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Sorry if this question is below the level of this site: I've read that the quotient of a Hausdorff topological group by a closed subgroup is again Hausdorff. I've thought about it but can't seem to figure out why. Is it obvious? A simple yes or no (with reference is possible) is all I need.

flag	asked Dec 15 2009 at 18:41 Dyke Acland 462●3●10

jvp · Accepted Answer · 2009-12-16 10:19:00Z UTC

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Edit: Below I expand my crude original answer "Yes" as requested by the community.

Yes. Let $G$ be the group and $H$ be the closed subgroup. The kernel of the quotient map $G \to G/H$ is equal to $\Delta^{-1}(H)$ where $\Delta : G \times G \to G$ is the continuous function $\Delta(x,y)= x- y$. Hence the kernel is closed. According to this $G/H$ is Hausdorff.

edited Dec 16 2009 at 10:19

answered Dec 15 2009 at 18:46

jvp
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	Given that the result is false without group structure, and given that your link is to the wiki article rather than a particular subsection, -1. Will upvote if you expand on this – Yemon Choi Dec 15 2009 at 19:12
	I am not sure a question like this needs much more explanation. See the discussion at mathoverflow.net/questions/9014/… – jvp Dec 15 2009 at 19:21
	But then don't answer because this link is useless :) – S1 Dec 15 2009 at 19:32
	Yes, the last line certainly implies G/H Hausd iff H closed. However, why is the last line true. (My apologies is this glaringly obvious.) – Dyke Acland Dec 15 2009 at 19:46
	+1 (i.e. revert now we have expanded on this). As people have said, the key is that in a top group a weak separation axiom boosts up to an a priori stronger one. – Yemon Choi Dec 15 2009 at 20:32

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S1 · Answer 2 · 2009-12-15 19:09:50Z UTC

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In fact, an even stronger statement holds: If $G$ is a topological group and $H$ is an (abstract) subgroup, then $G/H$ is Hausdorff if and only if $H$ is closed (cf Bourbaki, General Topology, III.2.5, prop 13). It's not hard to prove.

answered Dec 15 2009 at 19:09

S1
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Yes, before I posted the question I saw that H closed implies G/H T_0. What I couldn't see was that T_0 implies T_2 for a topological group. Is it obvious? – Dyke Acland Dec 15 2009 at 19:43

Quotient of a Hausdorff topological group by a closed subgroup

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