Proposition 9.2.2.label Let $G$ be a topological group and $H$ be a subgroup of $G$, then:
- (1)
The canonical map $\pi: G \to G/H$ is open.
- (2)
The coset space $G/H$ is Hausdorff if and only if $H$ is closed.
- (3)
The coset space $G/H$ is discrete if and only if $H$ is open.
Proof. (1): For each $U \subset G$, $\pi(U) = \pi(UH) = \pi^{-1}[\pi(U)]$.
(2): If $H$ is closed, then for any $g \in G \setminus H$, there exists $U \in \cn_{G}(1)$ such that $Ug \cap UH = \emptyset$, so $\pi(Ug) \cap \pi(UH) = \emptyset$.
On the other hand, if $G/H$ is Hausdorff, then $\ol H = \bigcup_{U \in \cn_G(1)}UH = H$, and $H$ is closed.$\square$
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