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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 590159, 9 pages
http://dx.doi.org/10.1155/2013/590159
Research Article

Extending Topological Abelian Groups by the Unit Circle

1Departamento de Física y Matemática Aplicada, University of Navarra, 31080 Pamplona, Spain
2Departamento de Métodos Matemáticos y de Representación, University of A Coruña, 15071 A Coruña, Spain

Received 26 June 2013; Accepted 30 August 2013

Academic Editor: Salvador Hernández

Copyright © 2013 Hugo J. Bello et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to . We study the class of topological groups G for which every twisted sum splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with 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 , we use the connection between extensions of the form 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 -space, which were interpreted for topological groups by Cabello.