Subgroup is Closed iff Quotient is Hausdorff

Theorem

Let $G$ be a topological group.

Let $H \le G$ be a subgroup.

Let $G / H$ be their quotient.

Then the following are equivalent:

$H$ is closed in $G$

$G / H$ is Hausdorff

Proof

This theorem requires a proof. In particular: use Group Acts by Homeomorphisms Implies Projection on Quotient Space is Open and Open Projection and Closed Graph Implies Quotient is Hausdorff You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.