locally closed subgroups of topological groups are closed
Let $G$ be a Hausdorff^{} topological group^{} and $H\subseteq G$ a subgroup (which is a topological group itself under the subspace topology).
Theorem  If $H$ is locally closed in $G$ then $H$ is closed.
In particular we see that if $H$ is either

•
open, or

•
discrete (http://planetmath.org/Discrete), or
 •
then $H$ is closed.
Title  locally closed subgroups of topological groups are closed 

Canonical name  LocallyClosedSubgroupsOfTopologicalGroupsAreClosed 
Date of creation  20130322 17:36:39 
Last modified on  20130322 17:36:39 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  11 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 22A05 