The free product of topological groups being Hausdorff is Hausdorff --- a new proof

TL;DR

This paper provides an explicit topological description of free products of topological groups and offers a new, concise proof that such free products are Hausdorff when they meet this condition.

Contribution

It introduces an explicit topology description for free products of topological groups and presents a new proof of Graev's theorem using this framework.

Findings

The topology of free products coincides with the $X_0$-topology.

A new short proof of Graev's theorem is provided.

The approach clarifies the topological structure of free products.

Abstract

The explicit description of the topology of the free product of topological groups being Hausdorff is given. In particular, it is shown that it coincides with the so-called $X_0$-topology for the corresponding colimit $X$ in the category of topological spaces. Applying this fact, a new short proof of the well-known Graev's theorem asserting that the free product of topological groups being Hausdorff is Hausdorff is given.