A note on Pontryagin duality and continuous logic

2022/04/24 by Nicolas Chavarria, Anand Pillay, Chavarria, Nicolas +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2204.11323

openalex publication_date 2022/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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