Embeddings into monothetic groups

TL;DR

This paper presents an elementary proof demonstrating that every bounded separable metric group can be embedded into a monothetic bounded metric group, highlighting the importance of the boundedness condition.

Contribution

It provides a simplified proof of a known embedding result and emphasizes the necessity of the boundedness assumption.

Findings

Every bounded separable metric group embeds into a monothetic bounded metric group

The boundedness assumption is essential for the embedding result

The proof simplifies understanding of the embedding property in topological groups

Abstract

We provide a very short elementary proof that every bounded separable metric group embeds into a monothetic bounded metric group, in such a way that the result of Morris and Pestov that every separable abelian topological group embeds into a monothetic group is an immediate corollary. We show that the boundedness assumption is essential.