d if there exis ts some n ∈ N such that the subgroup G [ n ] = { x ∈ G : nx = 0 } is op en. 20 Remark 4 .20 G is lo ca lly bounded iﬀ it has a neigh b ourho o d U in which all elements are of b ounded or der . Obviously , a metric ab elian top olo gical gr o up G is not lo cally b ounded iﬀ there e xists a null sequence x n → 0 such that o ( x n ) → ∞ . A lo cally compact ab elian gro up G is lo cally b ounded iﬀ it ha s a n op en compact subgroup of ﬁnite exp onent. Ideed, ass ume that G is a lo cally co mpact, lo cally b ounded ab elian group. F or suitable n ∈ N the subg r oup G [ n ] is op en. By the structure theo rem for lo cally compact ab elian g roups, G [ n ] co nt ains a n op en subgro up K . It is clear that K is op e n in G a nd o f ﬁnite e x po nent . The conv erse implica tion is trivial. Theorem 4. 2 1 L et ( G, d ) b e an ab elian, metrizable, non–discr ete gr oup. The fol lowing assertions ar e e qu ivalent: (i) G is not lo c al ly b ounde d; (ii) ther e ex ists a ﬁn er non–discr ete UFSS gr oup top olo gy T X on G ; (iii) t her e exists a s t rictly ﬁn er non–discr ete UFSS gr oup top olo gy T Y on G ; (iv) ther e exists an almost indep endent se qu enc e in G . Pr o of . (iii) = ⇒ (ii) is trivial. (ii) = ⇒ (i): Let G b e lo cally bounded, this means there exists n ≥ 1 s uch that the subg roup G [ n ] is open. Supp ose there exists a UFSS topo logy T X on G with distinguished neighborho o d X which is ﬁner tha n the topo logy τ induced by the metric. W e may ass ume that X ⊆ G [ n ], be c a use otherwis e w e ca n re place X by X ∩ G [ n ]. Then (1 /n ) X = X ∞ . Since we ass umed T X to be ﬁner than the origina l topo logy and hence Hausdorﬀ, { 0 } = X ∞ = (1 /n ) X . This implies that T X is dis crete. So the only ﬁner UFSS group top olo gy on G is the discrete one. (iv) = ⇒ (iii): this is covered by the pre c e ding lemma. (i) = ⇒ (iv): Assume now that G is not loca lly bounded. W e ha ve to show that there exists an almost independent sequence ( x n ) of elemen ts in G . This will b e done b y induction. F or n = 0 condition (4) is equiv alent to 3 x 0 6 = 0 6 = 4 x 0 . So ﬁx an element x 0 ∈ G o f o rder g reater than 4. Assume that x 0 , . . . , x n hav e alr e ady b een chosen to s a tisfy (4). Deﬁne ε n by (7). Then choose x n +1 with v ( x n +1 ) < ε n / 2 n +3 and o ( x n +1 ) > 2 n +3 . T o chec k that this works, let x = P n +1 j =0 a j x j with ( a j ) 6 = (0) and | a j | ≤ 2 j +2 for j = 0 , 1 , . . . , n + 1. If a 0 = . . . = a n = 0 then x 6 = 0, since | a n +1 | ≤ 2 n +3 < o( x n +1 ) . Otherwis e , we hav e v ( x ) ≥ v  P n j =0 a j x j  − v ( a n +1 x n +1 ) ≥ ε n − 2 n +3 v ( x n +1 ) > 0 , so x 6 = 0. By the c hoice of each x n , the seq ue nc e is almo s t indep endent. QED Remark 4 .22 Let us note that for the set Y constructed in the pro of, the strictly ﬁner non-discr ete UFSS top ology T Y is still lo cally unbounded. So to the gr oup ( G, T Y ) the same co nstruction ca n b e applied to provide an inﬁnite strictly incr easing chain of non-discr e te UFSS top o logies T Y = T Y 0 < T Y 1 < . . . < T Y n < . . . . Hence in the theorem one can a lso add a s tronger pr o p e r ty (v) claiming the existence o f such a chain. Corollary 4.23 L et ( G, d ) b e a c omplete ab elian, metrizable non lo c al ly b ounde d gr oup. Then ther e exists a c omp act total ly disc onne cte d GTG set X of G , such that T X is a ﬁner non-discr et e UFS S gr oup t op olo gy on G . Pr o of . According to the ab ov e theorem G admits an almost independent sequence ( x n ). QED E. Hewitt [20] obser ved tha t the gr o up T and the group R ha ve the prop erty that the only stronge r lo cally co mpa ct group top ologies are the discre te top olo gies. Since lo c a lly minimal top ologies genera lize the lo c a lly compa ct g roup top ologies, this suggests the following ques tion: Do the gr oups T and R admit str onger non-discr ete lo c al ly minimal top olo gies? The next co rollar y answers this question in a str o ngly po sitive wa y . Na mely , the class of a ll no n-totally disconnected lo cally compact metrizable ab elia n g roups (in place o f T a nd R o nly ) and for the smaller c la ss o f UFSS top ologies (in place of lo c a lly minimal topo lo gies). 21 Corollary 4.24 A lo c al ly c omp act ab elian metrizable gr oup G has a strict UFSS r eﬁnement iﬀ G c ontains no op en c omp act sub gr oup of ﬁnite exp onent. This happ ens for example, if G is not total ly disc onne ct e d. Pr o of . The ﬁrst assertio n is obvious when G is discrete, so we a ssume that G is non-discrete in the s equel. According to 4.2 1, G ha s a strict, non-discr ete UFSS reﬁnement iﬀ G is lo ca lly bo unded, which, by 4 .20, is equiv ale n t to the existence of a co mpact open subg roup of ﬁnite exp onent. In order to prov e the sec o nd statement it is suﬃcient to show that every gr oup H which ha s an op en compact subroup K of ﬁnite exp onent is totally dis c onnected. The connected co mpo ne nt C of H is contained in K and hence bo unded. O n the o ther hand side, as every compa ct ab elian connected group, C is divisible. This implies that C is trivial and he nce H is totally disconnected. QED Now comes the topo logy-free version of Theorem 4.21: Corollary 4.25 F or an ab elian gr oup G TF AE: (i) G is not b ounde d; (ii) G admits a non–discr ete UFSS gr oup top olo gy; (iii) t her e exists a n e arly indep endent se quenc e in G . Pr o of . The implication (iii) = ⇒ (ii) was pr ov ed in Lemma 4.14. T o prove the implication (ii) = ⇒ (i) assume G a dmits a non–discrete UFSS g r oup top olo gy T with distinguished neighborho o d U of 0. Then for every n ∈ N the set (1 / n ) U is a T -neighborho o d of 0, hence (1 /n ) U 6 = { 0 } . If nG were { 0 } for some n ∈ N , then (1 /n ) U = U ∞ = { 0 } which is a contradiction. So G is unbounded. T o prov e the implica tion (i) = ⇒ (iii) pick a countable s ubgroup H o f G that is still not b ounded. Since H is countable, there exists an injective homomo r phism j : H → T N . Denote by d the metric induced on H b y this embedding. Then ( H, d ) is a metric prec ompact group, hence it is not discr e te. Mo reov er, for no n ∈ N the subgroup H [ n ] is op en. Indeed, if H [ n ] were open, then by the precompactness of H it ha s ﬁnite index in H . Hence mH ⊆ H [ n ] for some m ∈ N . Therefore, mnH = 0, a contradiction. This a rgument pr ov es that no subgr oup H [ n ] ( n ∈ N ) is op en in H . Hence, ( H , d ) is not lo ca lly b o unded. Then H contains a n almost indep endent sequence ( x n ) by the ab ov e theorem. Clea rly , this is also a ne a rly independent sequence in H , and consequently , also in G . QED 5 Lo cally GTG groups 5.1 Lo cally GTG groups and their prop erties Deﬁnition 5.1 [V. T arieladze, oral communication] W e say that a Hausdor ﬀ top olo gical ab elian group G is lo c al ly GTG if it admits a basis of neighbor ho o ds of the identit y fo rmed by GTG subsets of G . Example 5. 2 (a) E very UFSS group is lo cally GTG. In par ticula r R and T ar e lo cally GTG. (b) Every lo cally conv ex space is loc ally GTG. (c) Assume that G is a b ounded ab elian group with exp onent m . If U is a GTG neighborho o d of 0 in some group top ology τ of G , then U ∞ = (1 /m ) U is a τ -neighbo rho o d of 0. Therefo r e, 22 (c 1 ) ( G, τ ) is lo cally GTG precisely when ( G, τ ) is linearly top ologized. (c 2 ) ( G, τ ) is UFSS precisely when ( G, τ ) is discrete. Example 5. 3 A top olog ical v ector space is said to b e lo c al ly pseudo c onvex if it has a basis of pseudo convex neigh- bo rho o ds of zero. A topo logical vector spa ce is lo cally GTG a s a topo logical ab elian gr oup if a nd only if it is loc a lly pseudo conv ex. Pr o of . Applying 4.5, it suﬃces to show that a top ologica l vector spa ce whic h is lo cally GTG has a neighborho o d basis consisting of balanced GTG se ts . So ﬁx a GTG neighborho o d A and deﬁne B := { a ∈ A : [0 , 1] a ⊆ A } . It is straightforward to prove that [0 , 1] B ⊆ B a nd it is a well known fac t that B is a neighborho o d of zero. Le t us pr ove that B is GTG. Since A was assumed to b e GTG, there exists n ∈ N such tha t (1 / n ) A + (1 /n ) A ⊆ A . Observe tha t (1 /n ) B = 1 n B . W e shall show that 1 n B + 1 n B ⊆ B . So ﬁx a, b ∈ 1 n B and t ∈ [0 , 1]. Let us see that t ( a + b ) b elo ng s to A : t ( a + b ) = ta + tb ∈ 1 n B + 1 n B ⊆ (1 /n ) A + (1 /n ) A ⊆ A . QED Example 5. 4 Lo ca l GTGnes s may seem to be a to o mild prop erty , but there exist na tural examples of a b elia n top ological groups lacking it. Co nsider the top olo gical vector s pace G = L 0 of all classes of Leb esg ue mea surable functions f on [0 , 1] (mo dulo almost everywhere equality) with the top ology of conv ergence in measur e. This to po logy can be deﬁned by the inv ariant metric d ( f , g ) = Z 1 0 min { 1 , | f ( t ) − g ( t ) |} dt (for details see for instance [22, Ch. 2]). It is known that L 0 is not lo cally pse udo conv ex and hence, b y Exa mple 5.3, it is not lo c a lly GTG a s a top olo gical g roup. Here w e collect several pro p er ties of lo cally GTG gro ups. Prop ositio n 5 .5 (a) Every su b gr oup of a lo c al ly GTG gr oup is lo c al ly GTG. (b) A gr oup with an op en lo c al ly GTG sub gr oup is lo c al ly GTG. (c) The pr o duct of lo c al ly GTG gr oups is lo c al ly GTG. (d) Q uotient gr oups of lo c al ly GTG gr oups n e e d not b e lo c al ly GTG. (e) Every gr oup lo c al ly isomorphi c to a lo c al ly GTG gr oup is lo c al ly GTG. In p articular, if a top olo gic al gr oup G admits a non-trivial lo c al ly GTG op en su b gr oup, then G is lo c al ly GTG. Pr o of . (a) is a consequence o f Lemma 4.9(a) a nd Example 4.3(b). (b) follows from the fact that a ny basis of neighborho o ds o f zero in the open subgroup is a bas is of neigh b orho o ds of zero in the whole group. (c) is a consequence of 4.9(c). (d) Let G b e a Hausdorﬀ group which is not lo cally GTG. G is a quotient of the free ab elia n top ological g r oup A ( G ) ([23]). The free lo cally conv ex space L ( G ) is loca lly GTG according to 5.2(b). According to a result of Usp enskij and Tk a chenko ([38] and [3 9]) the free a be lia n to po logical group A ( G ) is a subgro up of L ( G ) and he nce , due to (a), a lso loc ally GTG. This proves (d). (e) is straig htf orward using Lemma 4.9(a). QED Now we obtain another large c la ss of examples: Example 5. 6 (a) E very precompact ab elia n g roup is lo cally GTG. Indeed, every pr ecompact ab elian gr oup is (isomorphic to) a subgr oup of a p ow er o f T , s o items (a) a nd (c) of Pr op osition 5.5 a nd item (a) of Exa mple 5.2 a pply . 23 (b) Every locally compact ab elian gro up is lo cally GTG. I ndee d, every lo cally compa ct a belia n gro up has the form G = R n × G 0 , where n ∈ N and G 0 contains an op en co mpact subgroup. Then G 0 is lo cally GTG by item (a) and item (e) of Pr op osition 5 .5, while R n is UFSS, so lo cally GTG. No w item (c) o f Prop osition 5.5 applies. The connection betw een lo cally GTG and UFSS groups is the following: Theorem 5. 7 (a) If U is a GTG subset of an ab elian gr oup G , the quotient gr oup G U := ( G, T U ) /U ∞ is UFSS when e quipp e d with the quotient top olo gy of T U . (b) Every lo c al ly GTG gr oup G c an b e emb e dde d into a pr o du ct of UFSS gr oups. (c) A gr oup top olo gy τ on an ab elian gr oup G is a supr emum of U FSS top olo gies on G iﬀ τ is NSS and lo c al ly GTG. (d) If a gr oup top olo gy τ on an ab elian gr oup G is a supr emum of a family T = { τ i : i ∈ I } of UFSS top olo gies on G , then τ is UFSS iﬀ τ c oincide s with the supr emum of a ﬁnite subfamily of T . Pr o of . (a) Let U b e a GTG subset of G . Since U ∞ is the T U -closure of { 0 } , we can consider the Hausdorﬀ quotient group G U = ( G, T U ) /U ∞ , and the ca nonical epimorphism ϕ U : G → G U . Let m ∈ N be such that (1 / m ) U + (1 /m ) U ⊆ U. Let us show that fo r ev ery n ∈ N (1 /n ) ϕ U ((1 /m ) U ) ⊆ ϕ U ((1 /n ) U ) . Indeed, ﬁx an element ϕ U ( x ) ∈ (1 /n ) ϕ U ((1 /m ) U ) . Then fo r every k ∈ { 1 , . . . , n } , k x ∈ (1 /m ) U + U ∞ ⊆ U, hence ϕ U ( x ) ∈ ϕ U ((1 /n ) U ) . This shows that G U is a UFSS group w ith disting uis hed neighborho o d ϕ U ((1 /m ) U ). (b) Let U b e a basis of neighborho o ds of zero in G formed by GTG sets. The homomorphism Φ : G → Y U ∈U G U , Φ( x ) = ( ϕ U ( x )) U ∈U is injective and contin uous. Fix U ∈ U , a nd let m ∈ N b e such that (1 /m ) U + (1 /m ) U ⊆ U. Then (1 / m ) U + U ∞ ⊆ U, from whic h we deduce Φ( U ) ⊃ Φ( G ) ∩   Y U ′ ∈U \{ U } G U ′  × ϕ U ((1 /m ) U )  . This implies that Φ is op en onto its imag e. (c) It is clear that every s upremum of UFSS top ologies is b o th NSS and lo c a lly GTG. Co nv ersely , if G is lo cally GTG, its top ology is the supr emu m o f the family of top olo gies {T U } U ∈U where U is a basis of neighbor ho o ds of zer o. If moreover G is NSS, we may assume that no ne ig hborho o d in U contains nont rivia l subgr oups, and in particula r the topo logies T U are UFSS. (d) The suﬃciency is o b vious from Remark 3.13 (a). T o pro ve the necessity let us assume that τ = sup τ i is UFSS. Then there exis ts a distinguished τ - neighborho o d W of 0 such that τ = T W . The r e exis ts a ﬁnite subset J ⊆ I and τ j -neighborho o ds U j of 0 for each j ∈ J s uch that T j ∈ J U j ⊆ W . W e can assume without lo ss of genera lity that U j is a distinguished neig h b orho o d of 0 in τ j for each j ∈ J . Then (1 /n ) W ∈ sup j ∈ J τ j for every n . Hence τ = T W ≤ sup j ∈ J τ j . The inequality τ = sup j ∈ I ≥ sup j ∈ J τ j is trivial. This pr ov es that τ = sup j ∈ I τ j . QED Remark 5 .8 Note that “NSS” is needed in (c) ab ove; any no nmetrizable compac t a be lia n gr oup is lo cally GTG (see Example 5.6(a)) but its topolo gy is no t a s upr emum o f UFSS top olo gies. 24 Corollary 5.9 The class of lo c al ly GTG ab elian gr oups is stable u n der taking c ompletions. Pr o of . By T he o rem 5.7(b), every lo cally GTG gro up G can b e embedded int o a pro duct Q i G i of UFSS groups G i . By Prop osition 3.12 (a), the completion e G i of the UFSS group G i is UFSS. So the completion e G of G e mbeds into the pro duct P = Q i e G i of UFSS gr oups. By Prop os itio n 5.5 (c) P is lo c a lly GTG, so e G is lo cally GTG by Prop osition 5.5 (a). QED Theorem 5. 1 0 A Hausdorﬀ ab elian top olo gic al gr oup ( G, τ ) is a UFSS gr oup if and only if ( G, τ ) is lo c al ly minimal, lo c al ly GTG and NSS . Pr o of . Suppos e that ( G, τ ) is a UFSS gr oup with distinguished neighbo rho o d U . Then ( G, τ ) is U –lo cally minimal according to F acts 3.3(a), lo cally GTG acco rding to Example 5.2(a) and U do es not contain any nontrivial s ubgroup. Conv ersely , let ( G, τ ) b e lo cally minimal, lo cally GTG and NSS. There exists a neighbor ho o d of zero U which is a GTG s et, witnesses lo cal minimality a nd do es no t co n tain nontrivial subgro ups . The g roup top o lo gy T U generated by U is Hausdo rﬀ a nd coarser tha n τ ; since U is one of its zero neig hborho o ds, it coincides with τ . QED 5.2 Lo cally minimal, lo cally GTG groups In this section we will g ive v a rious prop erties of lo ca lly minimal lo ca lly GTG g roups. Most of o ur results are based on the following prop ositio n which a llows us to ﬁnd la r ge, in appro priate sense, minimal subg roups in a lo cally minimal group. Prop ositio n 5 .11 ( [12] ) Le t G b e a U –lo c al ly m inimal gr oup and let H b e a close d c entr al sub gr oup of G such that H + V ⊆ U for some neighb orho o d V of 0 in G . Then H is m inimal. Theorem 5. 1 2 If G is a U –lo c al ly minimal ab elian gr oup wher e U is a GTG set, then U ∞ is a m inimal sub gr oup. Pr o of . Prop osition 4.4 implies U ∞ + (1 /m ) U ⊆ U for some m ∈ N . Then, Prop ositio n 5.11 immediately gives us that U ∞ is a minimal subgr oup. QED One may ask whether GTG is needed in the ab ov e c o rollar y (see Ques tion 6.2). The pro blem is that without this assumption, the intersection U ∞ need no t b e a subgroup (a ltho ugh it is alwa ys a union of cyclic subgroups), as it happ ens in E xample 5.4. It ea sily follows from Theorem 5.12 that every lo cally minimal lo cally GTG a b elia n gr oup contains a minimal, hence pr ecompact, G δ -subgroup (note that the subgroup U ∞ in Theor em 5.12 is a G δ -set). Now we pro vide a diﬀeren t pro of of this fact, that makes no re course to lo cal GTG-ness. Prop ositio n 5 .13 Every lo c al ly minimal ab elian gr oup c ont ains a minimal, henc e pr e c omp act, G δ -sub gr ou p. Pr o of . Le t U witness lo cal minimality o f the gro up G . As in the pro of of P rop osition 2.13, it is p ossible to co nstruct inductively a s e quence ( V n ) of symmetric neig h b orho o ds of 0 in τ whic h satisfy V n + V n ⊆ V n − 1 (where V 0 := U ∩ − U ). It is easy to s ee tha t H = T n ∈ N V n is a subgroup of G , cont ained in each V n . In pa rticular, H + V 1 ⊆ V 0 ⊆ U . Now Prop ositio n 5.11 immediately gives us that H is a minimal subgr o up. QED Let us note that the minimal G δ -subgroup o btained in this pro of is cer tainly contained in the subgr o up U ∞ , provided U is a GTG set (as H ⊆ U and U ∞ is the la rgest subgr oup contained in U ). Ho wev er, this argument ha s the adv antage to require weaker hypo theses. The next cor ollary shows tha t no n- metrizable complete lo cally minimal ab elian g roups contain larg e compa ct subgroups. 25 Corollary 5.14 Every c omplete lo c al ly minimal ab elian gr ou p c ontains a c omp act G δ -sub gr ou p. Pr o of . F ollows directly from Prop osition 5.13. QED Corollary 5.15 L et ( G, τ ) b e either (a) a line arly top olo gize d ab elian gr oup, or (b) a b ounde d lo c al ly GTG ab elian gr oup. Then G is lo c al ly minimal iﬀ G has an op en minimal su b gr oup. Pr o of . If G has an open minimal subgro up, then G is lo cally minimal (Prop os ition 2 .4). Conversely , suppos e that G is lo cally minimal. (a) Let V be a n op en s ubgroup o f G witnessing lo cal minimality o f G . Then V + V ⊆ V , so V is minimal by Prop ositio n 5.11. (b) Let G b e U –lo c a lly minimal for a GTG neighbor ho o d U . According to Theorem 5.12, U ∞ is a minimal subgroup of G . F or the exp o nent m of G , w e obtain (1 /m ) U = U ∞ and he nce U ∞ is o pe n. QED If the algebr aic structur e of a group is suﬃciently well understo o d, Theorem 5.12 helps to characterize lo cally minimal gro up topolo g ies. As an example we des crib e the lo cally minima l lo cally GTG top olo gies on Z . Le t us recall that the minimal topo logies on Z ar e pr ecisely the p -adic ones (Pro danov [2 9]). Example 5. 16 Let ( Z , τ ) b e a lo ca lly minima l lo cally GTG gro up top olo gy . Then either (a) it is UFSS; or (b) ( Z , τ ) has an op en minimal subgr oup; more precisely , there ex ists a pr ime num ber p and n ∈ Z such that ( np m Z ) m ∈ N forms a neig hborho o d ba sis of the neutral ele men t. Indeed, if τ is not UFSS Theor em 5 .10 g ives that it is not NSS, and then, Exa mple 2.11(d) says that τ is a non- discrete linear topo logy . W e a pply now Cor ollary 5.15 a nd we obtain tha t G contains an open minimal subgro up N . Let N = n Z for some n 6 = 0. Then the minimality of N implies that for a s uitable prime p , a neig hborho o d ba sis of 0 in n Z is given by the se q uence of s ubgroups ( np m Z ) m ∈ N ((2.5.6) in [13]). Prop ositio n 5 .17 Pr o ducts of lo c al ly m inimal (ab elian pr e c omp act) gr oups ar e in gener al not lo c al ly minimal, namely the gr oup of inte gers with the 2 -adic top olo gy ( Z , τ 2 ) is minimal and henc e lo c al ly minimal, but the pr o d- uct ( Z , τ 2 ) × ( Z , τ 2 ) is not lo c al ly minimal. Pr o of . Supp ose that ( Z , τ 2 ) × ( Z , τ 2 ) is U – lo cally minimal. W e may ass ume that U = 2 n Z × 2 n Z . By 5 .11, the closed subgroup U is minimal. But U is top ologically isomorphic to ( Z , τ 2 ) × ( Z , τ 2 ) , whic h yields a contradiction. QED According to Corolla ry 5.15(b) the b ounded lo cally minimal lo cally GTG ab elian g r oups hav e an op en minimal subgroup. Now we use this fact to descr ibe the b ounded ab elian g roups that supp ort a non-discrete lo cally minimal and lo cally GTG group topo logy: Theorem 5. 1 8 L et G b e a b ounde d ab elian gr oup. Then the fol lowing assertions ar e e quivalent: (a) | G | ≥ c ; 26 (b) G admits a non-discr et e lo c al ly m inimal and lo c al ly GTG gr oup top olo gy; (c) G admits a non-discr et e lo c al ly c omp act metr izable gr oup top olo gy. Pr o of . T o prove the implicatio n (a) ⇒ (c) use Pr ¨ ufer’s theorem to deduce that G is a direct sum of cyclic s ubg roups. Since G is b ounded, ther e e xists an m > 1 such that G has as a direct summand a subgroup H ∼ = L c Z ( m ) ∼ = Z ( m ) ω . Since Z ( m ) ω carries a metrizable compact gro up topo logy , one can build a no n-discrete lo cally compac t metrizable group top ology on G by putting o n H the top olo gy transp or ted by the isomorphis m H ∼ = Z ( m ) ω and letting H to be a n o pen s ubg roup o f G . (c) ⇒ (b) Let τ b e a non-discre te lo ca lly compact group top ology on the gro up G . Acco rding to 2.3(b) and 5.6, τ is lo c ally minimal and lo ca lly GTG. (b) ⇒ (a) Assume that | G | < c . By Coro llary 5.15 ther e exists an op en minimal subg roup H o f G . As | H | < c , we conclude that r p ( H ) < ∞ for all primes p (see [13, Cor. 5.1.5 ]). Since H is a bo unded ab elian g roup we c onclude that H is ﬁnite. Since H is op en in G , G is discr e te, a contradiction. QED 6 Op en questions Question 6. 1 Is the closure of every GTG set in a top o lo gical g roup again a GTG set? Question 6. 2 Is every lo ca lly minimal ab elian gr oup necessa rily lo ca lly GTG? According to Theorem 5.18, for a negativ e a nswer to Question 6 .2 it suﬃces to build a non-discrete locally minimal group to po logy on an inﬁnite b ounded ab elian group of size < c . T o emphasize b etter the situation let us formulate this question in the fo llowing very sp eciﬁc case: Question 6. 3 Do es the inﬁnite Bo olean g roup L ω Z (2) admit a no n-discrete lo cally minimal g roup top olog y ? A po sitive answer to this question implies a negative a nswer to Question 6.2. Actually , the following weaker version of Question 6.2 w ill s till be useful for Theor em 5 .12: Question 6. 4 If G is a U -lo ca lly minima l a be lian group for some U ∈ V (0) , do es there exist a GTG neighborho o d of 0 cont ained in U ? Theorem 5.10 suggests a lso another weaker version of Question 6.2: Question 6. 5 Is every lo ca lly minimal NSS ab elian gro up necessar ily lo cally GTG? A p ositive answer to this question will mo dify the equiv alence proved in Theor em 5.1 0 to equiv alence b etw een UFSS and the co njunction of lo cal minimalit y and NSS. Remark 6 .6 Pro p o s ition 5.11 shows that the s pace in Example 5.4 ca nnot provide an a nswer to Question 6 .2, since actually it is not lo cally minimal. [Suppose that for some ε ∈ (0 , 1) , W ε witnesses lo cal minimality of L 0 . Let h be the characteristic function of [0 , ε/ 2] , a nd H the subgroup h h i of L 0 . H is dis c rete, hence it cannot b e minimal; how ever, H ⊆ W ε/ 2 and thus H + W ε/ 2 ⊂ W ε , which contradicts Prop. 5.11.] Question 6. 7 Is ev ery lo cally minimal NSS group metrizable? Accor ding to Prop ositio n 2.13, this is true for abelia n groups. The next q ues tion is related to Prop osition 2.5 and Co r ollary 2 .6: Question 6. 8 Let H be a clo s ed subg roup o f a (lo cally) minimal group G . Is then H neces sarily loca lly minimal ? 27 References [1] Arha ng el ′ skij, A . V. 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