Let {Gi:i∈N} be a family of finite Abelian groups. We say that a subgroup G≤∏i∈NGi is order controllable if for every i∈N, there is ni∈N such that for each c∈G, there exists c1∈G satisfying c1|[1,i]=c|[1,i], supp(c1)⊆[1,ni], and order (c1) divides order (c|[1,ni]). In this paper, we investigate the structure of order-controllable group codes. It is proved that if G is an order controllable, shift invariant, group code over a finite abelian group H, then G possesses a finite canonical generating set. Furthermore, our construction also yields that G is algebraically conjugate to a full group shift.

We further investigate the weak topology generated by the irreducible unitary representations of a group G. A deep result due to Ernest [13] and Hughes [22] asserts that every weakly compact subset of a locally compact (LC) group G is compact in the LC-topology, generalizing thereby a previous result of Glicksberg [19] for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countable compactness, pseudocompactness and functional boundedness.

It is a Theorem of W. W. Comfort and K. A. Ross that if G is a subgroup of a compact Abelian group, and S denotes those continuous homomorphisms from G to the one-dimensional torus, then the topology on G is the initial topology given by S. Assume that H is a subgroup of G. We study how the choice of S affects the topological placement and properties of H in G. Among other results, we have made significant progress toward the solution of the following specific questions: How many totally bounded group topologies does G admit such thatH is a closed (dense) subgroup? If CS denotes the poset of all subgroups of G that are S-closed, ordered by inclusion, does CS has a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an SC-group (topologically simple, resp.) if all its subgroups are closed (if G and {e} are its only possible closed normal subgroups, resp.) In addition, we investigate the following questions. How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group G admit?

Let G be a monoid that acts on a topological space X by homeomorphisms such that there is a point x0∈X with GU=X for each neighbourhood U of x0. A subset A of X is said to be G-bounded if for each neighbourhood U of x0 there is a finite subset F of G with A⊆FU. We prove that for a metrizable and separable G-space X, the bounded subsets of X are completely determined by the bounded subsets of any dense subspace. We also obtain sufficient conditions for a G-space X to be locally G-bounded, which apply to topological groups. Thereby, we extend some previous results accomplished for locally convex spaces and topological groups.

We provide some characterizations of precompact abelian groups G whose dual group $$G_p^\wedge $$ endowed with the pointwise convergence topology on elements of G contains a nontrivial convergent sequence. In the special case of precompact abelian torsion groups G, we characterize the existence of a nontrivial convergent sequence in $$G_p^\wedge $$ by the following property of G: No infinite Hausdorff quotient group of G is countable. Also, we present an example of a dense subgroup G of the compact metrizable group $${\mathbb {Z}}(2)^\omega $$ such that G is of the first category in itself, has measure zero, but the dual group $$G_p^\wedge $$ does not contain infinite compact subsets. This complements a result of J.E. Hart and K. Kunen (2005) on convergent sequences in dual groups. Making use of the group G we construct the first example of a precompact Pontryagin reflexive abelian group which is of the first Baire category.

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups C(X,R) of continuous real-valued functions on complete metric spaces X, with respect to the compact-open topology.

Let C be an epireflective subcategory of Top and let rC be the epireflective functor associated with C. If A denotes a (semi)topological algebraic subcategory of Top, we study when rC(A) is an epireflective subcategory of A. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. We also study when the epireflective functor preserves products, subspaces and other properties. In particular, we solve an open question about the coincidence of epireflections proposed by Echi and Lazaar in [5, Question 1.6] and repeated in [6, Question 1.9]. Finally, we apply our results in different specific topological algebraic structures.

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a Hausdorff precompact abelian group (G,w) is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of the topology w. As an application, an example of a MAP group (G,t) is given such that every closed, metrizable subgroup N of bG with N∩G={0} preserves compactness but (G,t) does not strongly respect compactness. Thereby, we respond to Questions 4.1 and 4.3 in [6].

The important rôle that W. W. Comfort played in the study of the Bohr topology is described.

Let G be a discrete group and let A and B be two subgroups of G-valued continuous functions defined on two 0-dimensional compact spaces X and Y. A group isomorphism H defined between A and B is called separating when, for each pair of maps f, g∈A satisfying that f-1eG∪g-1eG=X, it holds that Hf-1eG∪Hg-1eG=Y. We prove that under some mild conditions every biseparating isomorphism H:A→B can be represented by means of a continuous function h:Y→X as a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.

A subgroup G of a product ∏ _i∈ℕG_i is rectangular if there are subgroups H_i of G_i such that G=∏ _i∈ℕH_i . We say that G is weakly rectangular if there are finite subsets F_i⊆ℕ and subgroups H_i of ⊕ _j∈ F_i G_j that satisfy G=∏ _i∈ℕH_i . In this paper we discuss when a closed subgroup of a product is weakly rectangular. Some possible applications to the theory of group codes are also highlighted.

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of ℓ1. For that purpose, we transfer to general locally compact groups the notion of interpolation (I0) set, which was defined by Hartman and Ryll-Nardzewsky [24] for locally compact abelian groups. Thus we prove that for every sequence {gn}n<ω in a locally compact group G, then either {gn}n<ω has a weak Cauchy subsequence or contains a subsequence that is an I0 set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group G, an old question that remains open since 1974 (see [31] and [19]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.

A subgroup G of a product ∏_i∈ℕG_i is rectangular if there are subgroups H_i of G_i such that G=∏_i∈ℕH_i. We say that G is weakly rectangular if there are finite subsets F_i⊆ℕ and subgroups H_i of ⊕_j∈ F_i G_j that satisfy G=∏_i∈ℕH_i. product of the family {G_i}_i∈ I if G is weakly rectangular and this paper we discuss when a closed subgroup of a product is weakly rectangular. Some possible applications to the theory of group codes are also highlighted.

Let X and K be a Čech-complete topological group and a compact group, respectively. We prove that if G is a non-equicontinuous subset of CHom(X,K), the set of all continuous homomorphisms of X into K, then there is a countably infinite subset L⊆G such that L‾KX is canonically homeomorphic to βω, the Stone–Čech compactifcation of the natural numbers. As a consequence, if G is an infinite subset of CHom(X,K) such that for every countable subset L⊆G and compact separable subset Y⊆X it holds that either L‾KY has countable tightness or |L‾KY|≤c, then G is equicontinuous. Given a topological group G, denote by G+ the (algebraic) group G equipped with the Bohr topology. It is said that G respects a topological property P when G and G+ have the same subsets satisfying P. As an application of our main result, we prove that if G is an abelian, locally quasiconvex, locally kω group, then the following holds: (i) G respects any compact-like property P stronger than or equal to functional boundedness; (ii) G strongly respects compactness.

Combining ideas of Troallic [1] and Cascales, Namioka, and Vera [2], we prove several characterizations of almost equicontinuity and hereditarily almost equicontinuity for subsets of metric-valued continuous functions when they are defined on a Čech-complete space. We also obtain some applications of these results to topological groups and dynamical systems.

Let I be an infinite set, let {G(i) : i is an element of I} be a family of (topological) groups and let G = Pi(i is an element of J) Gi be its direct product. For J subset of I, p(J) : G Pi(i is an element of J) Gj denotes the projection. We say that a subgroup H of G is(i) uniformly controllable in G provided that for every finite set J subset of I there exists a finite set K subset of I such that pJ(H) = p(J)(H boolean AND circle plus (i is an element of K) G(i)),(ii) controllable in G provided that pJ(H) = p(J)(H boolean AND circle plus (i is an element of K) G(i)) for every finite set J subset of I,(iii) weakly controllable in G if H boolean AND circle plus (i is an element of I) G(i) is dense in H, when G is equipped with the Tychonoff product topology.One easily proves that ( i) double right arrow ( ii) double right arrow ( iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrowcan be reversed when H is compact, but the second arrow cannot be reversed even when H is compact. Both arrows can be reversed if all groups G(i) are finite. When G(i) = A for all i is an element of I, where A is an abelian group, we show that the first arrow can be reversed for all subgroups H of G if and only if A is finitely generated. We also describe compact groups topologically isomorphic to a direct product of countably many cyclic groups. Connections with coding theory are highlighted.

We prove that the group G = H o m ( Z N , Z ) G=\mathrm {Hom}(\mathbb {Z}^{\mathbb {N}}, \mathbb {Z}) of all homomorphisms from the Baer-Specker group Z N \mathbb {Z}^{\mathbb {N}} to the group Z \mathbb {Z} of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G G is discrete. As G G is non-discrete, it is not reflexive. Since G G can be viewed as a closed subgroup of the Tychonoff product Z c \mathbb {Z}^{\mathfrak {c}} of continuum many copies of the integers Z \mathbb {Z} , this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

We prove that the group G = Hom(Z(N), Z) of all homomorphisms from the Baer-Specker group Z(N) to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact sub-sets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product Z(c) of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Nunez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-) free discrete abelian groups need not be reflexive.

Abstract In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup.

Let F be a finite field (or discrete) and let A and B be vector spaces of F-valued continuous functions defined on locally compact spaces X and Y, respectively. We look at the representation of linear bijections H:A⟶B by continuous functions h:Y⟶X as weighted composition operators. In order to do it, we extend the notion of Hamming metric to infinite spaces. Our main result establishes that under some mild conditions, every Hamming isometry can be represented as a weighted composition operator. Connections to coding theory are also highlighted.