The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of
$k_\omega $
-groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the
$X_0$
-topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.