A note on Pontryagin duality and continuous logic

We exhibit Pontryagin duality as a special case of Stone in continuous logic setting. More specifically, given an abelian topological group A, and F the family (group) homomorphisms from A to circle T, then, viewing (A,+) equipped with collection structure M, we show that local type space SF(M) is precisely dual where latter considered discrete group. conclude, using (between compact groups), Bohr compactification A.