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Topology and its Applications 160 (2013) 2427–2442

Contents lists available at ScienceDirect

Topology and its Applications

www.elsevier.com/locate/topol

On characterized subgroups of compact abelian groups

D. Dikranjan a,∗, S.S. Gabriyelyan b

a Department of Mathematics and Computer Science, Udine University, via delle Scienze 206,33100 Udine, Italyb Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, P.O. 653, Israel

a r t i c l e i n f o a b s t r a c t

MSC:primary 22A10, 54H11secondary 43A05, 43A40

Keywords:T -sequenceTB-sequenceCharacterizing sequenceCharacterized subgroupBorel hierarchy

Let X be a compact abelian group. A subgroup H of X is called characterizedif there exists a sequence u = (un) of characters of X such that H = su(X),where su(X) := {x ∈ X: (un, x) → 0 in T}. Every characterized subgroup is anFσδ-subgroup of X. We show that every Gδ-subgroup of X is characterized. On theother hand, X has non-characterized Fσ-subgroups.A subgroup H of X is said to be countable modulo compact (CMC) if H has asubgroup K such that it is a compact Gδ-subgroup of X and H/K is countable.It is proved that every characterized subgroup H of X is CMC if and only if Xhas finite exponent. This result gives a complete description of the characterizedsubgroups of compact abelian groups of finite exponent.For every sequence u = (un) of characters of X we define a refinement Xu of X,that is a Čech complete locally quasi-convex (almost metrizable) group. With thesequence u we associate the closed subgroup Hu of Xu and the natural projectionπX : Xu → X such that πX(Hu) = su(X). This provides a description ofthe characterized subgroups of arbitrary compact abelian groups, extending thepreviously existing result from [25]. This description is new even in the case ofmetrizable compact groups.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

An element x of an abelian group X is called torsion if kx = 0 for some k ∈ N. The subgroup t(X) of alltorsion elements of X may play a key role in the understanding of the structure of the abelian group X.

Let X be an abelian topological group. For a prime p, Braconnier [11] and Vilenkin [41] defined topo-logically p-torsion elements as those elements x ∈ X such that the sequence pnx converges to zero in X.Robertson (see [1]) defined topologically torsion elements as the elements x of X such that n!x → 0 in X. Inorder to unify the notions of topologically torsion element and topologically p-torsion element the followinggeneralization was proposed in [21, §4.4.2]: for a sequence of natural numbers m = (mn) with mn|mn+1for every n ∈ N, an element x ∈ X is called topologically m-torsion if mnx → 0 in X. On the other hand,

* Corresponding author. Tel.: +390432558488; fax: +390432558499.E-mail addresses: [email protected] (D. Dikranjan), [email protected] (S.S. Gabriyelyan).

0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.topol.2013.07.037


2428 D. Dikranjan, S.S. Gabriyelyan / Topology and its Applications 160 (2013) 2427–2442

for the torus T the sequences of integers m may be considered as sequences of elements of the dual groupZ of T. Thus, one may modify the range of a sequence u = (un) for X considering a sequence u in thePontryagin dual X∧ of X following [20]:

Definition 1.1. For an abelian topological group X and a sequence u = (un) of elements of X∧ set

su(X) :={x ∈ X: (un, x) → 0 in T

}. (1)

A subgroup of X that is an intersection of subgroups of the form (1) is called g-closed [20]. It was shownin [20] that the class of all maximally almost periodic (MAP) groups can be characterized as follows: anabelian topological group X is MAP if and only if every cyclic subgroup of X is g-closed.

Here is another relevant question that also leads to the study of subgroups of the form (1) of compactabelian groups. Let u = (un)n∈ω be a non-trivial sequence in an abelian group G.

Question 1.2. Is there a Hausdorff group topology τ on G such that un → 0 in (G, τ)?

This question, especially for the integers, has been studied by many authors – Graev [29], Nienhuys [36],and others. We recall the relevant definitions from [38]:

Definition 1.3. ([38]) Let G be an abelian group and let u = (un) be a sequence in G. If there existsa Hausdroff group topology on G making u = (un) converge to 0, then we call u a T -sequence. For aT -sequence u there exists a finest Hausdroff group topology Tu that makes u = (un) converge to 0.

Protasov and Zelenyuk studied T -sequences thoroughly in [38,39], where a criterion giving the completeanswer to Question 1.2 was obtained. Nevertheless, the question of when a given sequence in an abeliangroup is a T -sequence remains non-trivial, as the examples given in [23,26,39] show.

The counterpart of Question 1.2 for precompact group topologies on Z is studied in [40,2,3]. We recallthe relevant definitions from [2], inspired by Definition 1.3:

Definition 1.4. ([2]) Let G be an abelian group and let u = (un) be a sequence in G. If there exists aprecompact group topology making u = (un) converge to 0, we call u a TB-sequence. For a TB-sequence uthere exists a finest precompact group topology τu making u = (un) converge to 0.

The topology τu was introduced in [2] and extensively studied in [3,18]. The idea to introduce thestronger concept of TB-sequence is very simple – while it is quite hard to check whether a given sequenceis a T -sequence, the case of TB-sequences is much easier to deal with due to a very simple criterion (seeFact 1.5).

For an abelian group G and an arbitrary subgroup H � G∧d , where Gd is the group G with discrete

topology, let TH be the weakest topology on G such that all characters of H are continuous with respectto TH . One can easily show [14] that TH is a totally bounded group topology on G, and it is Hausdorffif and only if H is dense in G∧

d . The next fact summarizes the most important properties concerning thetopology τu on G:

Fact 1.5. ([20]) A sequence u in a (discrete) abelian group G is a TB-sequence if and only if the subgroupsu(X) of the (compact) dual X = G∧ is dense. In this case τu = Tsu(X).

The subgroups of the form su(X) of a compact metrizable abelian group X have been studied by manyauthors, especially in the case X = T, see [1,4,6–8,10,22,33,35]. Borel [10] proved that every countablesubgroup of T has the form (1) for an appropriate u. It was proved by Larcher [35] (see also [2]) that for


D. Dikranjan, S.S. Gabriyelyan / Topology and its Applications 160 (2013) 2427–2442 2429

the celebrated Fibonacci sequence f = (fn), defined by f0 = f1 = 1 and fn+2 = fn+1 + fn for all n ∈ N, onehas sf (T) = 〈α〉, where α is the positive solution of the equation x2 −x− 1 = 0 (namely, the Golden Ratio)taken modulo 1 (i.e., as an element of T). In particular, sf (T) is infinite, hence dense (as α is irrational).Thus f is a TB-sequence by Fact 1.5.

The interest in the study of the groups of the form su(X) is motivated also by some applications toDiophantine approximation, dynamical systems and ergodic theory (see [9,37,42]).

Motivated by the above circumstances, the following notion was proposed in [20], making use of thesubgroups of the form su(X) of a topological abelian group X:

Definition 1.6. ([20]) Let H be a subgroup of a topological abelian group X. If there exists a sequenceu = (un)n∈ω of characters of X such that H = su(X) we say that H is characterized (by u) and that ucharacterizes H.

Let X be an abelian topological group and G = X∧. The next fact follows easily from Definition 1.6:

Fact 1.7. ([13,15]) Every characterized subgroup H of an abelian topological group X is an Fσδ-subgroupof X, and hence H is a Borel subset of X.

Indeed, if H is characterized by a sequence u = (un)n∈ω, this fact directly follows from the relation

su(X) =⋂

0<M<ω

⋃m∈ω

( ⋂n�m

Sn,M

), where Sn,M :=

{x ∈ X:

∣∣(un, x)∣∣ � 1

M

}.

In particular, if X is a compact abelian group and mX is a Haar measure on X, then every characterizedsubgroup H of X is measurable. If the characterizing sequence u is non-trivial (i.e., u contains infinitely manydifferent elements), then μ(H) = 0 [15]. This fact provides a formidable source of precompact topologieson G without non-trivial convergent sequences. Indeed, it is enough to take on a group G a precompacttopology τ such that H = (G, τ)∧ is a dense non-measurable subgroup of X [3,13]. Another importantrestraint on characterized subgroups of compact metrizable groups comes from the fact that a Borel subsetof such a group can either be countable or have size c.

The previous discussion and examples emphasize the importance of the following general problem (seealso Problem 4.4):

Problem 1.8. Describe the characterized subgroups of a compact abelian group.

Answering this problem it is natural to consider, at the beginning, the class of all metrizable compactgroups. Another natural restriction is to consider some classes of subgroups as countable or closed. Thefollowing theorem was proved by Kunen and the first named author [17] and by Beiglböck, Steineder, andWinkler [5] independently and almost simultaneously:

Theorem 1.9. ([5,17]) Let H be a countable subgroup of a compact metrizable abelian group. Then H ischaracterized.

It can be shown that metrizability is a necessary condition in this theorem (see Corollary 2.8).In this article we consider Problem 1.8 from the two quite different points of view. The first one is based

on a description of all Borel classes to which characterized subgroups may belong. Taking into accountFact 1.7, one can ask whether every Fσδ-subgroup is characterized, i.e., whether the class of all charac-terized subgroups coincides with the class of all Fσδ-subgroups. Answering a question of the first namedauthor, raised in March 2005, Biró [7] found the first example of a non-characterized Fσ-subgroup of T. In



Theorem D we prove a stronger result: namely, every infinite compact abelian group has a non-characterizedFσ-subgroup.

It is natural to consider other Borel classes of subgroups. For Gδ-subgroups we prove:

Theorem A. Every Gδ-subgroup of a compact abelian group X is characterized.

Taking into account this theorem we reduce Problem 1.8 to the metrizable case in the next theorem:

Theorem B. A subgroup H of a compact abelian group X is characterized if and only if H contains a closed(necessarily characterized) Gδ-subgroup K of X such that H/K is a characterized subgroup of the compactmetrizable group X/K.

From Theorems 1.9 and B we obtain:

Corollary B1. Let X be a compact abelian group and let H be a subgroup of X that contains a closedGδ-subgroup K such that H/K is countable. Then H is characterized.

We prove Theorems A and B in Section 2.Theorem B can be conveniently subsumed by the following exact sequence of characterized subgroups

0 → K → H → H/K → 0. (2)

More precisely, if H is a characterized subgroup of X, then H contains a closed (necessarily characterized)Gδ-subgroup K of X such that H/K is a characterized subgroup of the quotient X/K. Vice versa, if for aclosed subgroup K of X both ends of the exact sequence (2) are characterized subgroups (of the compactgroups X and X/K, respectively), then H is characterized. According to Corollary B1, this occurs whenH/K is countable. This motivates the following:

Definition 1.10. For a characterized subgroup H of a compact abelian group we say that

(a) H is countable modulo compact (CMC, for brevity), if H contains a compact subgroup K of X suchthat H/K is countable;

(b) H is countable torsion modulo compact (CTMC, for brevity), if H contains a compact subgroup K ofX such that H/K is countable and torsion.

One can show that the compact subgroups K in (a) and (b) are necessarily closed Gδ-subgroups (seeRemark 3.1). The group T has plenty of characterized subgroups that are not CMC. Indeed, if a sequencem = (mn)n∈ω satisfies mn+1

mn→ ∞, then sm(T) is uncountable [22]. Since every proper compact subgroup

of T is finite, sm(T) is not CMC. On the other hand, the infinite cyclic subgroups of T are CMC, butnot CTMC. Our interest in CTMC subgroups stems from Theorem 1.11 describing when a special kind ofFσ-subgroups has a characterizing sequence.

Theorem 1.11. ([17]) Suppose that X is a compact abelian group and H =⋃

n Fn, where each Fn � X isclosed and each Fn � Fn+1. Then H has a characterizing sequence if and only if X/Fm is metrizable forsome m and |Fn+1 : Fn| is finite for all n � m.

It is easy to see, that a subgroup H satisfies the equivalent conditions from the above theorem if andonly it is CTMC. Therefore, this theorem can be obtained also from Corollary B1.



The next theorem describes the compact abelian groups (as those of finite exponent) having all theircharacterized subgroups CMC (or equivalently, CTMC). This theorem gives a complete description of thecharacterized subgroups of compact abelian groups of finite exponent.

Theorem C. Let X be a compact abelian group. The following are equivalent:

(1) X has finite exponent;(2) every characterized subgroup H of X is CTMC (i.e., H has a compact Gδ-subgroup K such that H/K

is countable and torsion);(3) every characterized subgroup H of X is CMC (i.e., H has a compact Gδ-subgroup K such that H/K is

countable);(4) every characterized subgroup H of X has the form H =

⋃n Fn, where {Fn} is an increasing sequence

of closed Gδ-subgroups of X such that |Fn+1 : Fn| is finite for all n � m;(5) every characterized subgroup H of X admits a finer locally compact group topology with an open compact

subgroup K such that H/K is countable and X/K is metrizable.

Since every CMC subgroup is an Fσ-subgroup, we obtain:

Corollary C1. If a compact abelian group X has finite exponent, then every characterized subgroup of X isan Fσ-subgroup.

By Theorem A every Gδ-subgroup of a compact abelian group is characterized, but for Fσ-subgroups thesituation changes. Using (5) of Theorem C we prove:

Theorem D. Every infinite compact abelian group has a non-characterized Fσ-subgroup.

In the next theorem we summarize the obtained results concerning the Borel hierarchy of character-ized subgroups of compact abelian groups. For a compact abelian group X denote by Char(X) (respec-tively, SFσ(X), SFσδ(X) and SGδ(X)) the set of all characterized subgroups (respectively, Fσ-subgroups,Fσδ-subgroups and Gδ-subgroups) of X.

Theorem E. For every infinite compact abelian group X, the following inclusions hold:

SGδ(X) � Char(X) � SFσδ(X) and SFσ(X) � Char(X).

If in addition X has finite exponent, then

Char(X) � SFσ(X).

Theorems C, D and E are proved in Section 3.To explain our second approach to Problem 1.8 we need some notations. Let TN be the direct product

of a countable set of copies of T. Set T∞ := {ω = (zn)n∈ω ∈ TN: zn → 0}. The group T∞ under themetric d((z1

n), (z2n)) = sup{|z1

n − z2n|, n ∈ N} we denote by TH

0 . Then TH0 is a reflexive Polish group [24]

that will play a prominent role in the sequel. Let X be an arbitrary infinite compact abelian group. Forany (x, ω) = (x, (zn)) ∈ X × TH

0 , set πX(x, ω) := x and πn(x, (zn)) := zn. A general description of thecharacterized subgroups of metrizable abelian compact groups X was given by the second-named authorin [25]. We show that this characterization can be extended to the general case. However, our expositiongoes much beyond the limits of [25]. For a subgroup H of an abelian topological group X, we denote by



H⊥ its annihilator, i.e. H⊥ = {χ ∈ X∧: χ|H = 0}. Recall that H is called dually closed in X if for everyx ∈ X \H there exists a character χ ∈ H⊥ such that (χ, x) �= 0. The next theorem is one of the main resultsof Section 4 (see Theorem 4.3 for a more complete version of this theorem, providing a clearer picture evenin the metric case):

Theorem F. Let H be a subgroup of an infinite compact abelian group X and u = (un) be a sequence in X∧.Then the following are equivalent:

(a) H is characterized by u.(b) There exists a dually closed subgroup Hu of X × TH

0 such that(b1) the restriction to Hu of the projection πX is a bijection onto H, and(b2) un ◦ πX and πn coincide on Hu for every n.

2. Characterized subgroups vs Gδ-subgroups of a compact abelian group

To extend Theorem 1.9 to subgroups of arbitrary compact abelian groups we need first an easy necessarycondition satisfied by all characterized subgroups. To this end we assign now another subgroup Ku(X) of acompact abelian group X related to a sequence u = (un)n∈ω in the dual group of X by letting

Ku(X) :=⋂n∈ω

kerun.

In the sequel we omit X and write simply Ku when the group X is clear from the context. The subgroupKu may be much smaller than su(X), as one can see from the following example.

Example 2.1. In the group X = TN the subgroup H = T∞ of X is characterized by the sequence of thecoordinate projections pn : X → T [24]. While the subgroup H is quite large (e.g., it contains all finitepowers Tn × {0}N), Ku is trivial.

The next lemma is used repeatedly in the sequel.

Lemma 2.2. Let u = (un)n∈ω be a sequence of characters of a compact abelian group X. Then:

(i) The subgroup Ku is a closed Gδ-subgroup of X contained in su(X) and Ku = 〈u〉⊥.(ii) The subgroup su(X)/Ku is a characterized subgroup of X/Ku.

Proof. (i) Clearly, 〈u〉⊥ ⊆ Ku. Conversely, if h ∈ Ku, then (un, h) = 0 for every n ∈ ω. Hence (g, h) = 0 forevery g ∈ 〈u〉. Thus Ku ⊆ 〈u〉⊥. In particular, (X/Ku)∧ ∼= 〈u〉 is countable and hence X/Ku is metrizable.

To end the proof, it remains to recall that a closed normal subgroup N of a compact group K is a Gδ-setprecisely when the quotient group K/N is metrizable (due to the well known fact that a compact group ofcountable pseudocharacter is metrizable).

(ii) For the sake of simplicity we set H := su(X) and K := Ku. Let q : X → X/K be the quotient map.The adjoint homomorphism q∧ : (X/K)∧ → X∧ is injective and has image K⊥ in X∧. For every n ∈ ω,define the character un of X/K as follows: (un, q(x)) = (un, x) (un is well-defined since K ⊆ kerun). Thenu = (un)n∈ω is a sequence of characters of X/K such that q∧(un) = un.

We claim that H/K = su(X/K). Indeed, for every h + K ∈ H/K, by definition, we have (un, h + K) =(un, h) → 0. Thus H/K ⊆ su(X/K). If x + K ∈ su(X/K), then (un, x + K) = (un, x) → 0. This yieldsx ∈ H. Thus x + K ∈ H/K. �



The pair Ku ⊆ su(X) will play a prominent role in the sequel.

Proposition 2.3. A closed subgroup H of a compact abelian group X is characterized if and only if H is aGδ-subgroup of X.

Proof. Assume that H is characterized by a sequence u. Then H contains the Gδ-subgroup Ku of X. Sincequotient groups of compact metrizable groups are still metrizable, this shows that H is a closed Gδ-subgroupof X itself.

Conversely, let H be a closed Gδ-subgroup of X. Then X/H is metrizable. Hence (X/H)∧ is countable.Let q : X → X/H be the quotient map. The adjoint homomorphism q∧ : (X/H)∧ → X∧ is injective andhas image H⊥ in X∧. Let the sequence v = (vn)n∈ω enumerate the countable subgroup H⊥ in such a waythat each element appears infinitely many times. Then H = (H⊥)⊥ = sv(X). �

Now Theorem A immediately follows from Proposition 2.3 and the next one which is of independentinterest.

Proposition 2.4. Every Gδ-subgroup of a compact group X is closed.

Proof. Let H be a Gδ-subgroup of X. If H is finite the assertion is trivial. Let H be infinite. PuttingH = clX(H) we see that H is also a Gδ-subgroup of the compact group H. So we can assume that H

is dense in X. Being topologically complete H has a compact subgroup K such that H/K is metrizableand topologically complete [12]. Clearly, K is a compact subgroup of X. Thus X/K is metrizable as thecompletion of H/K. It is known that every dense Gδ-subset of a compact metrizable group must be of secondcategory [34, Chapter 3, §34(V), Theorem 2]. By the Banach–Kuratowski–Pettis theorem [32, Chapter 6(P)]H has a non-empty interior. Hence H is open and closed in X. The density of H implies that H = X. �

For a sequence u = (un)n∈ω in the dual group of a compact abelian group X and every m ∈ ω, we setum = (un)n�m. Clearly, sum(X) = su(X). In spite of the subgroup Ku may be even trivial (see Example 2.1)the next proposition shows that every compact subgroup of su(X) is contained in Kum for all sufficientlylarge natural number m.

Proposition 2.5. Let u = (un)n∈ω be a sequence in the dual group of a compact abelian group X. If a compactsubgroup K of X is contained in su(X), then K is contained in Kum for all sufficiently large naturals m.

Proof. In this proof the group T will be identified with the unit circle on the complex plane, writtenmultiplicatively. Therefore, su(X) = {x ∈ X: (un, x) → 1}. Denote by mK the unique normalized Haarmeasure on K. Since (un, x) → 1 for every x ∈ K, the Lebesgue dominated convergence theorem gives∫

K

(un, x) dmK →∫K

1 · dmK = 1. (3)

Since∫K

(u, x) dmK = 0 for every non-trivial character u of K [30, 23.19], (3) implies that (un, x) = 1 forall x ∈ K for all sufficiently large natural numbers n. This means that K ⊆ Kum for every m � m0 andsome natural m0. �

To prove Theorem B we need the following lemma that will be used in the sequel:

Lemma 2.6. Let K be a closed subgroup of a compact abelian group X and q : X → X/K be the quotientmap. Then H is a characterized subgroup of X/K if and only if q−1(H) is a characterized subgroup of X.



Proof. Set H := q−1(H).Assume that H is a characterized subgroup of X/K via the sequence u = (un)n∈ω. We have to show

that H is a characterized subgroup of X. The adjoint homomorphism q∧ : (X/K)∧ → X∧ is injective andhas image K⊥ in X∧. Set u = (un)n∈ω, where un = q∧(un). It is enough to show that H = su(X). Thisfollows from the following chain of equivalences. By definition, x ∈ su(X) if and only if

(un, x) → 0 ⇐⇒(un, q(x)

)→ 0 ⇐⇒ q(x) ∈ H = H/K ⇐⇒ x ∈ H.

The last equivalence is due to the inclusion K ⊆ H.Conversely, assume that H is a characterized subgroup of X and let u = (un)n∈ω be a sequence

characterizing H. Since K ⊆ H = su(X), K is contained in Kum for some sufficiently large naturalnumber m, by Proposition 2.5. Denote by qu the quotient homomorphism from X/K onto X/Kum andlet q′ = qu ◦ q : X → X/Kum . Then H/Kum = q′(H) = qu(H) and H = q−1

u (H/Kum). Moreover,H = su(X) = sum(X). Therefore, H/Kum is a characterized subgroup of X/Kum , by Lemma 2.2(ii). ThenH is a characterized subgroup of X/K by the first part of the proof. �

Now we are in position to prove Theorem B:

Proof of Theorem B. Let H be characterized by a sequence u = (un)n∈ω. Set K := Ku. Since K is a closedGδ-subgroup of X by Lemma 2.2(i), X/K is metrizable. By Lemma 2.2(ii) H/K is a characterized subgroupof X/K and K is characterized in X by Theorem A.

Conversely, let H contain a closed Gδ-subgroup K of X such that H/K is a characterized subgroup ofthe compact metrizable group X/K. Then H is a characterized subgroup of X by Lemma 2.6. �Corollary 2.7. Every characterized subgroup H of a compact abelian group X contains a closed Gδ-subgroupK of X such that H/K is either countable or has size c.

Proof. Apply Theorem B to find a closed Gδ-subgroup K of X contained in H such that H/K is a charac-terized subgroup of X/K. Then H/K is a Borel subgroup of the compact metrizable group X/K. So H/K

is either countable or has size c. �It was mentioned already in [16] that if all countable subgroups of a compact abelian group are char-

acterized, then the group is metrizable. The next corollary shows that metrizability in Theorem 1.9 is anecessary condition in a more precise way.

Corollary 2.8. If a compact abelian group X has a countable characterized subgroup, then X is metrizable.

Proof. Assume that H is a countable characterized subgroup of X. By Theorem B, H contains a closedGδ-subgroup K of X. Since countable compact groups are finite, we deduce that K is finite. Since thequotient X/K is metrizable, we conclude that X is metrizable as well. �3. Proof of Theorems C, D and E

Let us start with a discussion on characterized subgroups that are CMC.

Remark 3.1.

(a) Let a characterized subgroup H of a compact abelian group X be CMC. So that H contains a compactsubgroup K of X such that H/K is countable. Let us see that K is a closed Gδ-subgroup of X. Indeed,



assume H is characterized by a sequence u. Then the countable quotient H/K contains a subgroupisomorphic to the compact quotient group (Ku+K)/K ∼= Ku/(K∩Ku), this is possible only if the latteris finite. Hence K ∩Ku is a finite-index subgroup of the Gδ-subgroup Ku, hence K is a Gδ-subgroupitself.

(b) If a characterized subgroup H is CMC, and this is witnessed by two compact subgroups K ′ and K ′′

of H (so that H/K ′ and H/K ′′ are countable), then K ′ ∩ K ′′ is a finite-index subgroup of both K ′

and K ′′, i.e., K ′ and K ′′ are commensurable. (Indeed, the compact quotient group (K ′ + K ′′)/K ′ isisomorphic to a subgroup of the countable quotient H/K ′, hence K ′′/(K ′ ∩ K ′′) ∼= (K ′ + K ′′)/K ′ isfinite. Analogously, K ′/(K ′ ∩K ′′) ∼= (K ′ + K ′′)/K ′′ is finite.)

To prove Theorem C we need the following lemma:

Lemma 3.2. If every characterized subgroup a compact abelian group X is CMC, then every characterizedsubgroup of each quotient group of X is CMC as well.

Proof. Let Y = X/K be a quotient of X and q : X → X/K be the quotient map. We have to prove thatevery characterized subgroup H0 of Y is CMC; that is, H0 contains a compact Gδ-subgroup K0 of Y suchthat H0/K0 is countable. Set H := q−1(H0). Then H is a characterized subgroup of X by Lemma 2.6. So H

contains a compact Gδ-subgroup K ′ of X such that H/K ′ is countable. Since K ′ +K ⊆ H and H/(K ′ +K)is a quotient of H/K ′, we can assume that K ⊆ K ′. Set K0 = q(K ′) ⊆ H0. As Y/K0 ∼= X/K ′, K0 isa Gδ-subgroup of Y (we use here the fact formulated at the end of the proof of item (i) of Lemma 2.2).Clearly, H0/K0 ∼= H/K ′ is countable. �

In the sequel we denote by Z(m) the cyclic group of order m.

Proof of Theorem C. (1) ⇒ (2) Assume that H is characterized. By Theorem B, there exists a compactGδ-subgroup K0 of H such that H/K0 is characterized in X/K0. By [27, Lemma 3.1 and Theorem 2.26],the group H/K0 contains a compact subgroup K ′ such that (H/K0)/K ′ is countable. Set K := q−1(K ′),where q : X → X/K0 is the quotient map. Then K is a compact subgroup of X such that H/K is countable.Since X has finite exponent, we obtain that H/K is a countable abelian group of finite exponent. HenceH/K is the increasing union of finite subgroups. Thus, H is CTMC.

(2) ⇔ (4) follows from the fact that every countable torsion abelian group is the union of an increasingchain of finite subgroups.

(2) ⇒ (3) is obvious.(3) ⇒ (5) Let H be a characterized subgroup of X. By hypothesis, H has a compact Gδ-subgroup K of

X such that H/K is countable. Let τ be the induced topology on K. Denote by τ the topology on H witha base formed by all translates in H of τ -open subsets of K. Then τ is a locally compact group topologyon H such that K is a compact open subgroup of H. Clearly, τ is finer than the induced one on H from X.Since K is a Gδ-subgroup of X, the quotient group X/K is metrizable.

(5) ⇒ (3) Let a characterized subgroup H of X admit a finer locally compact group topology ν withan open compact subgroup K such that H/K is countable. Then K is compact also in the induced by X

topology, so H is CMC.The equivalence (5) ⇔ (3) obtained so far will be used in the proof of the last remaining implication.(5) ⇒ (1) Assume that X is of infinite exponent. To obtain a contradiction with the hypothesis (5), it is

enough to find a characterized subgroup H of X that does not satisfy (5). When necessary or convenient,we shall use also the equivalent form (3).

Since X has infinite exponent, G = X∧ is of infinite exponent as well. It is well known that G containsa countably infinite subgroup S of one of the following form:



(a) S ∼= Z;(b) S ∼= Z(p∞);(c) S ∼=

⊕n∈ω Z(bn), where b0 < b1 < · · · .

Fix such a subgroup S and put K = S⊥. Then (X/K)∧ ∼= S and K is a Gδ-subgroup of X. Let q : X → X/K

be the quotient map. So, by Lemma 3.2, to obtain a contradiction it is enough to build a proper densecharacterized subgroup H0 of X/K that does not satisfy (5).

In all three cases (a)–(c) there exists a TB-sequence u in S such that (S, Tu) is reflexive and non-discrete[26]. Set H0 := su(X/K) and assume for a contradiction that H0 satisfies (5). Then

(i) H0 admits a finer Polish group topology τ as the dual group of (S, Tu) [25];(ii) H0 admits a finer locally compact group topology τ0 satisfying assumption (5).

Let us show that τ0 is also Polish. Indeed, according to assumption (5), (H0, τ0) has an open compactsubgroup K0 such that H0/K0 is countable. So K0 is a compact subgroup of the metrizable group X/K.Hence K0 is a metrizable compact group. Since H0/K0 is countable, τ0 is Polish.

By the uniqueness of the Polish group topology we have τ = τ0. So by (i), (S, Tu)∧ = (H0, τ0) is locallycompact, and hence (S, Tu)∧∧ is locally compact as well. As (S, Tu) is reflexive, (S, Tu) ∼= (S, Tu)∧∧ is alsolocally compact. Being countable, (S, Tu) must necessarily be discrete. This contradicts the non-discretenessof (S, Tu). Thus H0 does not satisfy (5). �

The converse in Corollary C1 does not hold true, for example the group T has characterized subgroupsthat are not Fσ.1 We do not know whether this remains true for all compact abelian groups X of infiniteexponent, namely:

Problem 3.3. Does there exist a compact abelian group X of infinite exponent whose all characterizedsubgroups are Fσ?

As it was noticed in the introduction, the first example of a non-characterized Fσ-subgroup of T was givenby Biró in [7]. He showed that the subgroup generated by a Kronecker subset of T is not characterized.Recall that a subset K of an infinite compact metrizable abelian group X is called a Kronecker set if itis non-empty, compact, and every continuous function f : K → T can be uniformly approximated by acontinuous character. To prove Theorem D, we need the following two facts. The first one generalizes Biró’sexample:

Fact 3.4. ([25, Theorem 2]) Let F be an uncountable Kronecker subset of an infinite compact abelianmetrizable group X. Then the subgroup 〈F 〉 generated by F is a non-characterized Fσ-subgroup of X.

The following folklore fact is an easy consequence of Pontryagin duality.

Fact 3.5. Every infinite compact abelian group X contains a closed subgroup K such that X/K is an infinitecompact metrizable group.

It follows from Proposition 2.3 and its proof that one can choose such K in the form Ku for a non-trivialsequence u in X∧, see Lemma 2.2(i).

1 It was conjectured by the first named author that su(T), for un = n!, is such a group. The second named author found a proofof this fact that he will publish separately in a forthcoming paper [28].



Proof of Theorem D. By Fact 3.5, X contains a closed subgroup K such that X/K is an infinite compactmetrizable group. Clearly, if H0 is an Fσ-subgroup of X/K and q : X → X/K is the quotient map, thenH := q−1(H0) is an Fσ-subgroup of X. Moreover, if H0 is not a characterized subgroup of X/K, then H isnot characterized in X by Lemma 2.6. Hence without loss of generality we can assume that X is an infinitemetrizable compact group.

We distinguish between two cases.

Case 1. Assume that X is of infinite exponent. By [31, 41.5], X has an uncountable Kronecker set F . Thenthe group 〈F 〉 generated by F is a non-characterized Fσ-subgroup of X by Fact 3.4.

Case 2. Assume that X has finite exponent. By [30, 25.9] and the above reduction, we can assume withoutloss of generality that X ∼= Z(a)N for some a > 1, hence we can write X in the following form

X =∞∏i=1

Xi, where Xi = Z(a)N for all i ∈ N.

Clearly, the subgroup H :=⋃∞

n=1(∏n

i=1 Xi) is an Fσ-subgroup of X. Let us show that H is not characterized.

Arguing for a contradiction, assume that H is characterized. By Theorem C(5), we can find a locallycompact Polish group topology τ on H which is finer than the induced one from X. By the Baire theorem,there exists k such that the subgroup U :=

∏ki=1 Xi is open in (H, τ). Since τ is Polish, the discrete quotient

group H/U is countable. But

H/U ∼=∞⋃

n=k+1

(n∏

i=k+1Xi

)

is uncountable. This contradiction shows that H is not characterized. �To prove Theorem E we need the next lemma.

Lemma 3.6. Every infinite compact abelian group X has a proper dense characterized subgroup H which isCMC.

Proof. By Fact 3.5, X contains a closed subgroup K such that X/K is an infinite compact metrizable group.Let H be a dense countable subgroup of X/K. Set H := q−1(H), where q : X → X/K is the quotient map.So H is CMC. Now Corollary B1 implies that H is a proper dense characterized subgroup of X. �

Now we are in position to prove Theorem E.

Proof of Theorem E. Let X be an infinite compact abelian group. We have to check the following (non-)in-clusions:

SGδ(X) � Char(X) � SFσδ(X) and SFσ(X) � Char(X). (4)

Moreover, we have to see that

Char(X) � SFσ(X), (5)

in case X has finite exponent.



The inclusion SGδ(X) � Char(X) immediately follows from Theorem A, Proposition 2.4 and Lemma 3.6,while the inclusion Char(X) � SFσδ(X) follows from Fact 1.7 and Theorem D. This proves the first part of(4). Finally, Theorem D exactly means that SFσ(X) � Char(X). This proves completely (4).

If X has finite exponent, (5) follows from Corollary C1 and Theorem D. �4. The u-refinement Xu of a compact abelian group X

Here we define the u-refinement Xu of a compact abelian group X and describe the characterized sub-groups of X “through the looking glass” of this refinement.

A convenient way to obtain the subgroup su(X) via maps from X to the direct product TN of countablymany copies of T was proposed in [16, §5]. Recall that we denote by T∞ the subgroup of TN consisting ofthose elements (zn) ∈ TN such that zn → 0 in T. Every sequence u = (un) in X∧ gives rise to a morphismdu ∈ Hom(X,TN), namely, du : X → TN is defined by du(x) = (un, x)n∈N ∈ TN, for x ∈ X. The subgroupd−1u (T∞) of X obviously coincides with the subgroup su(X) defined in (1), namely,

su(X) = d−1u (T∞). (6)

The formula (6) from [16] alone does not suffice to obtain substantial results on characterized subgroupsof compact abelian groups. Nevertheless, it suggests a deeper analysis of the product TN. In the sequel TN

carries the usual (compact) product topology and also the (finer) topology u defined by the metric

d((zn),

(z′n))

= sup{∣∣zn − z′n

∣∣, n ∈ N}

(here u suggests the fact that the metric topology of d coincides with the uniform topology). We shall oftenabbreviate the topological group (TN, u) to TN

u . It is complete, but not separable [19]. Clearly, TH0 = (T∞, u)

and it is a closed subgroup of (TN, u).Let X be an arbitrary infinite compact abelian group. Define the projections

πX : X × TN → X, πT : X × TN → TN and πn : X × TN → T

by πX(x, (zn)) := x, πX(x, (zn)) := (zn) and πn(x, (zn)) := zn for (x, (zn)) ∈ X × TN and n ∈ N.From now on we fix arbitrarily a sequence u = (un) in X∧. Note that du : X → TN, defined above, is

continuous when TN carries the product topology (but du : X → TNu is discontinuous in general).

Obviously, the subgroup Ku =⋂

n kerun of X coincides with ker du. Using du, one can define the followingsubgroup of X × TN

Γu :={(

x, du(x))∈ X × TN: x ∈ X

}.

Since Γu is the graph of du, Γu is closed in X×TN by the closed graph theorem of the continuity of du. ThusΓu is also closed in X ×TN

u . We will denote by Γ compu (resp. Γ unif

u ) the group Γu endowed with the inducedtopology from X×TN (resp. X×TN

u ). Then Γ compu is a compact group and Γ unif

u is a Čech-complete locallyquasi-convex group (see below Proposition 4.1).

Since X is compact, the standard continuous isomorphism

fu : X → Γ compu , defined by fu(x) =

(x, du(x)

),

is a topological isomorphism. Note that fu is the right inverse of πX (i.e., πX ◦ fu = idX) and

un = πn ◦ fu(i.e., f∧

u (πn) = un

), for all n ∈ N. (7)



Clearly, the map fu : X → Γ unifu is discontinuous in general. Denote by Xu the group X equipped with

the topology that makes fu : Xu → Γ unifu a topological isomorphism (i.e., the topology transported by f−1

ufrom Γ unif

u ).Define the subgroup Hu of X × TH

0 as follows

Hu = Γ unifu ∩

(X × TH

0).

Clearly, Hu = sπ(Γ compu ) algebraically, where π = (πn). Since both Γ unif

u and X ×TH0 are closed in X ×TN

u ,Hu is a closed subgroup of X × TH

0 . In other words, Hu = Γu ∩ (X × TH0 ) is the intersection of the graph

of du and X × TH0 endowed with the induced topology from X × TH

0 (or from X × TNu ). It is clear that

Ku × {0} ⊆ Hu ⊆ Γ unifu . So, in the sequel we shall identify Ku with Ku × {0} and consider Ku as a

topological subgroup of both X and Xu.To resume: in this way, for every infinite compact abelian group X and every sequence u = (un) in X∧

we assigned the following data:

(i) the map du : X → TN, its graph Γu and its kernel Ku;(ii) the topological isomorphism fu : X → Γ comp

u ;(iii) the closed subgroup Hu = Γu ∩ (X × TH

0 ) of X × TH0 ;

(iv) the finer complete almost metrizable topology on X defining Xu, so that fu : Xu → Γ unifu is a

topological isomorphism.

The subgroup Hu will play a central role in the sequel, it was introduced in a different way in [25], withoutany recourse to the map du and its graph (see item (c) of Theorem 4.3).

Proposition 4.1. For every infinite compact abelian group X and every sequence u = (un)n∈N in X∧ thefollowing hold in terms of (i)–(iv) above:

(a) su(X) = πX(Hu) = f−1u (X × TH

0 ) = f−1u (sπ(Γ comp

u )) algebraically, where π = (πn).(b) Xu and Hu are locally quasi-convex Čech-complete groups. Moreover, Hu/Ku is isomorphic a closed

subgroup of TH0 and hence it is Polish. In particular, Hu is a Polish group whenever X is metrizable.

(c) Γu is a dually closed subgroup of X × TN.(d) Hu is a dually closed subgroup of X × TH

0 .

In particular, a subgroup of X is characterized by the sequence u if and only if it coincides with πX(Hu).

Proof. (a) By (6) we have (algebraically)

su(X) = d−1u (T∞) = f−1

u(X × TH

0)

= πX(Hu).

Finally, the equality f−1u (sπ(Γ comp

u )) = su(X) follows from the equality (7).(b) According to [19, Proposition 3.1 (b)], TN

u is a complete metrizable locally quasi-convex group. Takinginto account that X is compact, we obtain that the product X ×TN

u is a complete almost metrizable group.The group Xu is isomorphic to the closed subgroup Γ unif

u of X × TNu , so it is also a locally quasi-convex

Čech-complete group. Being a closed subgroup of X × TNu , the group Hu is a locally quasi-convex Čech-

complete group as well. Clearly, Hu ∩ (X ×{0}) = Ku. The restriction πrT : X ×TH

0 → TH0 of the projection

πT has kerπrT = Ku. Since X is compact, πr

T is a closed map [30, 5.18]. Thus πrT (Hu) ∼= Hu/Ku is a closed

subgroup of the Polish group TH0 . If X is metrizable, then X×TH

0 is Polish. So Hu, being closed in X×TH0 ,

is a Polish group as well.



(c) To prove that Γu is dually closed in X ×TN pick an x = (x, (zn)) /∈ Γu and pick an index i such thatzi �= (ui, x). Let

ω =(ui; (0, . . . , 0,−1, 0, . . .)

)∈(X × TH

0)∧

,

where −1 is placed in position i. Then (ω,x) = (ui, x) − zi �= 0. Hence Γu is dually closed in X × TN.(d) follows from item (c). �By item (a) of Proposition 4.1, a subgroup H of X is characterized by u if and only if H = πX(Hu).

Note that the projection πX restricted to Γu is injective, so practically πX(Hu) is just taking a coarser(precompact) topology induced by the compact group X. Hence we obtain the following simple necessarycondition for a subgroup to be characterized, which was noticed in [27]:

Proposition 4.2. Each characterized subgroup H = su(X) of a compact (metrizable) group X admits a finerlocally quasi-convex Čech-complete (resp., Polish) group topology.

As an immediate corollary of the above technique we obtain the following criterion for a subgroup of acompact abelian group to be characterized by a sequence u:

Theorem 4.3. Let H be a subgroup of an infinite compact abelian group X and u = (un) be a sequence inX∧. Then the following are equivalent:

(a) H is characterized by u;(b) H = πX(Hu);(c) there exists a dually closed subgroup Hu of X × TH

0 such that(c1) the restriction to Hu of the projection πX is a bijection onto H, and(c2) un ◦ πX and πn coincide on Hu for every n.

Proof. The equivalence of (a) and (b) follows from Proposition 4.1(a).Setting Hu := Hu, the implication (b) → (c) follows from Proposition 4.1.To prove the implication (c) → (b) we show now that the conjunction (c1) and (c2) alone implies the

equality H = πX(Hu) (so the dual closeness of Hu is not necessary to prove this implication, but providesa useful information about the characterized subgroups).

We are going to use a simple algebraic fact: since Hu is a subgroup of the direct product X×TH0 with (c1),

there exists a group homomorphism h : H → TH0 such that Hu coincides with the graph Gh = {(x, h(x)): x ∈

H} of h. Indeed, by (c1) for every x ∈ H there exists the unique b ∈ TH0 such that (x, b) ∈ Hu. Denote by

h(x) this unique b ∈ TH0 with (x, h(x)) ∈ Hu. It is easy to check that h is a group homomorphism. Since

πX |Hu is a bijection onto H we obtain that Hu = Gh.Note that (c2) implies that the homomorphism h coincides with the restriction du|H . Therefore, Hu = Hu.

Now (c1) yields H = πX(Hu), as desired. �Note that the equivalence of (a) and (c) was proved in [25] for metrizable X.

Proof of Theorem F. The equivalence of (a) and (c) in Theorem 4.3 is exactly Theorem F. �Taking into account numerous applications the following particular case of Problem 1.8 is of independent

interest:

Problem 4.4. Describe the characterized subgroups of T.



Unfortunately, Theorem F essentially depends on a T -sequence u. So it is natural to ask:

Problem 4.5. Describe the characterized subgroups of compact abelian groups without making use of thespecific sequence u.

Acknowledgement

It is a pleasure to thank Vaja Tarieladze for an essential hint that led to Proposition 2.4.

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