Characterized subgroups of topological abelian groups

A subgroup $H$ of a topological abelian group $X$ is said to be characterized by a sequence $\mathbf v =(v_n)$ of characters of $X$ if $H=\{x\in X:v_n(x)\to 0\ \text{in}\ \mathbb T\}$. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of autochacaracterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proved to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.


Introduction
For a topological abelian group X, we denote by X its dual group, that is, the group of all characters of X (i.e., continuous homomorphisms X → T). Following [16], for a sequence of characters v = (vn) ∈ X N , let sv(X) := {x ∈ X : vn(x) → 0} , which is always a subgroup of X. A subgroup H of X is said to be characterized if H = sv(X) for some v = (vn) ∈ X N .
Historically, characterized subgroups were studied exclusively in the case of the circle group T = R/Z (see [1,4,17,31]), also in relation with Diophantine approximation, Dynamical systems and Ergodic theory (see [4,35,36]). (One can find more on this topic in the nice survey [24], as well as in the more recent [13,14,25,33].) Some general results were then obtained in the case of metrizable compact abelian groups; for example, it is known that every countable subgroup of a metrizable compact abelian group is characterized (see [15,Theorem 1.4] and [3]), and it was pointed out in [3,10] that the metrizability is necessary, as a compact abelian group with a countable characterized subgroup is necessarily metrizable. Only recently, the case of general compact abelian groups was given full attention in [11] and a reduction theorem (to the metrizable case) was obtained.
The few exceptions [5,24] (concerning respectively characterized subgroups of R and of compact non-abelian groups), only confirm the tendency to study the characterized subgroups of T or, more recently, of compact abelian groups. To say the least, even the simplest case of characterized subgroups of discrete abelian groups has never been considered in the literature to the best of our knowledge.
The aim of these notes is to develop a general approach to characterized subgroups of arbitrary topological abelian groups, collecting the basic properties so far established in the compact case.
We isolate three special types of characterized subgroups, namely T -characterized, K-characterized and N -characterized subgroups (see Definition 5.1). Of those, T -characterized subgroups were introduced by For a topological space X = (X, τ ) the weight w(X) of X is the minimum cardinality of a base for τ . For a subset A of X we denote by A τ the closure of A in (X, τ ) (sometimes we write only A when there is no possibility of confusion). A topological abelian group X is totally bounded if for every open subset U of 0 in X there exists a finite subset F of X such that U + F = X. If X is totally bounded and Hausdorff we say that X is precompact. We denote by X the two-sided completion of X; in case X is precompact X coincides with the Weil completion.
For a subset A of X, the annihilator of A in X is A ⊥ = {χ ∈ X : χ(A) = {0}}, and for a subset B of X, the annihilator of B in X is B ⊥ = {x ∈ X : χ(x) = 0 for every χ ∈ B}.
We say that a sequence v ∈ X N is non-trivial if it is not eventually null.
2 Background on topological groups

Basic definitions
Let G be an abelian group and H a subgroup of Hom(G, T). Let TH be the weakest group topology on G such that all characters of H are continuous with respect to TH ; then TH is totally bounded. Viceversa, Comfort and Ross proved that any totally bounded group topology is of this type (see [8,Theorem 1.2]). (b) TH is metrizable if and only if H is countable.
The following two notions will be often used in the paper.
Definition 2.2. A topological abelian group X is said to be: (i) maximally almost periodic (briefly, MAP ) if X separates the points of X; (ii) minimally almost periodic (briefly, MinAP ) if X = {0}.
We recall that two group topologies τ1 and τ2 on an abelian group X are compatible if they have the same characters, that is, (X, τ1) = (X, τ2).
If X = (X, τ ) is a topological abelian group, denote by τ + its Bohr topology, that is, the finest totally bounded group topology on X coarser than τ (indeed, τ + = T X ); we denote X endowed with its Bohr topology also by X + and we call τ + also the Bohr modification of τ . Clearly, τ and τ + are compatible. Moreover, (i) τ is MAP if and only if τ + is Hausdorff; (ii) τ is MinAP if and only if τ + is indiscrete.
A subgroup H of (X, τ ) is: (a) dually closed if H is τ + -closed (or, equivalently, X/H is MAP); (b) dually embedded if every χ ∈ H can be extended to X. (ii) in particular, every locally compact abelian group is MAP, and (iii) X and X + have the same closed subgroups; (iv) consequently, X is separable if and only if X + is separable.
For a topological abelian group X and a subgroup L of X, the weak topology σ( X, L) of the dual X is the totally bounded group topology of X generated by the elements of L considered as characters of X; namely, for every x ∈ L, consider ξx : X → T defined by ξx(χ) = χ(x) for every χ ∈ X. A local base of σ( X, L) is given by the finite intersections of the sets ξ −1 x (U ), where x ∈ L and U is an open neighborhood of 0 in T. Clearly, if L1 ≤ L2, then σ( X, L1) ≤ σ( X, L2).
Note that the weak topology σ( X, X) is coarser than the compact-open topology on X. If L separates the points of X (e.g, when L is dense in X, or when L = X), then σ( X, L) is precompact.
Fact 2.4. If X is a reflexive topological abelian group, then σ( X, X) coincides with the Bohr topology of X.
We recall that a sequence v in an abelian group G is a T -sequence (respectively, T B-sequence) if there exists a Hausdorff (respectively, precompact) group topology τ on G such that v is a null sequence in (G, τ ). Lemma 2.5. Let X be a topological abelian group and v ∈ X N . Then: (i) for a subgroup L of X, vn(x) → 0 in T for every x ∈ L if and only if vn → 0 in σ( X, L); Proof. (i) follows from the definition of σ( X, L).
Let G be a discrete abelian group. For a sequence v ∈ G N , the group topology is the finest totally bounded group topology on G such that v is a null sequence in (G, σv).
Fact 2.6. [16, Lemma 3.1, Proposition 3.2] Let G be a discrete abelian group and v ∈ G N . The following conditions are equivalent: (ii) σv is Hausdorff;

Useful folklore results
We recall the following basic properties that will be used in the paper. Although most of them are well known, we offer proofs for reader's convenience.
Lemma 2.7. Let X be a topological abelian group and H a subgroup of X. Then: where π : X → X/H is the canonical projection. Then ψ is injective and its image is H ⊥ . (ii) Let ρ : X → H be defined by χ → χ ↾H . Then ker ρ = H ⊥ and so we get the thesis. Proof. The inequality w(K) ≤ c holds for every separable regular topological space K. Assume that w(K) ≤ c. The discrete abelian group X = K has size |X| = w(K) ≤ c. Consider the embedding i : X → D(X), where D(X) is the divisible hull of X. Then |D(X)| ≤ c and D(X) = i∈I Di, for some countable divisible abelian groups Di and a set of indices I with |I| ≤ c. Therefore, i : i∈I Di → X = K is a surjective continuous homomorphism and each Di is a metrizable compact abelian group. By Hewitt-Marczewski-Pondiczery Theorem, since |I| ≤ c, we have that i∈I Di is separable, hence K is separable as well.
Lemma 2.9. Let X be a precompact abelian group. Then {0} is G δ if and only if there exists a continuous injection X → T N .
Assume now that {0} = n∈N Un, where each Un is an open subset of X, and we can assume that Un is in the prebase of the neighborhoods of 0 in X. So for every n ∈ N there exist vn ∈ X and an open neighborhood Vn of 0 in T such that Vn does not contain any non trivial subgroup of T such that Un = v −1 n (Vn). Then {0} = n∈N ker vn. Hence, j : X → T N defined by j(x) = (vn(x)) n∈N is a continuous injective homomorphism.
Theorem 2.10. Let X be a locally compact abelian group. Then X is metrizable with |X| ≤ c if and only if there exists a continuous injective homomorphism X → T N .
Proof. If there exists a continuous injective homomorphism X → T N , then clearly X is metrizable and |X| ≤ |T N | = c.
Suppose now that X is metrizable and has cardinality at most c. It is well known (for example, see [29]) that X = R n × X0, where n ∈ N and X0 is a locally compact abelian group admitting an open compact (metrizable) subgroup K. Clearly, there exist two continuous injective homomorphisms j1 : R n ֒→ T N and j2 : K ֒→ T N . Therefore, j3 = (j1, j2) : R n × K → T N × T N ∼ = T N is an injective continuous homomorphism too. Since T N is divisible and R n × K is open in X, j3 extends continuously to j3 : X → T N . Let π : X → X/(R n × K) be the canonical projection. Since X/(R n × K) is discrete, there exists a continuous injective homomorphism j4 : for every x ∈ X. Then j is continuous since ϕ and j3 are continuous. Moreover, j is injective as j(x) = 0 for some x ∈ X implies ϕ(x) = 0 and j3(x) = 0; therefore, x ∈ R n × K, and so, since j3 ↾ R n ×K = j3 is injective, one has x = 0.

General permanence properties of characterized subgroups
Let X be a topological abelian group and denote by Char(X) the family of all subgroups of X that are characterized.
We start by observing that if v ∈ X N is eventually null, then X = sv(X). (3.1) The following are basic facts on characterized subgroups (see [9,11,15,16]), we give a proof for reader's convenience.  (iv) sv(X) is an F σδ -set (i.e., countable intersection of countable unions of closed subgroups).
Proof. Items (i) and (ii) are obvious. To prove (iii), if u, v ∈ X N , define w = (wn), where w2n = un and w2n+1 = vn for every n ∈ N, hence su(X) ∩ sv(X) = sw(X). To prove (iv), note that sv(X) = m k n≥k Sn,m, where each Sn,m = x ∈ X : vn(x) ≤ 1 m is a closed subset of X. Now we prove that, under suitable hypotheses, the relation of being a characterized subgroup is transitive: Proposition 3.2. Let X be a topological abelian group and X0, X1, X2 subgroups of X with X0 ≤ X1 ≤ X2 and such that X1 is dually embedded in X2. If X0 ∈ Char(X1) and X1 ∈ Char(X2), then X0 ∈ Char(X2).

Proof. Let v ∈ X1
N such that X0 = sv(X1) and let w ∈ X2 N such that X1 = sw(X2). As X1 is dually embedded in X2, vn extends to a character v * n of X2 for every n ∈ N, N by letting w * 2n = v * n and w * 2n+1 = wn for every n ∈ N. Then, by Lemma 3.1(i), so X0 ∈ Char(X2), as required.
Clearly, two compatible group topologies have the same characterized subgroups: Lemma 3.3. If τ1 and τ2 are compatible group topologies on an abelian group X, then Char(X, τ1) = Char(X, τ2).

The Γ-radical
The motivation for the choice of the term Γ-radical is the special case Γ = X, when n(X) := n X (X) is usually called the von Neumann radical of X. Then n(X) = {0} (respectively, n(X) = X) precisely when X separates the points of X (respectively, X = {0}); in other words:   (ii) If Γ is countable, then nΓ(X) is a characterized subgroup of X (indeed, nΓ(X) = sv(X) for v ∈ X N such that each character in Γ appears infinitely many times in v).
For a given sequence v ∈ X N , the support Γv = {vn : n ∈ N} of v is the set of all characters appearing in v. We abbreviate the notation of the Γv-radical by writing nv(X) := nΓ v (X), and we call this subgroup the v-radical of X. Lemma 4.3. Let X be a topological abelian group and v ∈ X N . Then: Proof. (i) and (ii) are clear from the definitions and (iii) follows from Remark 4.2 (ii).
(v) Since X/nv(X) is algebraically isomorphic to ϕ(X) ≤ T N and |T N | = c, we conclude that [X : nv(X)] ≤ c.
Remark 4.4. Let X be a topological abelian group and v ∈ X N . Then nv(X) is closed and G δ in every group topology on X that makes vn continuous for every n ∈ N. In particular, nv(X) is closed and G δ in every group topology on X compatible with the topology of X, so in the Bohr topology of X. Lemma 4.3 gives a bound for the index of the characterized subgroups: Every characterized subgroup of a topological abelian group X has index at most c.
The set Γv can be partitioned as In other words, Γ ∞ v is the set of all characters appearing infinitely many times in v, while each character in its complement Γ 0 v appears finitely many times in v. Clearly, v is a finitely many-to-one sequence if and only if In other words, one can safely replace v by v ∞ . This is why from now on we shall always assume that We see now how we can obtain the subgroup of X characterized by v by considering separately the v ∞radical of X and the subgroup of X characterized by v 0 . Lemma 4.6. Let X be a topological abelian group and v ∈ X N satisfying (4.1).
. This concludes the proof.

A hierarchy for characterized subgroups
The following definition introduces three specific types of characterized subgroups.
Definition 5.1. Let X be a topological abelian group. A subgroup H of X is: In analogy to Definition 5.1(i), we introduce the following smaller class of characterized subgroups (see also Problem 10.2).
Every T B-characterized subgroup is obviously T -characterized. Moreover, every T -characterized subgroup is also K-characterized, since every T -sequence contains no constant subsequences. The N -characterized subgroups are clearly closed, and they are always characterized as noted above.
On the other hand, proper dense characterized subgroups are T B-characterized by Lemma 2.5(ii), so also T -characterized, and in particular K-characterized, but they are not N -characterized. We shall see below that closed (even open) subgroups need not be K-characterized in general.
Denote by Char K (X) (respectively, Char N (X), Char T (X), Char T B (X)) the family of all K-characterized (respectively, N -characterized, T -characterized, T B-characterized) subgroups of the topological abelian group X. Then we have the following strict inclusions We start giving some basic properties that can be proved immediately. Corollary 5.3. Let X be a topological abelian group and X0, X1, X2 subgroups of X with X0 ≤ X1 ≤ X2 and such that X1 is dually embedded in X2.
Proof. (i) It suffices to note that if in the proof of Proposition 3.2, v is one-to-one and w is one-to-one, then w * is finitely many-to-one.
By Lemma 4.6, we have directly the following Corollary 5.4. Every characterized subgroup of a topological abelian group X is the intersection of an Ncharacterized subgroup of X and a K-characterized subgroup of X.
The following stability property is clear for N -characterized subgroups, while it is not known for characterized subgroups.
The next correspondence theorem was proved in [11] for characterized subgroups of compact abelian groups.
(iv) If H is T -characterized, that is, if H = su(X/F ) and u is a T -sequence, it remains to verify that v is a T -sequence as well, since π −1 (H) = sv(X) by (i). Let τ be a Hausdorff group topology on X/F , such that un → 0 in ( X/F , τ ). By Lemma 2.7(i), one can identify X/F with the subgroup F ⊥ of X by the algebraic isomorphism ψ : X/F → X defined by χ → χ • π. Let τ * be the group topology on X having as a local base at 0 the open neighborhoods of 0 in ( X/F , τ ). Then τ * is a Hausdorff group topology on X and vn → 0 in ( X, τ * ), as vn = ψ(un) ∈ F ⊥ for every n ∈ N by definition.
Lemma 5.7. Let X be a topological abelian group and H a subgroup of X such that n(X) ≤ H. Then H is characterized (respectively, K-characterized, N -characterized, T -characterized) if and only if H/n(X) is characterized (respectively, K-characterized, N -characterized, T -characterized).
Proof. Let H = sv(X) for some v ∈ X N and denote by π : X → X/n(X) the canonical projection. For every n ∈ N, since n(X) ≤ ker vn, the character vn factorizes as vn = un • π, where un ∈ X/n(X). Then H/n(X) = su(X/n(X)). Viceversa, if H/n(X) = su(X/n(X)) for some u ∈ X/n(X) N , let vn = un • π for every n ∈ N. Hence, H = sv(X). Moreover, v is a finitely many-to-one sequence if and only if u is a finitely many-to-one sequence, so It remains to check that v is a T -sequence precisely when u is a T -sequence. This follows from the fact that the natural homomorphism X/n(X) → X sending (the members of) u to (the members of) v is an isomorphism, so certainly preserves the property of being a T -sequence.
The following lemma gives equivalent conditions for a subgroup to be characterized.
Lemma 5.8. Let X be a topological abelian group and H a subgroup of X. The following conditions are equivalent: (ii) there exists a closed subgroup F of X such that F ≤ H and H/F ∈ Char(X/F ); and each un is the factorization of vn through the canonical projection π : X → X/F .
Since F ≤ H one has H = π −1 (H/F ) and by Proposition 5.6 one can conclude.

Autocharacterized groups
The following consequence of [11,Proposition 2.5] motivates the introduction of the notion of autocharacterized group (see Definition 6.3).
Proposition 6.1. Let X be a compact abelian group. Then sv(X) = X for some v ∈ X N if and only if the sequence v is eventually null.
Proof. It is clear from (3.1) that sv(X) = X if v is eventually null. Assume now that X = sv(X) for some v ∈ X N . By [11,Proposition 2.5], being sv(X) compact, there exists m ∈ N such that X = sv(X) = nv (m) (X), and so vn = 0 for all n ≥ m.
If one drops the compactness, then the conclusion of Proposition 6.1 may fail, as shown in the next example. Example 6.2.
(i) Let N be an infinite countable subgroup of T. As mentioned in the introduction, N is characterized in T, (ii) Let X = R and v = (vn) ∈ R N such that v0 = 0 and vn(x) = π( x n ) ∈ T for every x ∈ R and n ∈ N+. Obviously, sv(R) = R, even if v is non-trivial. Items (ii) and (iii) of Example 6.2 show that R and Qp are autocharacterized.
Remark 6.4. (i) Let X be an autocharacterized topological abelian group, so let v ∈ X N be non-trivial and such that X = sv(X). Then there exists a one-to-one subsequence u of v such that un = 0 for every n ∈ N and X = su(X). Indeed, if χ ∈ Γ ∞ v , then X = sv(X) ≤ ker χ and so χ = 0; therefore, Γ ∞ v is either empty or {0}. Since v is non-trivial, Γ 0 v is infinite, hence X = sv(X) = s v 0 (X) by Lemma 4.6(ii). Let u be the one-to-one subsequence of v 0 such that Γu = Γ v 0 , therefore X = su(X).
(ii) The above item shows that autocharacterized groups are K-characterized subgroups of themselves. But one can prove actually that they are T -characterized subgroups of themselves (indeed, T B-characterized subgroups of themselves, see [12]).

Basic properties of autocharacterized groups
We start by a direct consequence of Lemma 5.7: Lemma 6.5. Let X be a topological abelian group and H a subgroup of X such that n(X) ≤ H. Then X is autocharacterized if and only if X/n(X) is autocharacterized.
The next proposition, describing an autocharacterized group in terms of the null sequences of its dual, follows from the definitions and Lemma 2.5: Proposition 6.6. A topological abelian group X is autocharacterized if and only if ( X, σ( X, X)) has a nontrivial null sequence v (in such a case X = sv(X)).
In the next lemma we see when a subgroup of an autocharacterized group is autocharacterized, and viceversa. Lemma 6.7. Let X be a topological abelian group and H a subgroup of X. (a) Let X = H × Z. For every n ∈ N, let un be the unique character of X that extends vn and such that un vanishes on Z. Then un = 0 for every n ∈ N, and X = su(X).
(b) Arguing by induction, we can assume without loss of generality that [X : H] = p is prime. Let X = H + x with x ∈ H and px ∈ H. If px = 0, then H is an open direct summand of X, so H is also a topological direct summand of X, hence item (a) applies. Assume now that px = 0, and let an = vn(px) for every n ∈ N. If an = 0 for infinitely many n, extend vn to un ∈ X N for those n by letting un(x) = 0. Then obviously the sequence u obtained in this way is not eventually null and X = su(X), so X is autocharacterized. Assume now that an = 0 for infinitely many n; for those n, pick an element bn ∈ T with pbn = an and extend vn by letting un(h + kx) = vn(h) + kbn. Let wn = pun. Then wn(x) = an = 0, so w is not eventually null. Moreover, X = su(X) as pX ≤ H. Lemma 6.8. Let X be a topological abelian group and v ∈ X N . If F is a subgroup of X such that F ≤ sv(X) and F is not autocharacterized, then F ≤ nv (m) (X) for some m ∈ N.
Proof. Let un = vn ↾F for every n ∈ N and u = (un) ∈ F N . Then F = su(F ), so the sequence u must be eventually null. Let m ∈ N such that un = 0 for every n ≥ m. Therefore, F ≤ nv (m) (X).
The following consequence of Lemma 6.8 is a generalization of [11,Lemma 2.6] where the group X is compact. Corollary 6.9. Let X be a topological abelian group, F and H subgroups of X such that F is compact and F ≤ H. Then H/F ∈ Char(X/F ) if and only if H ∈ Char(X).
Proof. Denote by π : X → X/F the canonical projection. If H/F is a characterized subgroup of X/F then H = π −1 (H/F ) is a characterized subgroup of X by Proposition 5.6. Assume now that H = sv(X) for some v ∈ X N . Since F is compact, Proposition 6.1 implies that F is not autocharacterized. By Lemma 6.8, F is contained in nv (m) (X) for a sufficiently large m ∈ N. Let π ′ : X/F → X/nv (m) (X) be the canonical projection and q = π ′ • π, then q −1 (H/nv (m) (X)) = H = sv(X) = sv (m) (X). Therefore, one deduces from Lemma 5.8 that H/nv (m) (X) is a characterized subgroup of X/nv (m) (X). Hence, by Proposition 5.6, H/F = (π ′ ) −1 (H/n v (m) (X) ) ∈ Char(X/F ).
The argument of the above proof fails in case F is not compact. For example, take F = H = X = N , where N is as in Example 6.2; then one cannot conclude that v ↾F is eventually null, and hence that F is contained in nv (m) (X).

Criteria describing autocharacterized groups
Here we give two criteria for a group to be autocharacterized. We start below with a criteria for locally compact abelian groups, while a general one, in terms of the Bohr compactification, will be given at the end of the section.
We established in Proposition 6.1 that no compact abelian group is autocharacterized, now we prove in Theorem 6.10 that this property describes the compact abelian groups within the larger class of all locally compact abelian groups. This follows easily from Lemma 6.7(ii) for the locally compact abelian groups that contain a copy of R, while the general cases requires the following deeper argument. Theorem 6.10. If X is a locally compact abelian group, then X is autocharacterized if and only if X is not compact.
Proof. If X is autocharacterized, then X is not compact according to Proposition 6.1. Assume now that X is not autocharacterized. Then by Fact 2.4 and Proposition 6.6 the dual X has no non-trivial null sequences in its Bohr topology. But since X is locally compact, it has the same null sequences as its Bohr modification X + . Therefore, X is a locally compact group without non-trivial null sequences. We have to conclude that X is compact.
This follows from the conjunction of several facts. The first one is the deep fact that non-discrete locally compact abelian groups have non-trivial null sequences. (This follows, in turn, from that fact that a non-discrete locally compact abelian group either contains a line R, or an infinite compact subgroup. Since compact groups are dyadic compacts, i.e., continuous images of Cantor cubes, they have non-trivial null sequences.) Now we can conclude that the locally compact group X is discrete. It is a well known fact that this implies compactness of X.
Remark 6.11. An alternative argument to prove that non-discrete locally compact abelian groups have nontrivial null sequences is based on a theorem by Hagler, Gerlits and Efimov (proved independently also by Efimov in [21]). It states that every infinite compact group K contains a copy of the Cantor cube {0, 1} w(K) , which obviously has a plenty of non-trivial null sequences.
In order to obtain a general criterion describing autocharacterized groups we need another relevant notion in the theory of characterized subgroups: Definition 6.12. [16] Let X be a topological abelian group and H a subgroup of X. Let A subgroup H of X is said to be: We write simply g(H) when there is no possibility of confusion. Clearly, g(H) is a subgroup of X containing H. Moreover, g({0}) is the intersection of all characterized subgroups of X and g({0}) ≤ n(X). Remark 6.13. Let X = (X, τ ) be a topological abelian group and H a subgroup of X.
, v is the constant sequence given by χ). Item (i) says, in terms of [10,20], that g is a closure operator in the category of topological abelian groups.
The inclusion g X (H) ≤ H τ + says that g is finer than the closure operator defined by H → H τ + . (iii) If H is dually closed, then H is g-closed by item (ii).
(iv) If (X, τ ) is a locally compact abelian group, then every closed subgroup of (X, τ ) is dually closed, and so (ii) implies that g(H) ≤ H τ for every subgroup H of X. Therefore, g-dense subgroups are also dense in this case. The next result shows that the autocharacterized precompact abelian groups are exactly the dense non-gdense subgroups of the compact abelian groups. Theorem 6.14. Let X be a precompact abelian group. The following conditions are equivalent: Proof. (ii)⇒(i) Assume that X is not g-dense in K := X. Then there exists a sequence v ∈ K N such that X ≤ sv(K) < K. By Proposition 6.1 (see also Remark 6.4), we may assume without loss of generality that vn = 0 for every n ∈ N. Let un = vn ↾X for every n ∈ N. Since X is dense in K, clearly un = 0 for every n ∈ N. Moreover, X = su(X), hence X is autocharacterized.
(i)⇒(ii) Suppose that X is autocharacterized, say X = su(X) for u ∈ X N such that un = 0 for every n ∈ N. For ever n ∈ N, let vn ∈ K be the extension of un to K. Then X ≤ sv(K) < K by Proposition 6.1, so X is not g-dense in K.
If X is a MAP abelian group, then τ + is precompact and the Bohr compactification of X is rX : X → bX, where bX is the completion of (X, τ + ) and rX is an injective homomorphism. If X is not MAP, then n(X) = {0}. Consider the quotient X/n(X), which is a MAP group. Then take the Bohr compactification r X/n(X) : X/n(X) → b(X/n(X)). The Bohr compactification of X is rX : X → bX, where bX := b(X/n(X)) and rX = r X/n(X) • π, where π : X → n(X) is the canonical projection. Corollary 6.15. Let X be a MAP abelian group. The following conditions are equivalent: Proof. Since X is MAP, X embeds in bX. By Lemma 3.3, X and X + have the same characterized subgroups. Moreover, X + is precompact and by definition bX is the completion of X + . Then it suffices to apply Theorem 6.14.
Theorem 6.16. Let X be a topological abelian group. The following conditions are equivalent: Proof. Since X is autocharacterized precisely when X/n(X) is autocharacterized by Lemma 6.5, apply Corollary 6.15 to conclude.

K-characterized subgroups
We start by recalling the following result from [9]: if X is a compact abelian group and v ∈ X N has a one-to-one subsequence, then sv(X) has Haar measure 0 in X. In our terms, it reads as follows: Lemma 7.1 cannot be inverted, take for example the constant sequence u = (1) in T N . Remark 7.2. If X is a connected compact abelian group, then the conclusion of Lemma 7.1 holds for all nontrivial sequences in X, since every measurable proper subgroup H of X has measure 0 (indeed, X is divisible, so the proper subgroup H of X has infinite index, hence the measure of H must be 0, as X has measure 1). Example 7.3. Here we provide examples of non-autocharacterized non-compact abelian groups.
(i) A relatively simple example can be obtained by taking a dense non-measurable subgroup X of a connected compact abelian group K. Since we intend to deduce that X is non-autocharacterized by using Theorem 6.14, we have to check that X is g-dense in K. Indeed, every measurable proper subgroup of K has measure 0 as noted in Remark 7.2, therefore every proper characterized (hence, every non-g-dense) subgroup of K has measure 0. Therefore, X is not contained in any proper characterized subgroup of K, i.e., X is g-dense in K.
(ii) More sophisticated examples of g-dense subgroups of metrizable compact abelian groups were given in [2] (under the assumption of Martin Axiom) and in [27] (in ZFC). These groups have the additional property of being of measure zero (so that the above elementary argument cannot be used to verify the g-density).

When closed subgroups of infinite index are K-characterized
The next theorem gives a sufficient condition (see item (iii)) for a closed subgroup of infinite index H to be K-characterized. This condition implies, as a by-product, that H is also N -characterized. The easier case of open subgroups will be tackled in Theorem 7.7, by applying Theorem 7.4.
Theorem 7.4. Let X be a topological abelian group and H a closed subgroup of X of infinite index. The following conditions are equivalent: Since H is N -characterized, then H is dually closed by Lemma 4.3 (ii), that is, X/H is MAP. Let H = nv(X) for some v ∈ X N and let π : X → X/H be the canonical projection. Since ker π = H ≤ ker vn for every n ∈ N, one can factorize vn : X → T through π, i.e., write vn =vn • π for appropriatevn ∈ X/H. It remains to verify that D = {vn : n ∈ N} is dense in ( X/H, σ( X/H, X/H)). To this end, letȳ = π(y) ∈ X/H; if ξȳ(D) = {0}, thenvn(ȳ) = vn(y) = 0 and so y ∈ H, that is,ȳ = 0.
(iii)⇒(i) Let Y = X/H equipped with the quotient topology. By hypotheses, Y is infinite and MAP, while Y is an infinite topological abelian group with a countably infinite dense subgroup D. According to Proposition 5.6 applied to the canonical projection π : X → Y , it suffices to prove that {0} is a K-characterized subgroup of Y . Let D = {vn : n ∈ N} be a one-to-one enumeration of D and v = (vn). To prove that sv(Y ) = {0}, we have to show that for every non-zero y ∈ Y there exists a neighborhood U of 0 in T such that vn(y) ∈ U for infinitely many n ∈ N. Actually, we show that U = T+ works for all non-zero y ∈ Y . In fact, for every y ∈ Y \ {0} one has that Ny := {d(y) : d ∈ D} = {vn(y) : n ∈ N} is a non-trivial subgroup of T, as Y is MAP and y = 0.
Let y ∈ Y \ {0}. If Ny is infinite, then Ny is dense in T, so Ny \ U is infinite and we are done. Now consider the case when Ny is finite. As Ny = {0} and U contains no non-trivial subgroups of T, there exists a ∈ Ny such that a ∈ U . Then the map fy : D → T defined by fy(d) = d(y) is a homomorphism with fy(D) = Ny finite. Therefore, K := ker fy is a finite-index subgroup of D, so K is infinite. Let a = vm(y) for some m ∈ N. Then vm + K = {d ∈ D : d(y) = a} is infinite as well. This means that vn(y) = a ∈ U for infinitely many n (namely, those n for which vn ∈ vm + K). Therefore, vn(y) → 0 and so y ∈ sv(Y ).
Finally, let us note that the above argument shows also that H is N -characterized as obviously H ≤ nv(X).
The following is an obvious consequence of Theorem 7.4.
Corollary 7.5. Let X be a topological abelian group and H a closed subgroup of X of infinite index. If H ∈ Char N (X), then H ∈ Char K (X).
Next we rewrite Theorem 7.4 in the case of locally compact abelian groups.
Corollary 7.6. Let X be a locally compact abelian group and H a subgroup of X. Then H ∈ Char N (X) if and only if H is closed and X/H is separable.
Proof. As both conditions imply that H is closed, we assume without loss of generality that H is closed. Since X/H and X/H are locally compact abelian groups, X/H is MAP and the Bohr topology on X/H coincides with σ( X/H, X/H) by Fact 2.4, so the separability of X/H is equivalent to the separability of X/H + by Fact 2.3. If H has infinite index in X, apply Theorem 7.4 to conclude. If H has finite index in X, then the equivalence is trivially satisfied; indeed, H is a finite intersection of kernels of characters, so it is N -characterized, and X/H is finite, so separable. (iv) X/H is separable.

Proof. (ii)⇒(i) is clear and (i)⇒(iii) is Corollary 4.5.
(iii)⇒(iv) Since X/H is a compact abelian group of weight at most c, it is separable by Lemma 2.8.
(iv)⇒(ii) As [X : H] is infinite, we can apply Theorem 7.4 to conclude that H is K-characterized.
The following is another direct consequence of Theorem 7.4.
Corollary 7.8. If X is a metrizable compact abelian group, then every closed non-open subgroup of X is K-characterized.

When closed subgroups of finite index are K-characterized
We start by giving the following useful technical lemma.
Lemma 7.9. Let X be a topological abelian group and H an open subgroup of X such that X = H + x for some x ∈ X \ H. If H is autocharacterized, then H ∈ Char K (X).
Proof. Let u ∈ H N such that H = su(H). By Remark 5.7 we can assume that u is one-to-one and that un = 0 for every n ∈ N.
Assume first that H ∩ x = {0}. If o(x) is infinite, then fix an irrational number α ∈ R and for every n ∈ N let vn(x) = α + Z and vn(h) = un(h) for every h ∈ H. If o(x) = k is finite, then for every n ∈ N let vn(x) = 1 k +Z and vn(h) = un(h) for every h ∈ H. In both cases, it is straightforward to prove that H = sv(X). Moreover, since u is one-to-one, then also v is one-to-one.
Suppose now that H ∩ x = mx for some m ∈ N, with m ≥ 1. As x ∈ H, one has m ≥ 2. For every n ∈ N, let an = un(mx) ∈ T. Since un(mx) → 0, there exists n0 such that an < 1 m 2 for every n ≥ n0. As su (n 0 ) (H) = H, we shall assume for simplicity that an < 1 m 2 every n ∈ N.
Claim 7.10. For every a ∈ T with a < 1 m 2 , there exists b ∈ T such that mb = a and kb > 1 m 2 for every k ∈ N, 1 ≤ k < m.
Proof. We tackle the problem in R, that is, identifying T with [0, 1). First assume that 0 ≤ a < 1 m 2 and let b = a m + 1 m .
Then mb = a + 1 ≡ Z a and 1 Therefore, kb > 1 m 2 . This establishes (7.1) in the current case. It remains to consider the case m 2 −1 m 2 < a < 1. Let a * = 1 − a, i.e., a * = −a in T. Then obviously a * < 1 m 2 and 0 ≤ a * < 1 m 2 . Hence, by the above case applied to a * , there exists b * ∈ T satisfying (7.1) with −a in place of a (i.e., mb * = −a). Let b = −b * ∈ T. Then (7.1) holds true for b and a, as k(−b) = kb for every k ∈ N with 1 ≤ k < m.
For every n ∈ N, apply Claim 7.10 to an to get bn as in (7.1), then define vn : X → T by letting vn(x) = bn for every n ∈ N and vn(h) = un(h) for every h ∈ H. As un(mx) = vn(mx) = mvn(x) = mbn = an, this definition is correct. Moreover, since H is open in X, vn ∈ X. Since u is one-to-one, then also v = (vn) is one-to-one.
We show that vn(kx) → 0 for k ∈ N if and only if k ∈ mZ.
Every open finite-index subgroup is a finite intersection of kernels of characters, so it is N -characterized. In the next theorem we describe when a proper open finite-index subgroup is K-characterized. Proof. Assume that H ∈ Char K (X). We can write H = su(X) for u ∈ X N one-to-one. Let vn = un ↾H for every n ∈ N. Then the map un → vn is finitely many-to-one, as X/H is finite. Therefore, v = (vn) is finitely many-to-one. Obviously, H = sv(H), so H is autocharacterized. Now assume that H is autocharacterized. Since H has finite index in X, there exist x1, . . . , xn ∈ X such that X = H + x1, . . . , xn and that, letting Xi := H + x1, . . . , xi for i = 1, . . . , n and X0 := H, the subgroup Xi−1 is a proper subgroup of Xi for i = 1, . . . , n. We shall prove by induction on i = 1, . . . , n, that H ∈ Char K (Xi). (7.4) As X = Xn, this will give H ∈ Char K (X), as desired. Before starting the induction, we note that according to Lemma 6.7(ii), all subgroups Xi, for i = 1, . . . , n, are autocharacterized, as each Xi−1 is open in Xi. For i = 1, the assertion in (7.4) follows from Lemma 7.9. Assume that 1 < i ≤ n and (7.4) holds true for i − 1, i.e., H ∈ Char K (Xi−1). Since Xi−1 is open in Xi, again Lemma 7.9 applied to Xi = Xi−1 + xi gives that Xi−1 ∈ Char K (Xi). As H ∈ Char K (Xi−1) by our inductive hypothesis, we conclude with Corollary 5.3(i) that H ∈ Char K (Xi).

Further results on K-characterized subgroups
The next corollary resolves an open question from [30]: Corollary 7.12. Let X be an infinite discrete abelian group and H a subgroup of X. The following conditions are equivalent: We give now sufficient conditions for a non-closed characterized subgroup to be K-characterized.
Theorem 7.13. Let X be a topological abelian group and H ∈ Char(X) a non-closed subgroup of X such that: (i) X/H is MAP and ( X/H , σ( X/H, X/H )) is separable; Then H ∈ Char K (X).
Proof. As H = H is dense in H and obviously H ∈ Char(H), we deduce that H ∈ Char K (H), as dense characterized subgroups are T B-characterized by Lemma 2.5 (ii). If H = X, we are done. So assume that H is proper.
Our aim now is to apply Corollary 5.3, so we need to check that H ∈ Char K (X). If H has finite index in X, then H is autocharacterized by hypothesis and so Theorem 7.11 yields H ∈ Char K (X). If H is has infinite index in X, then H ∈ Char K (X) by Theorem 7.4. Corollary 7.14. Let X be a divisible topological abelian group and H ∈ Char(X) a non-closed subgroup of X such that H is dually closed. If ( X/H , σ( X/H, X/H )) is separable, then H ∈ Char K (X). In particular, H ∈ Char K (X) whenever X/H is separable.
Proof. The first part of our hypothesis entails that X/H is MAP. Moreover, divisible topological abelian groups have no proper closed subgroup of finite index. Therefore, the first assertion follows directly from Theorem 7.13.
The topology σ( X/H, X/H ) of the dual X/H is coarser than the compact-open topology of X/H , so that separability of X/H yields separability of ( X/H , σ( X/H, X/H )). Hence, the second assertion can be deduced from the first one.
In the case of connected locally compact abelian groups one obtains the following stronger conclusion: Corollary 7.15. Let X be a connected locally compact abelian group. Then Proof. The group X is divisible, as connected locally compact abelian groups are divisible.
Let H ∈ Char(X). Our next aim will be to check that X/H is separable.
then one has the chain of subgroups nv(X) ≤ H ≤ H ≤ X. If H = X, then X/H is trivially separable. Otherwise H has infinite index in X since X/H is divisible, so X/nv (X) is separable by Corollary 7.6; since H contains nv(X), the quotient group X/H is a quotient group of X/nv(X). Therefore, X/H is isomorphic to a subgroup of the separable group X/nv(X), so X/H is separable as well (see [7]). Furthermore, H is dually closed (so X/H is MAP) by Fact 2.3.
If H is not closed, Corollary 7.14 gives that H ∈ Char K (X). If H is a proper closed subgroup of X, then H has infinite index as X/H is divisible, so H ∈ Char K (X) by Theorem 7.4. This proves the inclusion Char(X) \ {X} ⊆ Char K (X) \ {X}, which along with the obvious inclusion Char K (X) ⊆ Char(X), proves the equality Char(X) \ {X} = Char K (X) \ {X}.
It remains to consider the (closed) subgroup H = X which obviously belongs to Char(X). If X is compact, then X ∈ Char K (X), by Lemma 7.1, so Char K (X) = Char(X) \ {X}. If X is not compact, then H = X ∈ Char K (X) by Theorem 6.10 and Remark 6.4 (ii). Hence, Char K (X) = Char(X) in this case.
In particular, the above corollary yields Char K (T) = Char(T) \ {T} and Char K (R) = Char(R). Remark 7.16. As we shall see in Corollary 9.7, connectedness is necessary in this corollary.

N -characterized subgroups
The following consequence of Lemma 6.8 gives a sufficient condition for a characterized subgroup to be Ncharacterized: Corollary 8.1. Let X be a topological abelian group and H a subgroup of X which is not autocharacterized. If H ∈ Char(X), then H ∈ Char N (X).
Here comes an easy criterion establishing when a subgroup is N -characterized.
Proof. (i)⇒(ii) Suppose that there exists a continuous injection j : X/H → T N . Let π : X → X/H be the canonical projection. For every n ∈ N, let pn : T N → T be n-th projection and let vn = pn • j • π.
Therefore, vn ∈ X N for every n ∈ N and H = nv(X), where v = (vn).
(ii)⇒(i) Let H = nv(X) for v ∈ X N . Let π : X → X/nv(X) be the canonical projection and define j : X/nv(X) → T N by j(π(x)) = (vn(x)) n∈N for every x ∈ X. Since nv(X) = n∈N ker vn, then j is welldefined and injective. Moreover, j is continuous.
Finally, (i) and (   where G d denotes the abelian group G endowed with the discrete topology. Therefore, the following fact is an immediate consequence of (9.1). Recall that a topological abelian group X is almost maximally almost periodic (briefly, AMAP ) if n(X) is finite. Remark 9.2. In relation with Fact 9.1, Lukács in [32] found a T -sequence in Z(p ∞ ) that is not a T B-sequence, providing in this way an example of an AMAP group. More precisely, he found a characterizing sequence v for p m Jp ≤ Jp for a fixed m ∈ N+, i.e., sv(Jp) = p m Jp. In this way, being Jp/p m Jp finite, then sv(Jp) ⊥ = n(Z(p ∞ ), σv) = {0} is finite.
Therefore, (Jp, σv) is AMAP. Further results in this direction were obtained by Nguyen [34]. Finally, Gabriyelyan in [22] proved that an abelian group G admits an AMAP group topology if and only if G has non-trivial torsion elements.
The following theorem, due to Gabriyelyan, links the notions of T -characterized subgroup and MinAP topology. Following [17, §4], for a topological abelian group X and a prime number p, we denote by Tp(X) the closure of the subgroup Xp = {x ∈ X : p n x → 0}. In case X is compact, one can prove that Tp(X) = {mX : m ∈ N+, (m, p) = 1}. (9.2) In particular, Tp(X) contains the connected component c(X) of X. More precisely, if X/c(X) = p∈P (X(c(X))p is the topologically primary decomposition of the totally disconnected compact abelian group X/c(X), then Tp(X)/c(X) ∼ = (X/c(X))p = Tp(X/c(X)).
Following [18], we say that d ∈ N is a proper divisor of n ∈ N provided that d ∈ {0, n} and dm = n for some m ∈ N. Note that, according to our definition, each d ∈ N \ {0} is a proper divisor of 0. Definition 9.4. Let G be an abelian group.
(i) For n ∈ N the group G is said to be of exponent n (denoted by exp(G)) if nG = {0}, but dG = {0} for every proper divisor d of n. We say that G is bounded if exp(G) > 0, and otherwise that G is unbounded.
(ii) [26] If G is bounded, the essential order eo(G) of G is the smallest n ∈ N+ such that nG is finite. If G is unbounded, we define eo(G) = 0.
In the next theorem we aim to give a detailed description of the closed characterized subgroups H of X that are not T -characterized. As stated in Corollary 8.7, a closed subgroup H of a compact abelian group X is characterized if and only if H is G δ (i.e., X/H is metrizable). This explains the blanket condition imposed on H to be a G δ -subgroup of X.
Theorem 9.5. For a compact abelian group X and a G δ -subgroup H of X, the following conditions are equivalent: (i) H ∈ Char T (X);