On local connectedness of locally compact Abelian groups

Abstract

A well-known characterization of the local connectedness of a compact Abelian group in terms of a dual property is the following theorem of Pontryagin ([4], § 38, Theorem 48): The dual group G of a discrete Abelian group G is locally connected, if and only if every finite set in G is contained in a finitely generated subgroup H of G with torsion-free G/H. Combining this theorem with a result of Braconnier ([1], p. 19), Dixmier ([2], p. 38) derived that a locally compact Abelian group is locally connected if and only if it is a product of the form R" x E x/) , where R" is a vector group with n __> 0, E is a discrete Abelian group, and D is a discrete, torsion-free Abelian group in which every subgroup of finite rank is free. The purpose of the present note is to prove the following result which is another natural extension of Pontryagin's theorem to arbitrary locally compact Abelian groups.