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

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.