A proper short exact sequence 0→H →G→K→0 (*) in the category of locally compact abelian groups is said to be topologically pure if the induced sequence 0→nH→nG→nK→0 is proper short exact for all positive integers n. Some characterizations of topologically pure sequences in terms of direct decompositions, pure extensions and tensor products are established. A simple proof is given for a theorem on pure subgroups by Hartman and Hulanickl. Using topologically pure extensions, we characterize those splitting locally compact abelian groups whose torsion part is a direct sum of a compact group and a discrete group. We determine the compact and discrete groups H with the property that every topologically pure sequence (*) splits. Some structural information on topologically pure infectives and projectives is obtained.*
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Loth, Peter. "Topologically Pure Extensions." Contemporary Mathematics 273 (2001): 191-201.