A proper short exact sequence in the category of locally compact abelian groups is said to be t-pure if φ(A) is a topologically pure subgroup of B, that is, if for all positive integers n. We establish conditions under which t-pure exact sequences split and determine those locally compact abelian groups K ⊕ D (where K is compactly generated and D is discrete) which are t-pure injective or t-pure projective. Calling the extension (*) almost pure if for all positive integers n, we obtain a complete description of the almost pure injectives and almost pure projectives in the category of locally compact abelian groups.
Loth, P. (2006). On t-pure and almost pure exact sequences of LCA groups. Journal of Group Theory, 9(6), 799-808. Doi: 10.1515/JGT.2006.051