We say that a mapping w between two topological abelian groups G and H is a pseudo-homomorphism if the associated map (x, y) is an element of G x G (bar right arrow) w(x y) -w(x) -w(y) is an element of H is continuous. This notion appears naturally in connection with cross sections (continuous right inverses for quotient mappings): given an algebraically splitting, closed subgroup H of a topological group X such that the projection pi : X -> X/H admits a cross section, one obtains a pseudo-homomorphism of X/H to H, and conversely. We show that H splits as a topological subgroup if and only if the corresponding pseudo-homomorphism can be decomposed as a sum of a homomorphism and a continuous mapping. We also prove that pseudo-homomorphisms between Polish groups satisfy the closed graph theorem. Several examples are given. (c) 2017 Elsevier B.V. All rights reserved.

Let G be a precompact, bounded torsion abelian group and G(p)<^>, its dual group endowed with the topology of pointwise convergence. We prove that if G is Baire (resp., pseudocompact), then all compact (resp., countably compact) subsets of G(p)<^> are finite. We also prove that G is pseudocompact if and only if all countable subgroups of G(p)<^> are closed. We present other characterizations of pseudocompactness and the Baire property of G(p)<^> in terms of properties that express in different ways the abundance of continuous characters of G. Besides, we give an example of a precompact boolean group G with the Baire property such that the dual group G(p)<^> contains an infinite countably compact subspace without isolated points. (C) 2016 Elsevier Inc. All rights reserved.

This paper deals with the splitting of extensions of topological abelian groups. Given topological abelian groups G and H, we say that Ext(G,H) is trivial if every extension of topological abelian groups of the form 1¿H¿X¿G¿1 splits. We prove that Ext(A(Y),K) is trivial for any free abelian topological group A(Y) over a zero-dimensional k¿-space Y and every compact abelian group K. Moreover we show that if K is a compact subgroup of a topological abelian group X such that the quotient group X/K is a zero-dimensional k¿-space, then there exists a continuous cross section from X/K to X. In the second part of the article we prove that Ext(G,H) is trivial whenever G is a product of locally precompact abelian groups and H has the form T¿×Rß for arbitrary cardinal numbers ¿ and ß. An analogous result is true if G=¿i¿IGi where each Gi is a dense subgroup of a maximally almost periodic, ¿ech-complete group for which both Ext(Gi,R) and Ext(Gi,T) are trivial.

In this article we study the arcwise connected component in the Pontryagin dual of an abelian topological group. It is clear that the set of continuous characters that can be lifted over the reals is contained in the arcwise connected component of the dual group. We show that the converse is true if all arcs in the character group are equicontinuous sets. This property is present in Pontryagin duals of pseudocompact groups, of reflexive groups and of groups which are k-spaces as topological spaces. We study the meaning of such a property and its presence in groups and vector spaces endowed with weak topologies. We also characterize the image of the exponential mapping of a dual group as formed by those characters which can be lifted over the reals endowed with the Bohr topology.

It is proved that, for a wide class of topological abelian groups (locally quasi-convex groups for which the canonical evaluation from the group into its Pontryagin bidual group is onto) the arc-component of the group is exactly the union of the one-parameter subgroups. We also prove that for metrizable separable locally arc-connected reflexive groups, the exponential map from the Lie algebra into the group is open.

A twisted sum in the category of topological Abelian groups is a short exact sequence 0 -> Y -> X -> Z -> 0 where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to 0 -> Y -> Y x Z -> Z -> 0. We study the class S-TG(T) of topological groups G for which every twisted sum 0 -> T -> X -> G -> 0 splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with L-infinity topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses of S-TG(T), we use the connection between extensions of the form 0 -> T -> X -> G -> 0 and quasi-characters on G, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of K-space, which were interpreted for topological groups by Cabello.

The Pontryagin duality theorem for locally compact abelian groups (briefly. LCA groups) has been the starting point for many different routes of research in Mathematics. From its appearance there was a big interest to extend it in a context broader than LCA groups. Kaplan in the 40's proposed and it still remains open the problem of characterization of all abelian topological groups for which the canonical mapping into its bidual is a topological isomorphism, assuming that the dual and the bidual carry the compact-open topology. Such groups are called reflexive. In this survey we deal with results on reflexivity of certain classes of groups, with special emphasis on the smaller class which better reflects the properties of LCA groups, namely that of strongly reflexive groups. A topological abelian group is said to be strongly reflexive if all its closed subgroups and its Hausdorff quotients as well as the closed subgroups and the Hausdorff quotients of its dual group are reflexive. (C) 2012 Elsevier B.V. All rights reserved.

It is proved that a locally quasi-convex group is a Schwartz group if and only if every continuously convergent filter on its dual group converges locally uniformly. We also show that for metrizable separable groups a similar result remains true when filters are replaced by sequences. As an ingredient in the proofs of these results, we obtain a Schauder-type theorem on compact homomorphisms acting between the natural group analogues of normed spaces.

We study the compact-open topology on the character group of dense subgroups of topological abelian groups. Permanence properties concerning open subgroups, quotients and products are considered. We also present some representative examples. We prove that every compact abelian group G with w(G) >= c has a dense pseudocompact group which does not determine G: this provides (under CH) a negative answer to a question posed by S. Hernandez, S. Macario and the third listed author two years ago.

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G, tau) is called locally minimal if there exists a neighborhood U of 0 in tau such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than tau. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GIG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GIG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GIG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS ...