Lemma 3.7.5.label Let $G$ be a topological group then if $G$ is
- (1)
$T_{1}$ then $G$ is Hausdorff
- (2)
not $T_{1}$ then $G/\cl{\curl{1}}$ is a Hausdorff topological group
hence WLOG every topological group is Hausdorff unless stated otherwise.
Proof.
- (1)
By Definition 3.7.1(6c) with $H=\curl{1}$
- (2)
By Definition 3.7.1(6a,6d) $\cl{\curl{1}}$ is a normal subgroup hence $G/\cl{\curl{1}}$ is a Hausdorff topological group by Definition 3.7.1(6c,6d)
$\square$
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