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  "body": "\n## Definition\n\nA *Hausdorff topological group* is a [[topological group|92700QAB5Y]] $G$ whose topology is [[Hausdorff|0S1FJ8VEX9]].\nThat is, for any two distinct $x, y \\in G$ there are disjoint open sets $U \\ni x$ and $V \\ni y$.\n\n## Remarks\n\nFor a topological group $G$ with identity $e$, the following are equivalent: $G$ is Hausdorff; $G$ is $T_1$; $G$ is $T_0$; the set $\\set{e}$ is [[closed|0ZCK8DQYN1]].\nThey are also equivalent to: the intersection of all neighbourhoods of $e$ is $\\set{e}$.\nExample: $(\\R, +)$ with the usual topology, which comes from a metric and so is Hausdorff.\nNon-example: $\\Z/2\\Z$ with the indiscrete topology is a topological group, but its two points have no disjoint open neighbourhoods.\nIn fact a topological group is Hausdorff if and only if it is a Tychonoff space.\n",
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