Countable powers of compact Abelian groups in the uniform topology and cardinality of their dual groups

TL;DR

This paper studies the topological properties of countable products of compact Abelian groups with the uniform topology, focusing on the dual group's cardinality, especially for the circle group, which is 2^c.

Contribution

It determines the cardinality of the dual group of the product of countably many circle groups in the uniform topology.

Findings

The dual group of the product has cardinality 2^c.

The study extends understanding of dual groups in uniform topologies.

Provides explicit cardinality results for specific compact Abelian groups.

Abstract

We equip the product of countably many copies of a compact Abelian group X with the uniform topology, and study some properties of the topological group G thus obtained. In particular, we determine the cardinality of the dual group of G, when X is the circle group: it is precisely 2^c.