Talk:Discrete group

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New To Advanced Math[edit]

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as discrete groups, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

Would it be reasonable to say that a discrete group is one with a cardinality not greater than aleph-null?

No. A discrete group can have any cardinality. Charles Matthews 07:57, 19 November 2005 (UTC)

What is discrete group[edit]

Now is:

"topological group is discrete if and only if the singleton containing the identity is an open set."

Must be:

"topological group is discrete if and only if the singleton containing the identity is an closed set."

Maksim-e (talk) 16:18, 20 July 2008 (UTC) Maksim-e (talk) 16:34, 20 July 2008 (UTC)

I think you have already answered this, but just in case: "Discrete" means every set is open and every set is closed. Since arbitrary unions of open sets are open, to be discrete means that singletons are open. Since topological groups are "the same everywhere", it is enough to require that the singleton containing the identity is open.
On the other hand, "Every singleton is closed" is a more natural property enjoyed by many more topological spaces, for instance every T1 space and so every Hausdorff space. Many people require all topological groups to be Hausdorff, so you are very right to say every singleton is closed. A discrete group is special: every singleton is both open and closed. JackSchmidt (talk) 02:21, 21 July 2008 (UTC)
Thanks! Maksim-e (talk) 19:30, 24 July 2008 (UTC)

is this isomorphism a functor?[edit]

"...underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups..."

Is the isomorphism to the category of discrete groups a functor and is it adjoint to the inverse isomorphism? Woud the inverse isomorphism be a forgetful functor?76.218.104.120 (talk) 10:13, 12 October 2013 (UTC)