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Continuity, boundedness, connectedness and the Lindelöf property for topological groups. (English) Zbl 0724.22003
Let G be a locally compact abelian (LCA) group and \(G^+\) the group G equipped with the weakest topology such that each character of G remains continuous. The purpose of this paper is to compare properties of the topological groups G and \(G^+\). For instance, the following results are shown: (1) If G and H are LCA groups, then a homomorphism \(\Phi: G\to H\) is continuous if and only if \(\Phi: G^+\to H^+\) is continuous. (2) G is zero-dimensional if and only if so is \(G^+\). (3) The connected components of the neutral element of G and \(G^+\) are equal (4) \(F\subset G\) is Lindelöf or functionally bounded if and only if F has this property as subspace of \(G^+\). Some of these results remain valid for a larger class of topological groups than that consisting of all LCA groups. As the author points out, some of the results proved in the paper are well-known.
Reviewer: M.Voit (München)

MSC:
22B05 General properties and structure of LCA groups
54C05 Continuous maps
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D05 Connected and locally connected spaces (general aspects)
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