# Discrete quotient group from closed subgroup

Tags: group theory, quotient group
 P: 11 Hi All, I've come across a theorem that I'm trying to prove, which states that: The quotient group G/H is a discrete group iff the normal subgroup H is open. In fact I'm only really interested in the direction H open implies G/H discrete.. To a lesser extent I'm also interested in the H being closed iff G/H Haussdorf. Thanks!
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 for every g holds that gH is an open set. From the definition of the quotient topology, we get that {gH} is an open set. Thus the singletons of G/H are open, this means that the topology on G/H is discrete.
P: 11
 Quote by micromass for every g holds that gH is an open set. From the definition of the quotient topology, we get that {gH} is an open set. Thus the singletons of G/H are open, this means that the topology on G/H is discrete.
Hi thanks for your quick reply. I'm still a little confused - the singletons of G/H are open is fine - but why does that imply the topology on G/H is discrete?

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P: 15,673

## Discrete quotient group from closed subgroup

Because every set is the union of singletons. And the union of open sets is open...
P: 11
 Quote by micromass Because every set is the union of singletons. And the union of open sets is open...
Sorry, I'm quite new to this - I might just be missing a definition... but why is the topological group discrete if the set is open?
 PF Patron Sci Advisor Thanks Emeritus P: 15,673 OK, how did you define discrete?
 P: 11 I took the definition of discrete group to be one that is totally disconnected... I would have thought that if the set of singletons was open, one could always go to another group element by following some connected path? If the singletons were closed, I would have guessed that then you would have a discrete group... I'm obviously thinking about this completely incorrectly though.
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