I may be way off on what you're looking for (learned from Serre's Galois Cohomology):
If $M$ is a torsion abelian group, then its dual $Hom(M,\mathbb{Q}/\mathbb{Z})$ with the topology of pointwise convergence, is a commutative profinite group.
This gives a "Poincare Pontryagin duality": $\lbrace$torsion abelian groups$\rbrace\Longleftrightarrow \lbrace$commutative profinite groups$\rbrace$.
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I may be way off on what you're looking for (learned from Serre's Galois Cohomology): |
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