A note on Pontryagin duality and continuous logic

Abstract

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