1. Introduction
For a topological abelian group
X, we denote by
its dual group, that is the group of all characters of
X (
i.e., continuous homomorphisms
). Following [
1], for a sequence of characters
, let:
which is always a subgroup of
X. A subgroup
H of
X is said to be
characterized if
for some
.
Historically, characterized subgroups were studied exclusively in the case of the circle group
(see [
2,
3,
4,
5]), also in relation to Diophantine approximation, dynamical systems and ergodic theory (see [
3,
6,
7]; one can find more on this topic in the nice survey [
8], as well as in the more recent [
9,
10,
11,
12]). 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 [
13] and [
14]), and it was pointed out in [
14,
15] 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 [
16], and a reduction theorem (to the metrizable case) was obtained.
The few exceptions [
8,
17,
18] only confirm the tendency to study the characterized subgroups of
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 3). Of those,
T-characterized subgroups were introduced by Gabriyelyan in [
11,
K-characterized subgroups were substantially studied by Kunen and his coauthors in [
13,
19,
20], while
N-characterized subgroups, even if never introduced explicitly, have been frequently used in the theory of duality in topological abelian groups (being nothing else but the annihilators of countable sets of the dual group). One of the advantages of this articulation is the possibility to establish some general permanence properties that fail to be true in the whole class of characterized subgroups, but hold true in some of these subclasses. Moreover, we see that each characterized subgroup is either
N-characterized or coincides with the intersection of an
N-characterized subgroup and a
K-characterized subgroup (see Corollary 3).
Inspired by the notion of
T-characterized subgroup, we introduce also the stronger one of
-characterized subgroup (see Definition 4). The following implications hold, and none of them can be reversed in general (see
Section 5):
where (*) holds under the assumption that the subgroup is closed and has infinite index (see Corollary 6).
In
Section 6, we introduce the prominent class of auto-characterized groups (see Definition 5). These are the topological abelian groups that are characterized subgroups of themselves by means of a
non-trivial sequence of characters (see (
2)). The fact that compact abelian groups are not auto-characterized is equivalent to the well-known non-trivial fact that the Bohr topology of an infinite discrete abelian group has no non-trivial convergent sequences. Here, we generalize this fact by proving that the property of being non-auto-characterized describes the compact abelian groups within the class of all locally compact abelian groups (see Theorem 3). Moreover, in the general case, we describe the auto-characterized groups in terms of their Bohr compactification (see Theorem 5).
We study the basic properties of
K- and of
N-characterized subgroups respectively in
Section 7 and
Section 8. For the case of discrete abelian groups, which is considered here for the first time, we give a complete description of characterized subgroups by showing that these are precisely the subgroups of index at most
and that a subgroup is characterized precisely when it is
K- and
N-characterized (see Corollary 16).
In
Section 7, we describe when a closed subgroup of infinite index is both
K- and
N-characterized, and we see that this occurs precisely when it is only
N-characterized (see Theorem 6); then, we consider the special case of open subgroups, proving that proper open subgroups of infinite index (respectively, of finite index) are
K-characterized if and only if they are characterized (respectively, auto-characterized) (see Theorems 7 and 8). In particular, no proper open subgroup of a compact abelian group is
K-characterized.
In
Section 8, extending a criterion for compact abelian groups given in [
16], we show that for locally compact abelian groups one can reduce the study of characterized subgroups to the metrizable case (see Theorem 11). Moreover, we describe the closed characterized subgroups of the locally compact abelian groups by showing that they are precisely the
N-characterized subgroups (see Theorem 12). As a consequence, we add other equivalent conditions to the known fact from [
16] that a closed subgroup of a compact abelian group is characterized if and only if it is
, namely that the subgroup is
K- and
N-characterized (see Theorem 13).
Section 9 concerns
T-characterized subgroups of compact abelian groups. We establish a criterion to determine when a characterized subgroup of a compact abelian group is not
T-characterized (see Theorem 15), which extends results from [
11]. The impact on characterized subgroups of connected compact abelian groups is discussed.
The final
Section 10 contains various comments and open problems, both general and specific.
1.1. Notation and Terminology
The symbol is used to denote the cardinality of continuum. The symbols , , and are used for the set of integers, the set of primes, the set of non-negative integers and the set of positive integers, respectively. The circle group is identified with the quotient group of the reals and carries its usual compact topology. Let be the canonical projection; the usual group norm on is defined by for every . We denote by the image of in . If m is a positive integer, , and is the cyclic group of order m. Moreover, for , we denote by and , respectively, the Prüfer group and the p-adic integers.
We say that an abelian group G is torsion if every element of G is torsion (i.e., for every , there exists , such that ). If M is a subset of G, then is the smallest subgroup of G containing M. We denote by the group of the homomorphisms .
For a topological space , the weight of X is the minimum cardinality of a base for τ. For a subset A of X, we denote by the closure of A in (we write only 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 . If X is totally bounded and Hausdorff, we say that X is precompact. We denote by the two-sided completion of X; in case X is precompact, coincides with the Weil completion.
For a subset A of X, the annihilator of A in is , and for a subset B of , the annihilator of B in X is .
We say that a sequence is trivially null if there exists , such that for every , and we say that is non-trivial if it is not trivially null.
2. Background on Topological Groups
2.1. Basic Definitions
Let
G be an abelian group and
H a subgroup of
. Let
be the weakest group topology on
G, such that all elements of
H are continuous with respect to
; then
is totally bounded. The other way around, Comfort and Ross proved that any totally bounded group topology is of this type (see [
21, Theorem 1.2]).
Theorem 1. [21, Theorems 1.2, 1.3 and 1.11, Corollary 1.4] Let G be an abelian group and H a subgroup of .
Then, is totally bounded and:- (a)
is Hausdorff if and only if H separates the points of G;
- (b)
is metrizable if and only if H is countable.
The following two notions will be often used in the paper.
Definition 1. A topological abelian group
X is said to be:
- (i)
maximally almost periodic (MAP) if separates the points of X;
- (ii)
minimally almost periodic (MinAP) if .
We recall that two group topologies and on an abelian group X are compatible if they have the same characters, that is .
If
is a topological abelian group, denote by
its
Bohr topology, that is the finest totally bounded group topology on
X coarser than
τ (indeed,
); we denote
X endowed with its Bohr topology also by
, and we call
also the
Bohr modification of
τ. Clearly,
τ and
are compatible. Moreover,
- (i)
τ is MAP if and only if is Hausdorff;
- (ii)
τ is MinAP if and only if is indiscrete.
A subgroup
H of
is:
- (a)
dually closed if H is -closed (or, equivalently, is MAP);
- (b)
dually embedded if every can be extended to X.
Clearly, dually closed implies closed, since .
Fact 1. Let X be a locally compact abelian group. Then:- (i)
every closed subgroup H of X is dually closed, i.e., is MAP;
- (ii)
in particular, every locally compact abelian group is MAP;
- (iii)
X and have the same closed subgroups;
- (iv)
consequently, X is separable if and only if is separable.
For a topological abelian group X and a subgroup L of X, the weak topology of the dual is the totally bounded group topology of generated by the elements of L considered as characters of ; namely, for every , consider defined by for every . A local base of is given by the finite intersections of the sets , where and U is an open neighborhood of 0 in . Clearly, if , then .
Note that the weak topology is coarser than the compact-open topology on . If L separates the points of (e.g., when L is dense in X or when ), then is precompact.
Fact 2. If X is a reflexive topological abelian group, then coincides with the Bohr topology of .
We recall that a sequence in an abelian group G is a T-sequence (respectively, -sequence) if there exists a Hausdorff (respectively, precompact) group topology τ on G, such that is a null sequence in .
Lemma 1. Let X be a topological abelian group and .
Then:- (i)
for a subgroup L of X, in for every if and only if in ;
- (ii)
if is dense in X, then is a TB-sequence.
Proof. (i) follows from the definition of .
(ii) As is dense in X, then is precompact. By item (i), in ; hence, is a -sequence. ☐
Let
G be a discrete abelian group. For a sequence
, the group topology:
is the finest totally bounded group topology on
G, such that
is a null sequence in
.
Fact 3. [1, Lemma 3.1, Proposition 3.2] Let G be a discrete abelian group and .
The following conditions are equivalent:- (i)
is a -sequence;
- (ii)
is Hausdorff;
- (ii)
is dense in .
2.2. 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 the reader’s convenience.
Lemma 2. Let X be a topological abelian group and H a subgroup of X. Then:- (i)
is algebraically isomorphic to ;
- (ii)
is algebraically isomorphic to a subgroup of .
Proof. (i) Let be defined by , where is the canonical projection. Then, ψ is injective, and its image is .
(ii) Let be defined by . Then, , and so, we get the thesis. ☐
The following fact follows from the equivalence of items (a) and (e) of [
4, Exercise 3.8.25]. Since no proofs are given there, we offer a proof for the reader’s convenience.
Lemma 3. A compact abelian group K is separable if and only if .
Proof. The inequality holds for every separable regular topological space K.
Assume that . The discrete abelian group has size . Consider the embedding , where is the divisible hull of X. Then, and , for some countable divisible abelian groups and a set of indices I with . Therefore, is a surjective continuous homomorphism, and each is a metrizable compact abelian group. By the Hewitt–Marczewski–Pondiczery Theorem, since , we have that is separable; hence, K is separable, as well. ☐
Lemma 4. Let X be a precompact abelian group. Then, the singleton is if and only if there exists a continuous injection .
Proof. If there exists a continuous injection , then is in X, as it is in .
Assume now that , where each is an open subset of X, and we can assume that is in the prebase of the neighborhoods of 0 in X. Therefore, for every , there exist and an open neighborhood of 0 in containing no non-trivial subgroup of , such that . Then, . Hence, defined by is a continuous injective homomorphism. ☐
Theorem 2. Let X be a locally compact abelian group. Then, X is metrizable with if and only if there exists a continuous injective homomorphism .
Proof. If there exists a continuous injective homomorphism , then clearly X is metrizable and .
Suppose now that
X is metrizable and has cardinality at most
. It is well known (for example, see [
22]) that
, where
and
is a locally compact abelian group admitting an open compact (metrizable) subgroup
K. Clearly, there exist two continuous injective homomorphisms
and
. Therefore,
is an injective continuous homomorphism, too. Since
is divisible and
is open in
X,
extends continuously to
. Let
be the canonical projection. Since
is discrete, there exists a continuous injective homomorphism
. Let
.
Let now be defined by for every . Then, j is continuous, since φ and are continuous. Moreover, j is injective, as for some implies and ; therefore, , and so, since is injective, one has . ☐
3. General Permanence Properties of Characterized Subgroups
Let X be a topological abelian group, and denote by the family of all subgroups of X that are characterized.
We start by observing that:
The following are basic facts on characterized subgroups (see [
1,
13,
16,
23]); we give a proof for the reader’s convenience.
Lemma 5. Let X be a topological abelian group and .
Then:- (i)
for every subgroup J of X, , where for every ;
- (ii)
if is any permutation of ;
- (iii)
is stable under taking finite intersections;
- (iv)
is an -set (i.e., countable intersection of countable unions of closed subsets).
Proof. Items (i) and (ii) are obvious. To prove (iii), if , define , where and for every ; hence, . To prove (iv),note that , where each is a closed subset of X. ☐
Now, we prove that, under suitable hypotheses, the relation of being a characterized subgroup is transitive:
Proposition 1. Let X be a topological abelian group and , , subgroups of X with and such that is dually embedded in . If and , then .
Proof. Let
, such that
, and let
, such that
. As
is dually embedded in
,
extends to a character
of
for every
; so, let
. Define
by letting
and
for every
. Then, by Lemma 5(i),
so
, as required. ☐
Clearly, two compatible group topologies have the same characterized subgroups:
Lemma 6. If and are compatible group topologies on an abelian group X, then .
In particular, for a topological abelian group , since τ and its Bohr modification are compatible, .
4. The Γ-Radical
Definition 2. Let
X be a topological abelian group. For a subset Γ of
, define the
Γ-radical of
X by:
Clearly, is a closed subgroup of X.
The motivation for the choice of the term Γ-radical is the special case
, when
is usually called the
von Neumann radical of
X. Then,
(respectively,
) precisely when
separates the points of
X (respectively,
); in other words:
- (i)
X is MAP if and only if ;
- (ii)
X is MinAP if and only if .
Remark 1. Let
X be a topological abelian group and Γ a subset of
.
- (i)
If , then .
- (ii)
If Γ is countable, then is a characterized subgroup of X (indeed, for , such that each character in Γ appears infinitely many times in ).
For a given sequence
, the
support of
is the set of all characters appearing in
. We abbreviate the notation of the
-radical by writing:
and we call this subgroup the
-radical of
X.
Lemma 7. Let X be a topological abelian group and . Then:
- (i)
;
- (ii)
is dually closed;
- (iii)
is characterized;
- (iv)
is closed, and is in (so, is );
- (v)
.
Proof. (i) and (ii) are clear from the definitions, and (iii) follows from Remark 1(ii).
(iv) Let be defined by for every . Since is in , we conclude that is in . Moreover, , so is in X.
(v) Since is algebraically isomorphic to and , we conclude that . ☐
Remark 2. Let X be a topological abelian group and . Then, is closed and in every group topology on X that makes continuous for every . In particular, is closed and in every group topology on X compatible with the topology of X, so in the Bohr topology of X.
Lemma 7 gives a bound for the index of the characterized subgroups:
Corollary 1. Every characterized subgroup of a topological abelian group X has index at most .
Proof. Let . Since by Lemma 7(i); hence, by Lemma 7(v). ☐
The set
can be partitioned as
where:
- (i)
;
- (ii)
.
In other words, is the set of all characters appearing infinitely many times in , while each character in its complement appears finitely many times in . Clearly, is a finitely many-to-one sequence if and only if .
In case , let be the largest subsequence of with . Then, clearly, .
In case
is finite, the subsequence
of
is cofinite, so
. In other words, one can safely replace
by
. This is why from now on, we shall always assume that:
If
is infinite, we denote by
the subsequence of
such that
. If
, we have the partition
of
in the two subsequences
and
. Moreover, always
, and so, if
is infinite (equivalently,
by (
3)),
is a finitely many-to-one sequence and
, where
is a one-to-one subsequence of
such that
.
We see now how we can obtain the subgroup of X characterized by by considering separately the -radical of X and the subgroup of X characterized by .
Lemma 8. Let X be a topological abelian group and satisfying (
3).
- (i)
If , then , so and .
- (ii)
If is infinite and , then .
Proof. (i) Since , we have , and as observed above, .
(ii) Since and are subsequences of , it follows that . Let now . Since both and and since , we conclude that , that is . This concludes the proof. ☐
For
and
, let
Note that for every .
5. A Hierarchy for Characterized Subgroups
The following definition introduces three specific types of characterized subgroups.
Definition 3. Let
X be a topological abelian group. A subgroup
H of
X is:
- (i)
T-characterized if where is a non-trivial T-sequence;
- (ii)
K-characterized if for some finitely many-to-one sequence (i.e., );
- (iii)
N-characterized if for some .
In analogy to Definition 3(i), we introduce the following smaller class of characterized subgroups (see also Problem 1).
Definition 4. A subgroup H of a topological abelian group X is -characterized if , where is a non-trivial -sequence.
The N-characterized subgroups are clearly closed, and they are always characterized as noted above. Every -characterized subgroup is obviously T-characterized. Moreover, every T-characterized subgroup is also K-characterized. Indeed, let , where is a non-trivial T-sequence, and without loss of generality, assume that for all ; then, , that is contains no constant subsequences, and so, H is K-characterized.
Furthermore, proper dense characterized subgroups are
-characterized by Lemma 1(ii), so also
T-characterized and, in particular,
K-characterized, but they are not
N-characterized (as
N-characterized subgroups are necessarily closed). We shall see below that closed (even open) subgroups need not be
K-characterized in general. Denote by
(respectively,
,
,
) the family of all
K-characterized (respectively,
N-characterized,
T-characterized,
-characterized) subgroups of the topological abelian group
X. Then, we have the following strict inclusions:
We start giving some basic properties that can be proven immediately.
Corollary 2. Let X be a topological abelian group and ,
,
subgroups of X with and such that is dually embedded in .
- (i)
If and , then .
- (ii)
If and , then .
Proof. (i) It suffices to note that if in the proof of Proposition 1, is one-to-one and is one-to-one, then is finitely many-to-one.
(ii) It suffices to note that if in the proof of Proposition 1, and , then also . ☐
By Lemma 8, we have directly the following:
Corollary 3. Every characterized subgroup of a topological abelian group X is either N-characterized or it is the intersection of an N-characterized 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.
Lemma 9. Countable intersections of N-characterized subgroups are N-characterized.
The next correspondence theorem was proven in [
16] for characterized subgroups of compact abelian groups.
Proposition 2. Let X be a topological abelian group and F a closed subgroup of X, and let be the canonical projection. If H is a characterized (respectively, K-characterized, N-characterized, T-characterized) subgroup of , then is a characterized (respectively, K-characterized, N-characterized, T-characterized) subgroup of X.
Proof. Let , and consider . For every , let and .
(i) Assume that . Then, , as if and only if , and this occurs precisely when .
(ii) Assume now that H is K-characterized, that is assume that and that . By (i), , and moreover, .
(iii) If H is N-characterized, then assume that . By (i), , and moreover, , since for every precisely when .
(iv) If H is T-characterized, that is if and is a T-sequence, it remains to verify that is a T-sequence, as well, since by (i). Let τ be a Hausdorff group topology on , such that in . By Lemma 2(i), one can identify with the subgroup of by the algebraic monomorphism defined by . Let be the group topology on having as a local base at 0 the open neighborhoods of 0 in . Then, is a Hausdorff group topology on and in , as for every by definition. ☐
Lemma 10. Let X be a topological abelian group and H a subgroup of X, such that . Then, H is characterized (respectively, K-characterized, N-characterized, T-characterized) if and only if is characterized (respectively, K-characterized, N-characterized, T-characterized).
Proof. Let for some , and denote by the canonical projection. For every , since , the character factorizes as , where . Then, . Vice versa, if for some , let for every . Hence, . Moreover, is a finitely many-to-one sequence if and only if is a finitely many-to-one sequence, so H is K-characterized if and only if is K-characterized. Similarly, is finite, precisely when is finite; hence, H is N-characterized if and only if is N-characterized.
It remains to check that is a T-sequence precisely when is a T-sequence. This follows from the fact that the natural homomorphism sending (the members of) to (the members of) is an isomorphism, so certainly preserving the property of being a T-sequence.
The following lemma gives equivalent conditions for a subgroup to be characterized.
Lemma 11. Let X be a topological abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
;
- (ii)
there exists a closed subgroup F of X, such that and ;
- (iii)
there exists , such that for every closed , one has that , where , and each is the factorization of through the canonical projection .
Proof. (iii)⇒(ii) Take .
(ii)⇒(i) Since , one has , and one can conclude, by Proposition 2.
(i)⇒(iii) Let for some and . Let be the canonical projection. For every , let be the character of defined by . Then, is well defined, since . Hence, for every , and . Indeed, for every , we have , and hence, . Conversely, if , then . Hence, , and so, . ☐
6. Auto-Characterized Groups
The following consequence of [
16, Proposition 2.5] motivates the introduction of the notion of auto-characterized group (see Definition 5).
Proposition 3. Let X be a compact abelian group. Then, for some if and only if the sequence is trivially null.
Proof. It is clear from (
2) that
if
is trivially null. Assume now that
for some
. By [
16, Proposition 2.5], being
compact, there exists
, such that
, and so,
for all
.
If one drops the compactness, then the conclusion of Proposition 3 may fail, as shown in the next example.
Example 1.- (i)
Let N be an infinite countable subgroup of . As mentioned in the Introduction, N is characterized in ; hence, for a non-trivial sequence . If , then is non-trivial (since N is dense in ), and .
- (ii)
Let , let be the canonical projection and let , such that and for every and . Obviously, , even though is non-trivial.
- (iii)
Let , where p is a prime. For every , let . Obviously, , even though is non-trivial.
Motivated by Proposition 3 and Example 1, we give the following:
Definition 5. A topological abelian group X is auto-characterized if for some non-trivial .
Items (ii) and (iii) of Example 1 show that and are auto-characterized.
Remark 3. - (i)
Let X be an auto-characterized group, so let be non-trivial and such that . Then, there exists a one-to-one subsequence of , such that for every and .
Indeed, if , then , and so, ; therefore, is either empty or . Since is non-trivial, is infinite; hence, by Lemma 8(ii). Let be the one-to-one subsequence of , such that ; therefore, .
- (ii)
The above item shows that auto-characterized groups are
K-characterized subgroups of themselves. However, one can prove actually that they are
T-characterized subgroups of themselves (indeed,
-characterized subgroups of themselves; see [
24]).
6.1. Basic Properties of Auto-Characterized Groups
We start by a direct consequence of Lemma 10:
Lemma 12. Let X be a topological abelian group and H a subgroup of X, such that . Then, H is auto-characterized if and only if is auto-characterized.
The next proposition, describing an auto-characterized group in terms of the null sequences of its dual, follows from the definitions and Lemma 1:
Proposition 4. A topological abelian group X is auto-characterized if and only if has a non-trivial null sequence (in such a case, ).
In the next lemma, we see when a subgroup of an auto-characterized group is auto-characterized, and vice versa.
Lemma 13. Let X be a topological abelian group and H a subgroup of X.
- (i)
If X is auto-characterized and H is dense in X, then H is auto-characterized.
- (ii)
If H is auto-characterized and one of the following conditions holds, then X is auto-characterized:- (a)
H is a topological direct summand of X;
- (b)
H is open and has a finite index.
Proof. (i) Let for with for every , and let . Then, each is non-zero and .
(ii) Let for some with for every .
(a) Let . For every , let be the unique character of X that extends and such that vanishes on Z. Then, for every , and .
(b) Arguing by induction, we can assume without loss of generality that is prime. Let with and . If , 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 , and let for every . If for infinitely many n, extend to for those n by letting . Then, obviously, the sequence obtained in this way is not trivially null and , so X is auto-characterized. Assume now that for infinitely many ; for those n, pick an element with , and extend by letting . Let . Then, , so is not trivially null. Moreover, as . ☐
Lemma 14. Let X be a topological abelian group and . If F is a subgroup of X, such that and F is not auto-characterized, then for some .
Proof. Let for every and . Then, , so the sequence must be trivially null. Let , such that for every . Therefore, . ☐
The following consequence of Lemma 14 is a generalization of Lemma 2.6 in [
16], where the group
X is compact.
Corollary 4. Let X be a topological abelian group, F and H subgroups of X, such that F is compact and . Then, if and only if .
Proof. Denote by the canonical projection. If is a characterized subgroup of , then is a characterized subgroup of X by Proposition 2. Assume now that for some . Since F is compact, Proposition 3 implies that F is not auto-characterized. By Lemma 14, F is contained in for a sufficiently large . Let be the canonical projection and , then . Therefore, one deduces from Lemma 11 that is a characterized subgroup of . Hence, by Proposition 2, . ☐
The argument of the above proof fails in case F is not compact. For example, take , where N is as in Example 1; then, one cannot conclude that is trivially null and, hence, that F is contained in .
6.2. Criteria Describing Auto-Characterized Groups
Here, we give two criteria for a group to be auto-characterized. We start below with a criterion 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 3 that no compact abelian group is auto-characterized; now, we prove in Theorem 3 that this property describes the compact abelian groups within the larger class of all locally compact abelian groups. This follows easily from Lemma 13(ii) for the locally compact abelian groups that contain a copy of , while the general case requires the following deeper argument.
Theorem 3. If X is a locally compact abelian group, then X is auto-characterized if and only if X is not compact.
Proof. If X is auto-characterized, then X is not compact according to Proposition 3. Assume now that X is not auto-characterized. Then, by Fact 2 and Proposition 4, the dual has no non-trivial null sequences in its Bohr topology. However, since is locally compact, it has the same null sequences as its Bohr modification . Therefore, 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 result 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 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 is discrete. It is a well known fact that this implies the compactness of X. ☐
Remark 4. An alternative argument to prove that non-discrete locally compact abelian groups have non-trivial null sequences is based on a theorem by Hagler, Gerlits and Efimov (proven independently also by Efimov in [
25]). It states that every infinite compact group
K contains a copy of the Cantor cube
, which obviously has plenty of non-trivial null sequences.
In order to obtain a general criterion describing auto-characterized groups we need another relevant notion in the theory of characterized subgroups:
Definition 6. [
1] Let
X be a topological abelian group and
H a subgroup of
X. Let:
A subgroup
H of
X is said to be:
- (i)
g-dense if ;
- (ii)
g-closed if .
We write simply when there is no possibility of confusion. Clearly, is a subgroup of X containing H. Moreover, is the intersection of all characterized subgroups of X and .
Remark 5 Let
be a topological abelian group and
H a subgroup of
X.
- (i)
If
is another topological abelian group and
a continuous homomorphism, then
(see [
1, Proposition 2.6]).
- (ii)
Moreover,
. Indeed,
and
for
with
(
i.e.,
is the constant sequence given by
χ). Item (i) says, in terms of [
15,
26], that g is a closure operator in the category of topological abelian groups. The inclusion
says that g is finer than the closure operator defined by
.
- (iii)
If H is dually closed, then H is g-closed by item (ii).
- (iv)
If is a locally compact abelian group, then every closed subgroup of is dually closed, and so, (ii) implies that for every subgroup H of X. Therefore, g-dense subgroups are also dense in this case.
- (v)
The inclusion may fail if the group H is not MAP (e.g., if X is MinAP, then for every H, while X may have proper closed subgroups).
The next result shows that the auto-characterized precompact abelian groups are exactly the dense non-g-dense subgroups of the compact abelian groups.
Theorem 4. Let X be a precompact abelian group. The following conditions are equivalent:- (i)
X is auto-characterized;
- (ii)
X is not g-dense in its completion .
Proof. (ii)⇒(i) Assume that X is not g-dense in . Then, there exists a sequence , such that . By Proposition 3 (see also Remark 3), we may assume without loss of generality that for every . Let for every . Since X is dense in K, clearly for every . Moreover, ; hence, X is auto-characterized.
(i)⇒(ii) Suppose that X is auto-characterized, say for , such that for every . For every , let be the extension of to K. Then, by Proposition 3, so X is not g-dense in K. ☐
If X is a MAP abelian group, then is precompact, and the Bohr compactification of X is , where is the completion of and is an injective homomorphism. If X is not MAP, then . Consider the quotient , which is a MAP group. Then, take the Bohr compactification . The Bohr compactification of X is , where and , where is the canonical projection.
Corollary 5. Let X be a MAP abelian group. The following conditions are equivalent:
- (i)
X is auto-characterized;
- (ii)
X is not g-dense in .
Proof. Since X is MAP, X embeds in . By Lemma 6, X and have the same characterized subgroups. Moreover, is precompact, and by definition, is the completion of . Then, it suffices to apply Theorem 4. ☐
Theorem 5. Let X be a topological abelian group. The following conditions are equivalent:- (i)
X is auto-characterized;
- (ii)
is not g-dense in .
Proof. Since X is auto-characterized precisely when is auto-characterized by Lemma 12, apply Corollary 5 to conclude. ☐
7. K-Characterized Subgroups
We start by recalling [
23, Lemma 3.19]: if
X is a compact abelian group and
is a one-to-one sequence, then
has Haar measure zero in
X. Since
K-characterized subgroups are characterized by finitely many-to-one sequences (which obviously contain a one-to-one subsequence), this result applies to
K-characterized subgroups and gives the following (formally weaker) result, which will be necessary and more convenient to apply in the current paper:
Lemma 15. If X is a compact abelian group and , then H has Haar measure zero (hence, is infinite). In particular, no open subgroup of X is K-characterized.
Lemma 15 cannot be inverted; take, for example, the constant sequence in .
Remark 6. If X is a connected compact abelian group, then the conclusion of Lemma 15 holds for all non-trivial sequences in , since every measurable proper subgroup H of X has measure zero (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 2. Here, we provide examples of non-auto-characterized 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 not auto-characterized by using Theorem 4, we have to check that X is g-dense in K. Indeed, every measurable proper subgroup of K has measure zero as noted in Remark 6; therefore, every proper characterized (hence, every non-g-dense) subgroup of K has measure zero. 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 [
27] (under the assumption of Martin Axiom) and in [
19] (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).
7.1. 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, by applying Theorem 6.
Theorem 6. Let X be a topological abelian group and H a closed subgroup of X of infinite index. The following conditions are equivalent:- (i)
;
- (ii)
;
- (iii)
is MAP, and is separable.
Proof. (i)⇒(ii) is obvious.
(ii)⇒(iii) Since H is N-characterized, then H is dually closed by Lemma 7(ii), that is is MAP. Let for some , and let be the canonical projection. Since for every , one can factorize through π, i.e., write for appropriate . It remains to verify that is dense in . To this end, let ; if , then , and so, , that is, .
(iii)⇒(i) Let , equipped with the quotient topology. By hypotheses, Y is infinite and MAP, while is an infinite topological abelian group with a countably infinite dense subgroup D. According to Proposition 2 applied to the canonical projection , it suffices to prove that is a K-characterized subgroup of Y. Let be a one-to-one enumeration of D and . To prove that , we have to show that for every non-zero , there exists a neighborhood U of 0 in , such that for infinitely many . Actually, we show that works for all non-zero . In fact, for every , one has that is a non-trivial subgroup of , as Y is MAP and .
Let . If is infinite, then is dense in ; so, is infinite, and we are done. Now, consider the case when is finite. As and U contains no non-trivial subgroups of , there exists , such that . Then, the map defined by is a homomorphism with finite. Therefore, is a finite-index subgroup of D, so K is infinite. Let for some . Then, is infinite, as well. This means that for infinitely many n (namely, those n for which ). Therefore, , and so, .
Finally, let us note that the above argument shows also that H is N-characterized, as obviously . ☐
The following is an obvious consequence of Theorem 6.
Corollary 6. Let X be a topological abelian group and H a closed subgroup of X of infinite index. If , then .
Next, we rewrite Theorem 6 in the case of locally compact abelian groups.
Corollary 7. Let X be a locally compact abelian group and H a subgroup of X. Then, if and only if H is closed and is separable.
Proof. As both conditions imply that H is closed, we assume without loss of generality that H is closed. Since and are locally compact abelian groups, is MAP, and the Bohr topology on coincides with by Fact 2, so the separability of is equivalent to the separability of by Fact 1. If H has a infinite index in X, apply Theorem 6 to conclude. If H has a 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 is finite, so separable. ☐
As a consequence of Theorem 6, we find a sufficient condition for an open subgroup of infinite index to be K-characterized:
Theorem 7. Let X be a topological abelian group and H an open subgroup of X of infinite index. Then, the following conditions are equivalent:- (i)
;
- (ii)
;
- (iii)
;
- (iv)
is separable.
Proof. (ii)⇒(i) is clear, and (i)⇒(iii) is Corollary 1.
(iii)⇒(iv) Since is a compact abelian group of weight at most , it is separable by Lemma 3.
(iv)⇒(ii) As is infinite, we can apply Theorem 6 to conclude that H is K-characterized. ☐
The following is another direct consequence of Theorem 6.
Corollary 8. If X is a metrizable compact abelian group, then every closed non-open subgroup of X is K-characterized.
7.2. When Closed Subgroups of Finite Index are K-Characterized
We start by giving the following useful technical lemma.
Lemma 16. Let X be a topological abelian group and H an open subgroup of X, such that for some . If H is auto-characterized, then .
Proof. Let , such that . By Remark 3, we can assume that is one-to-one and that for every .
Assume first that . If is infinite, then fix an irrational number , and for every , let and for every . If is finite, then for every , let and for every . In both cases, it is straightforward to prove that . Moreover, since is one-to-one, then also is one-to-one.
Suppose now that for some , with . As , one has .
For every , let . Since , there exists , such that for every . As , we shall assume for simplicity that every .
Claim 1. For every
with
, there exists
, such that
Proof. We tackle the problem in
, that is identifying
with
. First, assume that
, and let:
Then,
and
. Let now
with
, then:
Therefore,
. This establishes condition (
5) in the current case.
It remains to consider the case
. Let
,
i.e.,
in
. Then, obviously,
and
. Hence, by the above case applied to
, there exists
satisfying condition (
5) with
in place of
a (
i.e.,
). Let
. Then, condition (
5) holds true for
b and
a, as
for every
with
. ☐
For every
, apply Claim 1 to
to get
as in (
5), then define
by letting
for every
and
for every
. As
, this definition is correct. Moreover, since
H is open in
X,
. Since
is one-to-one, then
is one-to-one, too.
We show that
In fact, if
for some
,
The other way around, assume that
, where
and
. Then,
since
and
by (
5).
We deduce finally that
. Indeed,
, so it remains to prove that
. To this end, let
, that is
for some
and
with
. Then,
Since
, that is,
, while
by condition (
7), we conclude that
, that is,
. Hence,
.
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.
Theorem 8. Let X be a topological abelian group and H a proper open subgroup of X of finite index. Then, if and only if H is auto-characterized.
Proof. Assume that . We can write for one-to-one. Let for every . Then, the map is finitely many-to-one, as is finite. Therefore, is finitely many-to-one. Obviously, , so H is auto-characterized.
Now, assume that
H is auto-characterized. Since
H has finite index in
X, there exist
, such that
and that, letting
for
and
, the subgroup
is a proper subgroup of
for
. We shall prove by induction on
, that
As
, this will give
, as desired.
Before starting the induction, we note that according to Lemma 13(ii), all subgroups
, for
, are auto-characterized, as each
is open in
. For
, the assertion in condition (
8) follows from Lemma 16. Assume that
and condition (
8) holds true for
,
i.e.,
. Since
is open in
, again Lemma 16 applied to
gives that
. As
by our inductive hypothesis, we conclude with Corollary 2(i) that
. ☐
7.3. Further Results on K-Characterized Subgroups
The next corollary resolves an open question from [
28]:
Corollary 9. Let X be an infinite discrete abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
;
- (ii)
;
- (iii)
.
Proof. (ii)⇒(i) is clear, and (i)⇒(iii) is Corollary 1.
(iii)⇒(ii) If is infinite, then by Theorem 7. Therefore, assume that is finite, then H is infinite, and hence, H is auto-characterized by Theorem 3; therefore, by Theorem 8. ☐
We give now sufficient conditions for a non-closed characterized subgroup to be K-characterized.
Theorem 9. Let X be a topological abelian group and a non-closed subgroup of X, such that:- (i)
is MAP, and is separable;
- (ii)
if , then is auto-characterized.
Then,
.
Proof. As is dense in and obviously , we deduce that , as dense characterized subgroups are -characterized by Lemma 1(ii). If , we are done. Therefore, assume that is proper.
Our aim now is to apply Corollary 2, so we need to check that . If has finite index in X, then is auto-characterized by hypothesis, and so, Theorem 8 yields . If has infinite index in X, then by Theorem 6.
Corollary 10. Let X be a divisible topological abelian group and a non-closed subgroup of X, such that is dually closed. If is separable, then . In particular, whenever is separable.
Proof. The first part of our hypothesis entails that is MAP. Moreover, divisible topological abelian groups have no proper closed subgroup of finite index. Therefore, the first assertion follows directly from Theorem 9.
The topology of the dual is coarser than the compact-open topology of , so that the separability of yields the separability of . 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 11. 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
. Our next aim will be to check that
is separable. Indeed, if
for
, then one has the chain of subgroups
. If
, then
is trivially separable. Otherwise,
has infinite index in
X, since
is divisible, so
is separable by Corollary 7; since
contains
, the quotient group
is a quotient group of
. Therefore,
is isomorphic to a subgroup of the separable group
, so
is separable, as well (see [
29]). Furthermore,
is dually closed (so
is MAP) by Fact 1.
If H is not closed, Corollary 10 gives that . If H is a proper closed subgroup of X, then H has infinite index, as is divisible, so by Theorem 6. This proves the inclusion , which, along with the obvious inclusion , proves the equality .
It remains to consider the (closed) subgroup , which obviously belongs to . If X is compact, then , by Lemma 15, so . If X is not compact, then by Theorem 3 and Remark 3(ii). Hence, in this case. ☐
In particular, the above corollary yields and .
Remark 7. As we shall see in Corollary 18, connectedness is necessary in this corollary.
8. N-Characterized Subgroups
The following consequence of Lemma 14 gives a sufficient condition for a characterized subgroup to be N-characterized:
Corollary 12. Let X be a topological abelian group and H a subgroup of X, which is not auto-characterized. If , then .
Proof. Let for some . Then, for some by Lemma 14. Since , we deduce that .
Here comes an easy criterion establishing when a subgroup is N-characterized.
Theorem 10. Let X be a topological abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
H is closed, and there exists a continuous injection ;
- (ii)
;
- (iii)
H is closed, and is in .
Proof. (i)⇒(ii) Suppose that there exists a continuous injection
. Let
be the canonical projection. For every
, let
be
n-th projection, and let
.
Therefore,
for every
and
, where
.
(ii)⇒(i) Let for . Let be the canonical projection, and define by for every . Since , then j is well defined and injective. Moreover, j is continuous.
Finally, (i) and (iii) are obviously equivalent. ☐
The above criterion simplifies in the case of open subgroups:
Corollary 13. Let X be a topological abelian group and H an open subgroup of X. The following conditions are equivalent:- (i)
;
- (ii)
;
- (iii)
.
Proof. (ii)⇒(i) is obvious, and (i)⇒(iii) is Corollary 1.
(iii)⇒(ii) Since H is open, is discrete. By hypothesis , so there exists a continuous injection . Hence, by Theorem 10.
The next is another consequence of Theorem 10.
Corollary 14. Let X be a metrizable precompact abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
H is closed;
- (ii)
H is closed and ;
- (iii)
.
Proof. (iii)⇒(ii) and (ii)⇒(i) are clear.
(i)⇒(iii) Since X is metrizable, is metrizable, as well, and so, is in . By Lemma 4, there exists a continuous injective homomorphism . Therefore, H is N-characterized by Theorem 10. ☐
According to Theorem 6, one can add to the equivalent conditions in Corollaries 13 and 14 also “”, in case .
In the sequel, we consider the case of locally compact abelian groups. The following theorem was proven in Theorem B of [
16] for compact abelian groups.
Theorem 11. Let H be a locally compact abelian group and H a subgroup of X. Then, if and only if H contains a closed -subgroup K of X, such that , where is a metrizable locally compact abelian group.
Proof. The equivalence follows from Lemma 11, since the subgroup is closed and for every , by Lemma 7(iv). Since K is closed and , is a metrizable locally compact abelian group. ☐
By Lemma 7, is always closed and characterized. Theorem 12 describes the closed characterized subgroups of the locally compact abelian groups X by showing that these are precisely the N-characterized subgroups of X.
Theorem 12. Let X be a locally compact abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
H is closed and ;
- (ii)
;
- (iii)
H is closed and in the Bohr topology;
- (iv)
H is closed, and .
- (v)
H is closed, and is separable.
Proof. (iii)⇒(ii) follows from Theorem 10, (ii)⇒(i) by Lemma 7, and (ii)⇔(v) is Corollary 7.
(iv)⇒(iii) The group is locally compact, metrizable and has cardinality at most ; therefore, by Theorem 2, there exists a continuous injective homomorphism . Then, Theorem 10 gives the thesis.
(i)⇒(iv) Let for , and let be the canonical projection. By Corollary 1, . By Lemma 7, , and is closed and . Then, is a metrizable locally compact abelian group, and by hypothesis, is closed. Therefore, is closed and, hence, in . Therefore, is closed and in X. ☐
We see now the following result from [
16] as a consequence of Theorem 12.
Corollary 15. [16 Let X be a compact abelian group and H a closed subgroup of X. Then, if and only if H is .
Proof. If H is characterized, then H is by Theorem 12. Vice versa, assume that H is . Then, is compact and metrizable, hence . Therefore, again Theorem 12 implies that H is characterized. ☐
In the following theorem, we use that the
-subgroups of a locally compact abelian group are always closed (see [
28] (Theorem A.2.14)).
Theorem 13. Let X be a compact abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
and H is closed;
- (ii)
H is and non-open;
- (iii)
and H is non-open;
- (iv)
and H is closed and non-open.
Proof. (i)⇒(iv) Since Lemma 15 implies that H is non-open, (iv)⇔(iii) by Theorem 12, and (iv)⇔(ii) by Corollary 15.
(ii)⇒(i) Since is a metrizable compact non-discrete (hence infinite) abelian group, is closed and non-open in ; hence, is K-characterized in by Corollary 8. Therefore, H is K-characterized by Proposition 2. ☐
This theorem generalizes Corollary 15 as it implies that, for a closed non-open subgroup
H of a compact abelian group
X, one has:
The following immediate consequence of Corollaries 13 and 9 shows that for a discrete abelian group all characterized subgroups are N-characterized.
Corollary 16. Let X be an infinite discrete abelian group and H a subgroup of X. The following conditions are equivalent:- (i)
;
- (ii)
;
- (iii)
;
- (iv)
.
9. T-Characterized Closed Subgroups of Compact Abelian Groups
In [
30, Theorem 4], Gabriyelyan observed that if
is a
T-sequence of an infinite countable abelian group
G, then (see formula (
1) for the definition of
)
where
denotes the abelian group
G endowed with the discrete topology. Therefore, the following fact is an immediate corollary of this result.
Fact 4. Let be a T-sequence of an infinite countable abelian group G. Then:- (i)
is MAP if and only if is a -sequence;
- (ii)
is MinAP if and only if and .
Recall that a topological abelian group X is almost maximally almost periodic (AMAP) if is finite.
Remark 8. In relation to Fact 4, Lukács in [
31] found a
T-sequence in
that is not a
-sequence, providing in this way an example of a non-trivial AMAP group. More precisely, he found a characterizing sequence
for
for a fixed
,
i.e.,
. In this way, being
finite, then
Therefore,
is AMAP. Further results in this direction were obtained by Nguyen in [
32]. Finally, Gabriyelyan in [
33] 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.
Theorem 14. [
11]
Let X be a compact abelian group and H a closed subgroup of X. Then, if and only if H is and carries a MinAP topology. Following [
4, §4], for a topological abelian group
X and a prime number
p, we denote by
the closure of the subgroup
. In case
X is compact, one can prove that
In particular,
contains the connected component
of
X. More precisely, if
is the topologically primary decomposition of the totally disconnected compact abelian group
, then:
Following [
34], we say that
is a
proper divisor of
, provided that
and
for some
. Note that, according to our definition, each
is a proper divisor of zero.
Definition 7. Let
G be an abelian group.
- (i)
For the group G is said to be of exponentn (denoted by ) if , but for every proper divisor d of n. We say that G is bounded if and, otherwise, that G is unbounded.
- (ii)
[
35] If
G is bounded, the
essential order of
G is the smallest
, such that
is finite. If
G is unbounded, we define
.
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 15, a closed subgroup H of a compact abelian group X is characterized if and only if H is (i.e., is metrizable). This explains the blanket condition imposed on H to be a -subgroup of X.
Theorem 15. For a compact abelian group X and a -subgroup H of X, the following conditions are equivalent:- (i)
;
- (ii)
does not admit a MinAP group topology;
- (iii)
there exists , such that is finite and non-trivial;
- (iv)
;
- (v)
there exists a finite set P of primes, so that:- (a)
for all ,
- (b)
for every there exist with ,
- (c)
there exists , such that and has finite index in ;
- (vi)
there exists a finite set P of primes, so that , where each is a compact p-group, and there exist some and , such that is finite and non-trivial.
Proof. (i)⇔(ii) is Theorem 14, and (iii)⇔(iv) is clear from the definition.
(ii)⇔(iii) The main theorem in [
36] states that an abelian group
G does not admit a MinAP group topology precisely when there exists
, such that
is finite and non-trivial. In our case,
is topologically isomorphic to
, so
G does not admit a MinAP group topology if and only if
is finite and non-trivial, and this occurs precisely when
is finite and non-trivial.
(vi)⇒(v) Write , where each is a compact p-group. Let for every , and let be finite and non-trivial for and .
Obviously, all
are coprime to
. As
, we deduce from the equality (9) that
This proves (a). From (
10), we deduce that that
.
The quotient groups
and
are totally disconnected; hence,
and
. Here,
Furthermore,
. From (
10), we deduce that
for all
. Therefore,
and
. Hence,
, and consequently,
for all
. Thus,
for all
. Equivalently,
for
. This proves (b).
As
is finite and non-trivial, we deduce that
. Therefore,
is still finite and non-trivial. Hence,
, and so,
To prove the second assertion in (c), note that the finiteness of
yields that:
is finite. Hence, from (
11), we deduce that
has finite index in
. Therefore,
has finite index in
.
(v)⇒(iii) Let be the product of all when p runs over P, and let . Then, an argument similar to the above argument shows that is finite. ☐
Corollary 17. Let X be a compact abelian group and H a closed subgroup of X that does not contain the connected component of X. Then, if and only if .
Proof. Clearly, H T-characterized implies H characterized. Therefore, assume that H is characterized. Then, H is in X by Corollary 15. Let be the canonical projection. Then, is a non-trivial connected subgroup of ; hence, is unbounded, as its connected component is a non-trivial divisible subgroup. According to Theorem 15, H is T-characterized. ☐
By Corollary 17, for a connected compact abelian group
X and
H a closed subgroup of
X,
This result was obtained in [
11]; actually, the following more precise form holds (the equivalence (i)⇔(ii) is proven in [
11]):
Corollary 18. For a compact abelian group X, the following conditions are equivalent:- (i)
X is connected;
- (ii)
for every closed subgroup H of G;
- (iii)
for every closed subgroup H of G.
Proof. (i)⇒(ii) by (
12), and (ii)⇒(iii) is obvious.
(iii)⇒(i) Assume that
X is not connected. Then,
X has a proper open subgroup
H, as the connected component of
X is an intersection of clopen subgroups (see [
22]). Then,
, but
by Lemma 15. ☐
Obviously, each one of the above equivalent conditions implies that
for every closed subgroup
H of a compact abelian group
X. To see that in general the latter property is strictly weaker than
, consider the group
. Then, for the closed subgroup
of
X, one has
by Theorem 13, while
by Theorem 15, as
is finite and non-trivial (moreover,
does not admit any MinAP group topology, as noticed by Remus; see [
37]).
10. Final Comments and Open Questions
In this section, we collect various open questions arising throughout the paper.
For a topological abelian group
X and
, we defined for each
the closed subgroup
in (
4). Since these subgroups are contained one in each other, the increasing union
is an
-subgroup of
X, and it is contained in
. We do not know whether
is characterized or not:
Question 1. For a topological abelian group X and , is a characterized subgroup of X?
This question is motivated by [
16, Theorem 1.11], where it is proven that under some additional restraint, the union of a countably infinite increasing chain of closed characterized subgroups of a metrizable compact abelian group is still characterized. On the other hand, it is known that every characterized subgroup is
(see Lemma 5(iv)) and that the characterized subgroup need not be
.
In analogy to T-characterized subgroups, we have introduced here the notion of the -characterized subgroup (see Definition 4). In relation to what is already known for T-characterized subgroups, one could consider the following general problem.
Problem 1. Study the -characterized subgroups of topological abelian groups.
Next comes a more precise question on the properties of T- and -characterized subgroups. In fact, we do not know whether the counterpart of Corollary 2 is true for T- and -characterized subgroups:
Question 2. Let X be a topological abelian group and , , subgroups of X with , such that is dually embedded in .
- (i)
If and , is then ?
- (ii)
If and , is then ?
A full description of open K-characterized subgroups is given in Theorems 7 and 8, while Theorem 6 describes the closed K-characterized subgroups of infinite index that are also N-characterized. Moreover, N-characterized closed subgroups of infinite index are K-characterized. This leaves open the following general problem and question.
Problem 2. For a topological abelian group X, describe .
Question 3. Let X be a topological abelian group. Can one add “” as an equivalent condition in Theorem 6? Equivalently, does there exist a closed subgroup H of X, such that ?
In Theorem 8, we have seen in particular that a proper open finite-index subgroup H of a topological abelian group X is auto-characterized precisely when . We do not know whether also the stronger condition is equivalent:
Question 4. Let H be a topological abelian group and H an open subgroup of X of finite index. Does whenever H is auto-characterized? What about the case when H is a topological direct summand of X?
By looking at Theorem 10 and Corollary 14, the following natural question arises:
Question 5. Are the closed -subgroups of a precompact abelian groups always N-characterized?
This amounts to asking whether there exists a continuous injection from into for every closed subgroup F of a precompact abelian group X; in other words, we are asking for a generalization of Lemma 4.