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.

中文翻译：

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关于 HAUSDORFF 拓扑群自由积的 GRAEV 定理

本文给出了 Graev 定理的简单证明（断言 Hausdorff 拓扑群的自由乘积是 Hausdorff），具体情况包括可数情况$k_\欧米茄$-群和林德洛夫的可数情况磷-团体。为此表明，在这些特殊情况下，豪斯多夫拓扑群的自由积的拓扑与$X_0$- Mal'cev 意义上的拓扑，其中X是标识其单位的拓扑群的不相交并集。